Optimal. Leaf size=56 \[ \frac{x^2}{2}-\frac{1}{6} \log \left (x^2+1\right )+\frac{1}{12} \log \left (x^4-x^2+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
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Rubi [A] time = 0.0428234, antiderivative size = 56, normalized size of antiderivative = 1., 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 = {275, 321, 200, 31, 634, 618, 204, 628} \[ \frac{x^2}{2}-\frac{1}{6} \log \left (x^2+1\right )+\frac{1}{12} \log \left (x^4-x^2+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 275
Rule 321
Rule 200
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^7}{1+x^6} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{1+x^3} \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^3} \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^2\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{2-x}{1-x+x^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{1}{6} \log \left (1+x^2\right )+\frac{1}{12} \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,x^2\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{1}{6} \log \left (1+x^2\right )+\frac{1}{12} \log \left (1-x^2+x^4\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x^2\right )\\ &=\frac{x^2}{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 = 0.0141678, size = 79, normalized size = 1.41 \[ \frac{1}{12} \left (6 x^2-2 \log \left (x^2+1\right )+\log \left (x^2-\sqrt{3} x+1\right )+\log \left (x^2+\sqrt{3} x+1\right )+2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-2 x\right )+2 \sqrt{3} \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 46, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}+{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ({x}^{2}+1 \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50142, size = 61, normalized size = 1.09 \begin{align*} \frac{1}{2} \, x^{2} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46882, size = 135, normalized size = 2.41 \begin{align*} \frac{1}{2} \, x^{2} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.132635, size = 51, normalized size = 0.91 \begin{align*} \frac{x^{2}}{2} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12547, size = 61, normalized size = 1.09 \begin{align*} \frac{1}{2} \, x^{2} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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