Optimal. Leaf size=56 \[ -\frac {1}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (1+x^2\right )-\frac {1}{12} \log \left (1-x^2+x^4\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {281, 331, 298,
31, 648, 632, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{2 x^2}+\frac {1}{6} \log \left (x^2+1\right )-\frac {1}{12} \log \left (x^4-x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 210
Rule 281
Rule 298
Rule 331
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1+x^6\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (1+x^3\right )} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}-\frac {1}{2} \text {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{6} \log \left (1+x^2\right )-\frac {1}{12} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,x^2\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{6} \log \left (1+x^2\right )-\frac {1}{12} \log \left (1-x^2+x^4\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (1+x^2\right )-\frac {1}{12} \log \left (1-x^2+x^4\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 83, normalized size = 1.48 \begin {gather*} \frac {1}{12} \left (-\frac {6}{x^2}+2 \sqrt {3} \tan ^{-1}\left (\sqrt {3}-2 x\right )+2 \sqrt {3} \tan ^{-1}\left (\sqrt {3}+2 x\right )+2 \log \left (1+x^2\right )-\log \left (1-\sqrt {3} x+x^2\right )-\log \left (1+\sqrt {3} x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.17, size = 46, normalized size = 0.82
method | result | size |
risch | \(-\frac {1}{2 x^{2}}+\frac {\ln \left (x^{2}+1\right )}{6}-\frac {\ln \left (x^{4}-x^{2}+1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{2}-\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}\) | \(44\) |
default | \(-\frac {1}{2 x^{2}}-\frac {\ln \left (x^{4}-x^{2}+1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\ln \left (x^{2}+1\right )}{6}\) | \(46\) |
meijerg | \(-\frac {1}{2 x^{2}}-\frac {x^{4} \left (-\frac {\ln \left (1+\left (x^{6}\right )^{\frac {1}{3}}\right )}{\left (x^{6}\right )^{\frac {2}{3}}}+\frac {\ln \left (1-\left (x^{6}\right )^{\frac {1}{3}}+\left (x^{6}\right )^{\frac {2}{3}}\right )}{2 \left (x^{6}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{2-\left (x^{6}\right )^{\frac {1}{3}}}\right )}{\left (x^{6}\right )^{\frac {2}{3}}}\right )}{6}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.53, size = 45, normalized size = 0.80 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) - \frac {1}{2 \, x^{2}} - \frac {1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 54, normalized size = 0.96 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + x^{2} \log \left (x^{4} - x^{2} + 1\right ) - 2 \, x^{2} \log \left (x^{2} + 1\right ) + 6}{12 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.06, size = 53, normalized size = 0.95 \begin {gather*} \frac {\log {\left (x^{2} + 1 \right )}}{6} - \frac {\log {\left (x^{4} - x^{2} + 1 \right )}}{12} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{2}}{3} - \frac {\sqrt {3}}{3} \right )}}{6} - \frac {1}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.95, size = 45, normalized size = 0.80 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) - \frac {1}{2 \, x^{2}} - \frac {1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.07, size = 57, normalized size = 1.02 \begin {gather*} \frac {\ln \left (x^2+1\right )}{6}-\frac {1}{2\,x^2}+\ln \left (x^2-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}-\frac {1}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x^2+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}-\frac {1}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________