Optimal. Leaf size=49 \[ \frac {\tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1-x^2\right )+\frac {1}{12} \log \left (1+x^2+x^4\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {281, 206, 31,
648, 632, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1-x^2\right )+\frac {1}{12} \log \left (x^4+x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 206
Rule 210
Rule 281
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x}{1-x^6} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^3} \, dx,x,x^2\right )\\ &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,x^2\right )+\frac {1}{6} \text {Subst}\left (\int \frac {2+x}{1+x+x^2} \, dx,x,x^2\right )\\ &=-\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}{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 {\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 = 0.01, size = 73, normalized size = 1.49 \begin {gather*} \frac {1}{12} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-2 \log (1-x)-2 \log (1+x)+\log \left (1-x+x^2\right )+\log \left (1+x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.18, size = 66, normalized size = 1.35
method | result | size |
risch | \(-\frac {\ln \left (x^{2}-1\right )}{6}+\frac {\ln \left (x^{4}+x^{2}+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{2}+\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}\) | \(37\) |
meijerg | \(-\frac {x^{2} \left (\ln \left (1-\left (x^{6}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{6}\right )^{\frac {1}{3}}+\left (x^{6}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{2+\left (x^{6}\right )^{\frac {1}{3}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{3}}}\) | \(64\) |
default | \(\frac {\ln \left (x^{2}+x +1\right )}{12}-\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}-\frac {\ln \left (x -1\right )}{6}-\frac {\ln \left (x +1\right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 38, normalized size = 0.78 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \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 = 0.39, size = 38, normalized size = 0.78 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \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]
________________________________________________________________________________________
Sympy [A]
time = 0.05, size = 46, normalized size = 0.94 \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.42, size = 39, 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}{12} \, \log \left (x^{4} + x^{2} + 1\right ) - \frac {1}{6} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 52, normalized size = 1.06 \begin {gather*} -\frac {\ln \left (x^2-1\right )}{6}-\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]
________________________________________________________________________________________