Optimal. Leaf size=81 \[ \frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+x+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.172358, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {321, 209, 634, 618, 204, 628, 203} \[ \frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+x+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 321
Rule 209
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rubi steps
\begin{align*} \int \frac{x^6}{1+x^6} \, dx &=x-\int \frac{1}{1+x^6} \, dx\\ &=x-\frac{1}{3} \int \frac{1-\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx-\frac{1}{3} \int \frac{1+\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx-\frac{1}{3} \int \frac{1}{1+x^2} \, dx\\ &=x-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{12} \int \frac{1}{1-\sqrt{3} x+x^2} \, dx-\frac{1}{12} \int \frac{1}{1+\sqrt{3} x+x^2} \, dx+\frac{\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx}{4 \sqrt{3}}-\frac{\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx}{4 \sqrt{3}}\\ &=x-\frac{1}{3} \tan ^{-1}(x)+\frac{\log \left (1-\sqrt{3} x+x^2\right )}{4 \sqrt{3}}-\frac{\log \left (1+\sqrt{3} x+x^2\right )}{4 \sqrt{3}}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 x\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 x\right )\\ &=x+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (\sqrt{3}+2 x\right )+\frac{\log \left (1-\sqrt{3} x+x^2\right )}{4 \sqrt{3}}-\frac{\log \left (1+\sqrt{3} x+x^2\right )}{4 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0132297, size = 76, normalized size = 0.94 \[ \frac{1}{12} \left (\sqrt{3} \log \left (x^2-\sqrt{3} x+1\right )-\sqrt{3} \log \left (x^2+\sqrt{3} x+1\right )+12 x+2 \tan ^{-1}\left (\sqrt{3}-2 x\right )-4 \tan ^{-1}(x)-2 \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.018, size = 62, normalized size = 0.8 \begin{align*} x-{\frac{\arctan \left ( x \right ) }{3}}-{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{6}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{6}}+{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}-{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.4994, size = 82, normalized size = 1.01 \begin{align*} -\frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) + x - \frac{1}{6} \, \arctan \left (2 \, x + \sqrt{3}\right ) - \frac{1}{6} \, \arctan \left (2 \, x - \sqrt{3}\right ) - \frac{1}{3} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.60829, size = 294, normalized size = 3.63 \begin{align*} -\frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) + x - \frac{1}{3} \, \arctan \left (x\right ) + \frac{1}{3} \, \arctan \left (-2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}\right ) + \frac{1}{3} \, \arctan \left (-2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.198119, size = 70, normalized size = 0.86 \begin{align*} x + \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} - \frac{\operatorname{atan}{\left (x \right )}}{3} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{x^{6} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]