Optimal. Leaf size=185 \[ -\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1-\sqrt {5}\right ) x+1\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )+\frac {1}{5} \log (x+1)-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} x+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.35, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {1593, 293, 634, 618, 204, 628, 31} \begin {gather*} -\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1-\sqrt {5}\right ) x+1\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )+\frac {1}{5} \log (x+1)-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} x+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 204
Rule 293
Rule 618
Rule 628
Rule 634
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{\frac {1}{x^2}+x^3} \, dx &=\int \frac {x^2}{1+x^5} \, dx\\ &=\frac {2}{5} \int \frac {\frac {1}{4} \left (-1-\sqrt {5}\right )-\frac {1}{4} \left (1+\sqrt {5}\right ) x}{1-\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx+\frac {2}{5} \int \frac {\frac {1}{4} \left (-1+\sqrt {5}\right )-\frac {1}{4} \left (1-\sqrt {5}\right ) x}{1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{5} \int \frac {1}{1+x} \, dx\\ &=\frac {1}{5} \log (1+x)+\frac {\int \frac {1}{1+\frac {1}{2} \left (-1-\sqrt {5}\right ) x+x^2} \, dx}{2 \sqrt {5}}-\frac {\int \frac {1}{1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx}{2 \sqrt {5}}+\frac {1}{20} \left (-1-\sqrt {5}\right ) \int \frac {\frac {1}{2} \left (-1+\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{20} \left (-1+\sqrt {5}\right ) \int \frac {\frac {1}{2} \left (-1-\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (-1-\sqrt {5}\right ) x+x^2} \, dx\\ &=\frac {1}{5} \log (1+x)-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (2-x-\sqrt {5} x+2 x^2\right )-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (2-x+\sqrt {5} x+2 x^2\right )+\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5-\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (-1+\sqrt {5}\right )+2 x\right )}{\sqrt {5}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5+\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (-1-\sqrt {5}\right )+2 x\right )}{\sqrt {5}}\\ &=\sqrt {\frac {2}{5 \left (5+\sqrt {5}\right )}} \tan ^{-1}\left (\frac {1-\sqrt {5}-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}-4 x\right )\right )+\frac {1}{5} \log (1+x)-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (2-x-\sqrt {5} x+2 x^2\right )-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (2-x+\sqrt {5} x+2 x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 144, normalized size = 0.78 \begin {gather*} \frac {1}{20} \left (-\left (1+\sqrt {5}\right ) \log \left (x^2+\frac {1}{2} \left (\sqrt {5}-1\right ) x+1\right )+\left (\sqrt {5}-1\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )+4 \log (x+1)-2 \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {-4 x+\sqrt {5}+1}{\sqrt {10-2 \sqrt {5}}}\right )-2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {4 x+\sqrt {5}-1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\frac {1}{x^2}+x^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.28, size = 637, normalized size = 3.44 \begin {gather*} -\frac {1}{20} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )} \log \left (\frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + x\right ) + \frac {1}{20} \, {\left (\sqrt {5} + 2 \, \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} - 1\right )} \log \left (-\frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} - \frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \frac {1}{2} \, \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} {\left (\sqrt {5} - 1\right )} + 2 \, x - 1\right ) + \frac {1}{20} \, {\left (\sqrt {5} - 2 \, \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} - 1\right )} \log \left (-\frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} - \frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} - \frac {1}{2} \, \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} {\left (\sqrt {5} - 1\right )} + 2 \, x - 1\right ) + \frac {1}{20} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} \log \left (\frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + x\right ) + \frac {1}{5} \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 112, normalized size = 0.61 \begin {gather*} \frac {1}{20} \, {\left (\sqrt {5} - 1\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) - \frac {1}{20} \, {\left (\sqrt {5} + 1\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) - \frac {1}{10} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{10} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) + \frac {1}{5} \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.11, size = 156, normalized size = 0.84 \begin {gather*} \frac {2 \sqrt {5}\, \arctan \left (\frac {4 x -\sqrt {5}-1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}-\frac {2 \sqrt {5}\, \arctan \left (\frac {4 x +\sqrt {5}-1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {\ln \left (x +1\right )}{5}+\frac {\sqrt {5}\, \ln \left (2 x^{2}-\sqrt {5}\, x -x +2\right )}{20}-\frac {\ln \left (2 x^{2}-\sqrt {5}\, x -x +2\right )}{20}-\frac {\sqrt {5}\, \ln \left (2 x^{2}+\sqrt {5}\, x -x +2\right )}{20}-\frac {\ln \left (2 x^{2}+\sqrt {5}\, x -x +2\right )}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.90, size = 124, normalized size = 0.67 \begin {gather*} -\frac {2 \, \sqrt {5} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {2 \, \sqrt {5} + 10}} + \frac {2 \, \sqrt {5} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} + 10}} + \frac {\log \left (2 \, x^{2} - x {\left (\sqrt {5} + 1\right )} + 2\right )}{5 \, {\left (\sqrt {5} + 1\right )}} - \frac {\log \left (2 \, x^{2} + x {\left (\sqrt {5} - 1\right )} + 2\right )}{5 \, {\left (\sqrt {5} - 1\right )}} + \frac {1}{5} \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.91, size = 197, normalized size = 1.06 \begin {gather*} \frac {\ln \left (x+1\right )}{5}-\ln \left (1-\frac {x\,{\left (\sqrt {2}\,\sqrt {-\sqrt {5}-5}-\sqrt {5}+1\right )}^3}{64}\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )+\ln \left (\frac {x\,{\left (\sqrt {2}\,\sqrt {-\sqrt {5}-5}+\sqrt {5}-1\right )}^3}{64}+1\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}+\frac {\sqrt {5}}{20}-\frac {1}{20}\right )-\ln \left (1-\frac {x\,{\left (\sqrt {5}+\sqrt {2}\,\sqrt {\sqrt {5}-5}+1\right )}^3}{64}\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )-\ln \left (1-\frac {x\,{\left (\sqrt {5}-\sqrt {2}\,\sqrt {\sqrt {5}-5}+1\right )}^3}{64}\right )\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.54, size = 36, normalized size = 0.19 \begin {gather*} \frac {\log {\left (x + 1 \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} + 125 t^{3} + 25 t^{2} + 5 t + 1, \left (t \mapsto t \log {\left (25 t^{2} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________