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<caption>Sympy vs Sage integration routines</caption>
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<td valign="bottom" align="left" style=" font-size:10pt;">Legend:</td>
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<td valign="bottom" align="left" style="background:#FF00FF; font-size:10pt;">Disgraceful failure</td>
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<td valign="bottom" align="left" style="background:#FF9900; font-size:10pt;">Timeout or manual interupt</td>
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<td valign="bottom" align="left" style="background:#33CCCC; font-size:10pt;">Return an unevalued integral</td>
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<td valign="bottom" align="left" style="background:#99CC00; font-size:10pt;">Asking for assumptions</td>
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<td valign="bottom" align="left" style="background:#00FF00; font-size:10pt;">Solution with fancy special functions</td>
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<td colspan="1" rowspan="2" valign="center" align="center" style=" font-size:10pt; border-top:thin solid #000000; border-left:thin solid #000000; border-right:thin solid #000000;">Integrant</td>
<td colspan="2" rowspan="1" valign="center" align="center" style=" font-size:10pt; border-top:thin solid #000000;">Sympy `integrate`</td>
<td colspan="2" rowspan="1" valign="center" align="center" style=" font-size:10pt; border-top:thin solid #000000; border-left:thin solid #000000;">Sage `indefinite_integral`</td>
<td colspan="1" rowspan="2" valign="center" align="center" style=" font-size:10pt; border-top:thin solid #000000; border-left:thin solid #000000; border-right:thin solid #000000;">Comments</td>
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<td valign="center" align="center" style=" font-size:10pt; border-bottom:thin solid #000000;">Result</td>
<td valign="center" align="center" style=" font-size:10pt; border-bottom:thin solid #000000;">cpu time</td>
<td valign="center" align="center" style=" font-size:10pt; border-bottom:thin solid #000000; border-left:thin solid #000000;">Result</td>
<td valign="center" align="center" style=" font-size:10pt; border-bottom:thin solid #000000;">cpu time</td>
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<td valign="center" align="left" style=" font-size:10pt; border-top:thin solid #000000; border-left:thin solid #000000; border-right:thin solid #000000;">x</td>
<td valign="center" align="left" style=" font-size:10pt;">x**2/2</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">1/2*x^2</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-top:thin solid #000000; border-left:thin solid #000000; border-right:thin solid #000000;"></td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">a*x**n</td>
<td valign="center" align="left" style=" font-size:10pt;">a*x**(n + 1)/(n + 1)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style="background:#99CC00; font-size:10pt;">a*x^(n + 1)/(n + 1)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sage asks whether the denominator is zero before solving.</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">a*x**n + 1</td>
<td valign="center" align="left" style=" font-size:10pt;">a*x**(n + 1)/(n + 1) + x</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style="background:#99CC00; font-size:10pt;">a*x^(n + 1)/(n + 1) + x</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sage asks whether the denominator is zero before solving.</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">a*x**b + 1</td>
<td valign="center" align="left" style=" font-size:10pt;">a*x**(b + 1)/(b + 1) + x</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style="background:#99CC00; font-size:10pt;">a*x^(b + 1)/(b + 1) + x</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sage asks whether the denominator is zero before solving.</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/x</td>
<td valign="center" align="left" style=" font-size:10pt;">log(x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">log(x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/(x + 1)</td>
<td valign="center" align="left" style=" font-size:10pt;">log(x + 1)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">log(x + 1)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/(x**2 + 1)</td>
<td valign="center" align="left" style=" font-size:10pt;">atan(x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">arctan(x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/(x**3 + 1)</td>
<td valign="center" align="left" style=" font-size:10pt;">log(x + 1)/3 - log(x**2 - x + 1)/6 + sqrt(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/3</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">1/3*sqrt(3)*arctan(1/3*(2*x - 1)*sqrt(3)) + 1/3*log(x + 1) - 1/6*log(x^2 - x + 1)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">x**(-n)/a</td>
<td valign="center" align="left" style=" font-size:10pt;">x**(-n + 1)/(a*(-n + 1))</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style="background:#99CC00; font-size:10pt;">-x^(-n + 1)/((n - 1)*a)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sage asks whether the denominator is zero before solving.</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/(a*x**n + 1)</td>
<td valign="center" align="left" style="background:#00FF00; font-size:10pt;">x*gamma(1/n)*lerchphi(a*x**n*exp_polar(I*pi), 1, 1/n)/(n**2*gamma(1 + 1/n))</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">2</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">integrate(1/(x^n*a + 1), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">1</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sympy solves with special functions an integral that Sage cannot solve.</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/(a*x**b + 1)</td>
<td valign="center" align="left" style="background:#00FF00; font-size:10pt;">x*gamma(1/b)*lerchphi(a*x**b*exp_polar(I*pi), 1, 1/b)/(b**2*gamma(1 + 1/b))</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">1</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">integrate(1/(x^b*a + 1), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">1</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sympy solves with special functions an integral that Sage cannot solve.</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">a*x**2/(b*x**2 + 1)</td>
<td valign="center" align="left" style=" font-size:10pt;">a*(sqrt(-1/b**3)*log(-b*sqrt(-1/b**3) + x)/2 - sqrt(-1/b**3)*log(b*sqrt(-1/b**3) + x)/2 + x/b)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">(x/b - arctan(sqrt(b)*x)/b^(3/2))*a</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">a*x**3/(b*x**3 + 1)</td>
<td valign="center" align="left" style="background:#00FF00; font-size:10pt;">a*(RootSum(_t**3 + 1/(27*b**4), Lambda(_t, _t*log(-3*_t*b + x))) + x/b)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">-1/6*(2*sqrt(3)*arctan(1/3*(2*b^(2/3)*x - b^(1/3))*sqrt(3)/b^(1/3))/b^(4/3) - 6*x/b - log(b^(2/3)*x^2 - b^(1/3)*x + 1)/b^(4/3) + 2*log((b^(1/3)*x + 1)/b^(1/3))/b^(4/3))*a</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Interesting examples deserving more study as Sympy uses the sum of the roots of a high order polynomial while Sage uses elementary special functions.</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">a*x**n/(b*x**m + 1)</td>
<td valign="center" align="left" style="background:#00FF00; font-size:10pt;">a*(n*x*x**n*gamma(n/m + 1/m)*lerchphi(b*x**m*exp_polar(I*pi), 1, n/m + 1/m)/(m**2*gamma(1 + n/m + 1/m)) + x*x**n*gamma(n/m + 1/m)*lerchphi(b*x**m*exp_polar(I*pi), 1, n/m + 1/m)/(m**2*gamma(1 + n/m + 1/m)))</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">3</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">(m*integrate(x^n/((m - n - 1)*b^2*x^(2*m) + 2*(m - n - 1)*x^m*b + m - n - 1), x) - x^(n + 1)/((m - n - 1)*x^m*b + m - n - 1))*a</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sympy solves with special functions an integral that Sage cannot solve.</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">(a*x**2 + 1)/(b*x**2 + 1)</td>
<td valign="center" align="left" style=" font-size:10pt;">a*x/b + sqrt((-a**2 + 2*a*b - b**2)/b**3)*log(-b*sqrt((-a**2 + 2*a*b - b**2)/b**3)/(a - b) + x)/2 - sqrt((-a**2 + 2*a*b - b**2)/b**3)*log(b*sqrt((-a**2 + 2*a*b - b**2)/b**3)/(a - b) + x)/2</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">a*x/b - (a - b)*arctan(sqrt(b)*x)/b^(3/2)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sage symplifies better (log-to-trig formulas).</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">(a*x**3 + 1)/(b*x**3 + 1)</td>
<td valign="center" align="left" style="background:#00FF00; font-size:10pt;">a*x/b + RootSum(_t**3 + (a**3 - 3*a**2*b + 3*a*b**2 - b**3)/(27*b**4), Lambda(_t, _t*log(-3*_t*b/(a - b) + x)))</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">a*x/b - 1/3*(a*b - b^2)*sqrt(3)*arctan(1/3*(2*b^(2/3)*x - b^(1/3))*sqrt(3)/b^(1/3))/b^(7/3) + 1/6*(a*b^(2/3) - b^(5/3))*log(b^(2/3)*x^2 - b^(1/3)*x + 1)/b^2 - 1/3*(a*b^(2/3) - b^(5/3))*log((b^(1/3)*x + 1)/b^(1/3))/b^2</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Interesting examples deserving more study as Sympy uses the sum of the roots of a high order polynomial while Sage uses elementary special functions.</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">(a*x**n + 1)/(b*x**m + 1)</td>
<td valign="center" align="left" style="background:#00FF00; font-size:10pt;">a*(n*x*x**n*gamma(n/m + 1/m)*lerchphi(b*x**m*exp_polar(I*pi), 1, n/m + 1/m)/(m**2*gamma(1 + n/m + 1/m)) + x*x**n*gamma(n/m + 1/m)*lerchphi(b*x**m*exp_polar(I*pi), 1, n/m + 1/m)/(m**2*gamma(1 + n/m + 1/m))) + x*gamma(1/m)*lerchphi(b*x**m*exp_polar(I*pi), 1, 1/m)/(m**2*gamma(1 + 1/m))</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">5</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">a*m*integrate(x^n/((m - n - 1)*b^2*x^(2*m) + 2*(m - n - 1)*x^m*b + m - n - 1), x) - a*x^(n + 1)/((m - n - 1)*x^m*b + m - n - 1) + integrate(1/(x^m*b + 1), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">1</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sympy solves with special functions an integral that Sage cannot solve.</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">(a*x**5 + x**3 + 1)/(b*x**5 + x**3 + 1)</td>
<td valign="center" align="left" style="background:#00FF00; font-size:10pt;">a*x/b + RootSum(_t**5 + _t**3*(500*a**2*b**3 + 27*a**2 - 1000*a*b**4 - 54*a*b + 500*b**5 + 27*b**2)/(3125*b**6 + 108*b**3) + _t**2*(27*a**3 - 81*a**2*b + 81*a*b**2 - 27*b**3)/(3125*b**6 + 108*b**3) + _t*(9*a**4 - 36*a**3*b + 54*a**2*b**2 - 36*a*b**3 + 9*b**4)/(3125*b**6 + 108*b**3) + (a**5 - 5*a**4*b + 10*a**3*b**2 - 10*a**2*b**3 + 5*a*b**4 - b**5)/(3125*b**6 + 108*b**3), Lambda(_t, _t*log(x + (3662109375*_t**4*b**12 + 3986718750*_t**4*b**9 + 242757000*_t**4*b**6 + 3779136*_t**4*b**3 - 1054687500*_t**3*a*b**9 - 72900000*_t**3*a*b**6 - 1259712*_t**3*a*b**3 + 1054687500*_t**3*b**10 + 72900000*_t**3*b**7 + 1259712*_t**3*b**4 + 410156250*_t**2*a**2*b**9 + 655340625*_t**2*a**2*b**6 + 51267654*_t**2*a**2*b**3 + 944784*_t**2*a**2 - 820312500*_t**2*a*b**10 - 1310681250*_t**2*a*b**7 - 102535308*_t**2*a*b**4 - 1889568*_t**2*a*b + 410156250*_t**2*b**11 + 655340625*_t**2*b**8 + 51267654*_t**2*b**5 + 944784*_t**2*b**2 - 48828125*_t*a**3*b**9 - 186046875*_t*a**3*b**6 + 16774290*_t*a**3*b**3 + 629856*_t*a**3 + 146484375*_t*a**2*b**10 + 558140625*_t*a**2*b**7 - 50322870*_t*a**2*b**4 - 1889568*_t*a**2*b - 146484375*_t*a*b**11 - 558140625*_t*a*b**8 + 50322870*_t*a*b**5 + 1889568*_t*a*b**2 + 48828125*_t*b**12 + 186046875*_t*b**9 - 16774290*_t*b**6 - 629856*_t*b**3 - 2812500*a**4*b**6 + 3596400*a**4*b**3 + 104976*a**4 + 11250000*a**3*b**7 - 14385600*a**3*b**4 - 419904*a**3*b - 16875000*a**2*b**8 + 21578400*a**2*b**5 + 629856*a**2*b**2 + 11250000*a*b**9 - 14385600*a*b**6 - 419904*a*b**3 - 2812500*b**10 + 3596400*b**7 + 104976*b**4)/(9765625*a**4*b**8 + 26493750*a**4*b**5 + 746496*a**4*b**2 - 39062500*a**3*b**9 - 105975000*a**3*b**6 - 2985984*a**3*b**3 + 58593750*a**2*b**10 + 158962500*a**2*b**7 + 4478976*a**2*b**4 - 39062500*a*b**11 - 105975000*a*b**8 - 2985984*a*b**5 + 9765625*b**12 + 26493750*b**9 + 746496*b**6))))</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">106</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">-(a - b)*integrate((x^3 + 1)/(b*x^5 + x^3 + 1), x)/b + a*x/b</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sympy solves with special functions an integral that Sage cannot solve.</td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt(1/x)</td>
<td valign="center" align="left" style=" font-size:10pt;">2*x*sqrt(1/x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">2*x*sqrt(1/x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
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<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt(1/(x + 1))</td>
<td valign="center" align="left" style=" font-size:10pt;">2*x*sqrt(1/(x + 1)) + 2*sqrt(1/(x + 1))</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">2/sqrt(1/(x + 1))</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt(1/(x**2 + 1))</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">Integral(sqrt(1/(x**2 + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">arcsinh(x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sympy cannot solve this simple integral while Sage can.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt(1/(x**3 + 1))</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">Integral(sqrt(1/(x**3 + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">3</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">integrate(sqrt(1/(x^3 + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt(x**(-n)/a)</td>
<td valign="center" align="left" style=" font-size:10pt;">-2*x*sqrt(1/a)*sqrt(x**(-n))/(n - 2)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style="background:#99CC00; font-size:10pt;">-2*x*sqrt(x^(-n)/a)/(n-2)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sage asks whether the denominator is zero before solving.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt(1/(a*x**n + 1))</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">Integral(sqrt(1/(a*x**n + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">29</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">integrate(sqrt(1/(x^n*a + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt(1/(a*x**b + 1))</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">Integral(sqrt(1/(a*x**b + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">35</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">integrate(sqrt(1/(x^b*a + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">1</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt(a*x**2/(b*x**2 + 1))</td>
<td valign="center" align="left" style=" font-size:10pt;">sqrt(a)*x*sqrt(x**2)*sqrt(1/(b*x**2 + 1)) + sqrt(a)*sqrt(x**2)*sqrt(1/(b*x**2 + 1))/(b*x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">2</td>
<td valign="center" align="left" style=" font-size:10pt;">(sqrt(a)*b*x^2 + sqrt(a))/(sqrt(b*x^2 + 1)*b)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt(a*x**3/(b*x**3 + 1))</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">Integral(sqrt(a*x**3/(b*x**3 + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">7</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">integrate(sqrt(a*x^3/(b*x^3 + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt(a*x**n/(b*x**m + 1))</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">115</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">integrate(sqrt(x^n*a/(x^m*b + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">1</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt((a*x**2 + 1)/(b*x**2 + 1))</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">110</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">integrate(sqrt((a*x^2 + 1)/(b*x^2 + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt((a*x**3 + 1)/(b*x**3 + 1))</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">109</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">integrate(sqrt((a*x^3 + 1)/(b*x^3 + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt((a*x**n + 1)/(b*x**m + 1))</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">114</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">integrate(sqrt((x^n*a + 1)/(x^m*b + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">1</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sqrt((a*x**5 + x**3 + 1)/(b*x**5 + x**3 + 1))</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">104</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">integrate(sqrt((a*x^5 + x^3 + 1)/(b*x^5 + x^3 + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log(x)</td>
<td valign="center" align="left" style=" font-size:10pt;">x*log(x) - x</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">x*log(x) - x</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log(1/x)</td>
<td valign="center" align="left" style=" font-size:10pt;">-x*log(x) + x</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">-x*log(x) + x</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log(1/(x + 1))</td>
<td valign="center" align="left" style=" font-size:10pt;">-x*log(x + 1) + x - log(x + 1)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">-(x + 1)*log(x + 1) + x + 1</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log(1/(x**2 + 1))</td>
<td valign="center" align="left" style=" font-size:10pt;">-x*log(x**2 + 1) + 2*x - 2*I*log(x + I) + I*log(x**2 + 1)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">2</td>
<td valign="center" align="left" style=" font-size:10pt;">-x*log(x^2 + 1) + 2*x - 2*arctan(x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log(1/(x**3 + 1))</td>
<td valign="center" align="left" style=" font-size:10pt;">-x*log(x**3 + 1) + 3*x - 3*log(x + 1)/2 + sqrt(3)*I*log(x + 1)/2 + log(x**3 + 1)/2 - sqrt(3)*I*log(x**3 + 1)/2 + sqrt(3)*I*log(x - 1/2 - sqrt(3)*I/2)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">6</td>
<td valign="center" align="left" style=" font-size:10pt;">-x*log(x^3 + 1) - sqrt(3)*arctan(1/3*(2*x - 1)*sqrt(3)) + 3*x - log(x + 1) + 1/2*log(x^2 - x + 1)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log(x**(-n)/a)</td>
<td valign="center" align="left" style=" font-size:10pt;">-n*x*log(x) + n*x - x*log(a)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">n*x + x*log(x^(-n)/a)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log(1/(a*x**n + 1))</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">Integral(log(1/(a*x**n + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">68</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">n*x - n*integrate(1/(a*e^(n*log(x)) + 1), x) - x*log(x^n*a + 1)</td>
<td valign="center" align="right" style=" font-size:10pt;">1</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log(1/(a*x**b + 1))</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">91</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">b*x - b*integrate(1/(a*e^(b*log(x)) + 1), x) - x*log(a*e^(b*log(x)) + 1)</td>
<td valign="center" align="right" style=" font-size:10pt;">2</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log(a*x**2/(b*x**2 + 1))</td>
<td valign="center" align="left" style=" font-size:10pt;">x*log(a) + 2*x*log(x) - x*log(b*x**2 + 1) + 2*I*log(x - I*sqrt(1/b))/(b*sqrt(1/b)) - I*log(b*x**2 + 1)/(b*sqrt(1/b))</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">10</td>
<td valign="center" align="left" style=" font-size:10pt;">x*log(a*x^2/(b*x^2 + 1)) - 2*arctan(sqrt(b)*x)/sqrt(b)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log(a*x**3/(b*x**3 + 1))</td>
<td valign="center" align="left" style=" font-size:10pt;">-216*b**4*x**6*(1/b)**(7/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 216*(-1)**(2/3)*b**4*x**6*(1/b)**(7/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 72*(-1)**(1/6)*sqrt(3)*b**4*x**6*(1/b)**(7/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 72*sqrt(3)*I*b**4*x**6*(1/b)**(7/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 588*b**2*x**5*(1/b)**(2/3)*log(a)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 1764*b**2*x**5*(1/b)**(2/3)*log(x)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 588*b**2*x**5*(1/b)**(2/3)*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 630*b**2*x**5*(1/b)**(2/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 420*(-1)**(5/6)*sqrt(3)*b**2*x**5*(1/b)**(2/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 210*sqrt(3)*I*b**2*x**5*(1/b)**(2/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 981*b**2*x**3*(1/b)**(4/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 981*(-1)**(2/3)*b**2*x**3*(1/b)**(4/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 327*(-1)**(1/6)*sqrt(3)*b**2*x**3*(1/b)**(4/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 327*sqrt(3)*I*b**2*x**3*(1/b)**(4/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 135*b**2*(1/b)**(7/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 135*(-1)**(2/3)*b**2*(1/b)**(7/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 45*(-1)**(1/6)*sqrt(3)*b**2*(1/b)**(7/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 45*sqrt(3)*I*b**2*(1/b)**(7/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 147*(-1)**(2/3)*b*x**4*log(a)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 49*(-1)**(1/6)*sqrt(3)*b*x**4*log(a)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 49*(-1)**(5/6)*sqrt(3)*b*x**4*log(a)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 147*(-1)**(1/3)*b*x**4*log(a)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 882*(-1)**(2/3)*b*x**4*log(a)*log(x)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 294*(-1)**(1/6)*sqrt(3)*b*x**4*log(a)*log(x)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 294*(-1)**(5/6)*sqrt(3)*b*x**4*log(a)*log(x)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 882*(-1)**(1/3)*b*x**4*log(a)*log(x)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 294*(-1)**(1/3)*b*x**4*log(a)*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 98*(-1)**(5/6)*sqrt(3)*b*x**4*log(a)*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 98*(-1)**(1/6)*sqrt(3)*b*x**4*log(a)*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 294*(-1)**(2/3)*b*x**4*log(a)*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 588*(-1)**(2/3)*b*x**4*log(a)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 1323*(-1)**(2/3)*b*x**4*log(x)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 441*(-1)**(1/6)*sqrt(3)*b*x**4*log(x)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 441*(-1)**(5/6)*sqrt(3)*b*x**4*log(x)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 1323*(-1)**(1/3)*b*x**4*log(x)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 882*(-1)**(1/3)*b*x**4*log(x)*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 294*(-1)**(5/6)*sqrt(3)*b*x**4*log(x)*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 294*(-1)**(1/6)*sqrt(3)*b*x**4*log(x)*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 882*(-1)**(2/3)*b*x**4*log(x)*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 882*(-1)**(1/3)*b*x**4*log(x)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 294*(-1)**(5/6)*sqrt(3)*b*x**4*log(x)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 294*(-1)**(1/6)*sqrt(3)*b*x**4*log(x)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 882*(-1)**(2/3)*b*x**4*log(x)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 294*(-1)**(5/6)*sqrt(3)*b*x**4*log(x - (-1)**(1/3)*(1/b)**(1/3))/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 882*(-1)**(1/3)*b*x**4*log(x - (-1)**(1/3)*(1/b)**(1/3))/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 147*(-1)**(2/3)*b*x**4*log(b*x**3 + 1)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 49*(-1)**(1/6)*sqrt(3)*b*x**4*log(b*x**3 + 1)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 49*(-1)**(5/6)*sqrt(3)*b*x**4*log(b*x**3 + 1)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 147*(-1)**(1/3)*b*x**4*log(b*x**3 + 1)**2/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 588*(-1)**(2/3)*b*x**4*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 294*(-1)**(1/6)*sqrt(3)*b*x**4*log(x - (-1)**(1/3)*sqrt(3)*I*(1/b)**(1/3)/2 + (-1)**(1/3)*(1/b)**(1/3)/2)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 882*(-1)**(2/3)*b*x**4*log(x - (-1)**(1/3)*sqrt(3)*I*(1/b)**(1/3)/2 + (-1)**(1/3)*(1/b)**(1/3)/2)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 294*(-1)**(2/3)*b*x**4/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 98*(-1)**(1/6)*sqrt(3)*b*x**4/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 98*(-1)**(5/6)*sqrt(3)*b*x**4/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 294*(-1)**(1/3)*b*x**4/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 588*b*x**2*(1/b)**(2/3)*log(a)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 1764*b*x**2*(1/b)**(2/3)*log(x)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 588*b*x**2*(1/b)**(2/3)*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 945*b*x**2*(1/b)**(2/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 630*(-1)**(5/6)*sqrt(3)*b*x**2*(1/b)**(2/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 315*sqrt(3)*I*b*x**2*(1/b)**(2/3)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 98*(-1)**(1/6)*sqrt(3)*x*log(a)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 98*(-1)**(5/6)*sqrt(3)*x*log(a)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 294*(-1)**(2/3)*x*log(a)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 294*(-1)**(1/3)*x*log(a)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 294*(-1)**(5/6)*sqrt(3)*x*log(x - (-1)**(1/3)*(1/b)**(1/3))/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 882*(-1)**(1/3)*x*log(x - (-1)**(1/3)*(1/b)**(1/3))/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 294*(-1)**(1/3)*x*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 294*(-1)**(2/3)*x*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 98*(-1)**(5/6)*sqrt(3)*x*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 98*(-1)**(1/6)*sqrt(3)*x*log(b*x**3 + 1)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) - 294*(-1)**(1/6)*sqrt(3)*x*log(x - (-1)**(1/3)*sqrt(3)*I*(1/b)**(1/3)/2 + (-1)**(1/3)*(1/b)**(1/3)/2)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3)) + 882*(-1)**(2/3)*x*log(x - (-1)**(1/3)*sqrt(3)*I*(1/b)**(1/3)/2 + (-1)**(1/3)*(1/b)**(1/3)/2)/(588*b**2*x**4*(1/b)**(2/3) + 588*b*x*(1/b)**(2/3))</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">26</td>
<td valign="center" align="left" style=" font-size:10pt;">x*log(a*x^3/(b*x^3 + 1)) - 1/2*(2*sqrt(3)*a*arctan(1/3*(2*b^(2/3)*x - b^(1/3))*sqrt(3)/b^(1/3))/b^(1/3) - a*log(b^(2/3)*x^2 - b^(1/3)*x + 1)/b^(1/3) + 2*a*log((b^(1/3)*x + 1)/b^(1/3))/b^(1/3))/a</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Bizzare difference between the two results. Why are there no special functions in the Sympy solution while there are such in the Sage solution.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log(a*x**n/(b*x**m + 1))</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">96</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">(m - n + log(a))*x - m*integrate(1/(b*e^(m*log(x)) + 1), x) - x*log(x^m*b + 1) + x*log(x^n)</td>
<td valign="center" align="right" style=" font-size:10pt;">2</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log((a*x**2 + 1)/(b*x**2 + 1))</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">Integral(log((a*x**2 + 1)/(b*x**2 + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">72</td>
<td valign="center" align="left" style="background:#99CC00; font-size:10pt;">x*log((a*x^2 + 1)/(b*x^2 + 1)) + 2*arctan(sqrt(a)*x)/sqrt(a) - 2*arctan(sqrt(b)*x)/sqrt(b)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sage asks whether `a` and `b` are positive and then returns an answer. Sympy fails irrespective of the assumptions.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log((a*x**3 + 1)/(b*x**3 + 1))</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">89</td>
<td valign="center" align="left" style="background:#99CC00; font-size:10pt;">x*log((a*x^3 + 1)/(b*x^3 + 1)) + sqrt(3)*arctan(1/3*(2*a^(2/3)*x - a^(1/3))*sqrt(3)/a^(1/3))/a^(1/3) - sqrt(3)*arctan(1/3*(2*b^(2/3)*x - b^(1/3))*sqrt(3)/b^(1/3))/b^(1/3) - 1/2*log(a^(2/3)*x^2 - a^(1/3)*x + 1)/a^(1/3) + log((a^(1/3)*x + 1)/a^(1/3))/a^(1/3) + 1/2*log(b^(2/3)*x^2 - b^(1/3)*x + 1)/b^(1/3) - log((b^(1/3)*x + 1)/b^(1/3))/b^(1/3)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sage asks whether `a` and `b` are positive and then returns an answer. Sympy fails irrespective of the assumptions.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log((a*x**n + 1)/(b*x**m + 1))</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">89</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">(m - n)*x - m*integrate(1/(b*e^(m*log(x)) + 1), x) + n*integrate(1/(x^n*a + 1), x) - x*log(x^m*b + 1) + x*log(x^n*a + 1)</td>
<td valign="center" align="right" style=" font-size:10pt;">4</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">log((a*x**5 + x**3 + 1)/(b*x**5 + x**3 + 1))</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">Integral(log((a*x**5 + x**3 + 1)/(b*x**5 + x**3 + 1)), x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">42</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">-x*log(b*x^5 + x^3 + 1) + x*log(a*x^5 + x^3 + 1) - integrate((2*x^3 + 5)/(b*x^5 + x^3 + 1), x) + integrate((2*x^3 + 5)/(a*x^5 + x^3 + 1), x)</td>
<td valign="center" align="right" style=" font-size:10pt;">1</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">When both Sage and Sympy fail, Sage is quicker.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sin(x)</td>
<td valign="center" align="left" style=" font-size:10pt;">-cos(x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">-cos(x)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sin(x)**n*cos(x)**m</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">102</td>
<td valign="center" align="left" style="background:#FF00FF; font-size:10pt;">No result</td>
<td valign="center" align="right" style=" font-size:10pt;">110</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Disgraceful failure by Sage.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sin(a*x)**n*cos(b*x)**m</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">81</td>
<td valign="center" align="left" style="background:#FF00FF; font-size:10pt;">No result</td>
<td valign="center" align="right" style=" font-size:10pt;">112</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Disgraceful failure by Sage.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/sin(x)</td>
<td valign="center" align="left" style=" font-size:10pt;">log(cos(x) - 1)/2 - log(cos(x) + 1)/2</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">0</td>
<td valign="center" align="left" style=" font-size:10pt;">1/2*log(cos(x) - 1) - 1/2*log(cos(x) + 1)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/(sin(x) + 1)</td>
<td valign="center" align="left" style=" font-size:10pt;">-2/(tan(x/2) + 1)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">1</td>
<td valign="center" align="left" style=" font-size:10pt;">-2/(sin(x)/(cos(x) + 1) + 1)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/(sin(x)**2 + 1)</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">96</td>
<td valign="center" align="left" style=" font-size:10pt;">1/2*sqrt(2)*arctan(sqrt(2)*tan(x))</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sage simply beats Sympy.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/(sin(x)**3 + 1)</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">87</td>
<td valign="center" align="left" style="background:#FF00FF; font-size:10pt;">Maxima: `quotient' by `zero'</td>
<td valign="center" align="right" style=" font-size:10pt;">78</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Disgraceful failure by Sage.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">sin(x)**(-n)/a</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">Integral(sin(x)**(-n)/a, x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">36</td>
<td valign="center" align="left" style="background:#FF00FF; font-size:10pt;">No result</td>
<td valign="center" align="right" style=" font-size:10pt;">227</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Disgraceful failure by Sage.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/(a*sin(x)**n + 1)</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">98</td>
<td valign="center" align="left" style="background:#FF00FF; font-size:10pt;">Maxima: expt: undefined: 0 to a negative exponent.</td>
<td valign="center" align="right" style=" font-size:10pt;">1</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Disgraceful failure by Sage.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">1/(a*sin(x)**b + 1)</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">83</td>
<td valign="center" align="left" style="background:#FF00FF; font-size:10pt;">No result</td>
<td valign="center" align="right" style=" font-size:10pt;">140</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Disgraceful failure by Sage.</td>
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<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">a*sin(x)**2/(b*sin(x)**2 + 1)</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">93</td>
<td valign="center" align="left" style=" font-size:10pt;">(x/b - arctan(sqrt(b + 1)*tan(x))/(sqrt(b + 1)*b))*a</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sage simply beats Sympy.</td>
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<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">a*sin(x)**3/(b*sin(x)**3 + 1)</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">82</td>
<td valign="center" align="left" style="background:#FF00FF; font-size:10pt;">No result</td>
<td valign="center" align="right" style=" font-size:10pt;">568</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Disgraceful failure by Sage.</td>
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<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">a*sin(x)**n/(b*sin(x)**m + 1)</td>
<td valign="center" align="left" style="background:#33CCCC; font-size:10pt;">Integral(a*sin(x)**n/(b*sin(x)**m + 1), x)</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">24</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Manual Interupt</td>
<td valign="center" align="right" style=" font-size:10pt;">1527</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Both Sage and Sympy fail, however Sympy is quicker.</td>
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<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">(a*sin(x)**2 + 1)/(b*sin(x)**2 + 1)</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">98</td>
<td valign="center" align="left" style=" font-size:10pt;">a*x/b - (a - b)*arctan(sqrt(b + 1)*tan(x))/(sqrt(b + 1)*b)</td>
<td valign="center" align="right" style=" font-size:10pt;">0</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Sage simply beats Sympy.</td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">(a*sin(x)**3 + 1)/(b*sin(x)**3 + 1)</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">96</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Manual Interupt</td>
<td valign="center" align="right" style=" font-size:10pt;">203</td>
<td style=" border-left:thin solid #000000; border-right:thin solid #000000;"></td>
</tr>
<tr>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">(a*sin(x)**n + 1)/(b*sin(x)**m + 1)</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-right:thin solid #000000;">83</td>
<td valign="center" align="left" style="background:#FF00FF; font-size:10pt;">Maxima: expt: undefined: 0 to a negative exponent.</td>
<td valign="center" align="right" style=" font-size:10pt;">1</td>
<td valign="center" align="left" style=" font-size:10pt; border-left:thin solid #000000; border-right:thin solid #000000;">Disgraceful failure by Sage.</td>
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<tr>
<td valign="center" align="left" style=" font-size:10pt; border-bottom:thin solid #000000; border-left:thin solid #000000; border-right:thin solid #000000;">(a*sin(x)**5 + sin(x)**3 + 1)/(b*sin(x)**5 + sin(x)**3 + 1)</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt; border-bottom:thin solid #000000;">Timeout</td>
<td valign="center" align="right" style=" font-size:10pt; border-bottom:thin solid #000000; border-right:thin solid #000000;">89</td>
<td valign="center" align="left" style="background:#FF6600; font-size:10pt; border-bottom:thin solid #000000;">Manual Interupt</td>
<td valign="center" align="right" style=" font-size:10pt; border-bottom:thin solid #000000;">142</td>
<td style=" border-bottom:thin solid #000000; border-left:thin solid #000000; border-right:thin solid #000000;"></td>
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