Optimal. Leaf size=69 \[ x+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \tanh ^{-1}(x)+\frac {1}{6} \log \left (1-x+x^2\right )-\frac {1}{6} \log \left (1+x+x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1607, 1598,
396, 216, 648, 632, 210, 642, 212} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{6} \log \left (x^2-x+1\right )-\frac {1}{6} \log \left (x^2+x+1\right )+x-\frac {2}{3} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 212
Rule 216
Rule 396
Rule 632
Rule 642
Rule 648
Rule 1598
Rule 1607
Rubi steps
\begin {align*} \int \frac {\frac {1}{x^3}+x^3}{-\frac {1}{x^3}+x^3} \, dx &=\int \frac {x^3 \left (\frac {1}{x^3}+x^3\right )}{-1+x^6} \, dx\\ &=\int \frac {1+x^6}{-1+x^6} \, dx\\ &=x+2 \int \frac {1}{-1+x^6} \, dx\\ &=x-\frac {2}{3} \int \frac {1}{1-x^2} \, dx-\frac {2}{3} \int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx-\frac {2}{3} \int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx\\ &=x-\frac {2}{3} \tanh ^{-1}(x)+\frac {1}{6} \int \frac {-1+2 x}{1-x+x^2} \, dx-\frac {1}{6} \int \frac {1+2 x}{1+x+x^2} \, dx-\frac {1}{2} \int \frac {1}{1-x+x^2} \, dx-\frac {1}{2} \int \frac {1}{1+x+x^2} \, dx\\ &=x-\frac {2}{3} \tanh ^{-1}(x)+\frac {1}{6} \log \left (1-x+x^2\right )-\frac {1}{6} \log \left (1+x+x^2\right )+\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=x-\frac {\tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \tanh ^{-1}(x)+\frac {1}{6} \log \left (1-x+x^2\right )-\frac {1}{6} \log \left (1+x+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 78, normalized size = 1.13 \begin {gather*} \frac {1}{6} \left (6 x-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 = 0.03, size = 67, normalized size = 0.97
method | result | size |
risch | \(x -\frac {\ln \left (1+x \right )}{3}+\frac {\ln \left (x^{2}-x +1\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x -\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{3}-\frac {\ln \left (x^{2}+x +1\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x +\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{3}+\frac {\ln \left (-1+x \right )}{3}\) | \(63\) |
default | \(x -\frac {\ln \left (x^{2}+x +1\right )}{6}-\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\ln \left (-1+x \right )}{3}-\frac {\ln \left (1+x \right )}{3}+\frac {\ln \left (x^{2}-x +1\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 66, normalized size = 0.96 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + x - \frac {1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{3} \, \log \left (x + 1\right ) + \frac {1}{3} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 66, normalized size = 0.96 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + x - \frac {1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{3} \, \log \left (x + 1\right ) + \frac {1}{3} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.16, size = 85, normalized size = 1.23 \begin {gather*} x + \frac {\log {\left (x - 1 \right )}}{3} - \frac {\log {\left (x + 1 \right )}}{3} + \frac {\log {\left (x^{2} - x + 1 \right )}}{6} - \frac {\log {\left (x^{2} + x + 1 \right )}}{6} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.42, size = 68, normalized size = 0.99 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + x - \frac {1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{3} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{3} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 94, normalized size = 1.36 \begin {gather*} x+\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3}-\mathrm {atan}\left (\frac {x\,32{}\mathrm {i}}{-32+\sqrt {3}\,32{}\mathrm {i}}-\frac {32\,\sqrt {3}\,x}{-32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{3}-\frac {1}{3}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,32{}\mathrm {i}}{32+\sqrt {3}\,32{}\mathrm {i}}+\frac {32\,\sqrt {3}\,x}{32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{3}+\frac {1}{3}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________