Optimal. Leaf size=128 \[ -\frac{1}{8} \left (1-x^3\right )^{8/3}+\frac{1}{5} \left (1-x^3\right )^{5/3}-\frac{1}{2} \left (1-x^3\right )^{2/3}+\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0879538, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {446, 88, 55, 617, 204, 31} \[ -\frac{1}{8} \left (1-x^3\right )^{8/3}+\frac{1}{5} \left (1-x^3\right )^{5/3}-\frac{1}{2} \left (1-x^3\right )^{2/3}+\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 88
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^{11}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt [3]{1-x}}-(1-x)^{2/3}+(1-x)^{5/3}-\frac{1}{\sqrt [3]{1-x} (1+x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{1}{2} \left (1-x^3\right )^{2/3}+\frac{1}{5} \left (1-x^3\right )^{5/3}-\frac{1}{8} \left (1-x^3\right )^{8/3}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=-\frac{1}{2} \left (1-x^3\right )^{2/3}+\frac{1}{5} \left (1-x^3\right )^{5/3}-\frac{1}{8} \left (1-x^3\right )^{8/3}+\frac{\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=-\frac{1}{2} \left (1-x^3\right )^{2/3}+\frac{1}{5} \left (1-x^3\right )^{5/3}-\frac{1}{8} \left (1-x^3\right )^{8/3}+\frac{\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}\\ &=-\frac{1}{2} \left (1-x^3\right )^{2/3}+\frac{1}{5} \left (1-x^3\right )^{5/3}-\frac{1}{8} \left (1-x^3\right )^{8/3}-\frac{\tan ^{-1}\left (\frac{1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ \end{align*}
Mathematica [A] time = 0.0508381, size = 128, normalized size = 1. \[ \frac{1}{120} \left (-15 \left (1-x^3\right )^{2/3} x^6+6 \left (1-x^3\right )^{2/3} x^3-51 \left (1-x^3\right )^{2/3}+10\ 2^{2/3} \log \left (x^3+1\right )-30\ 2^{2/3} \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )-20\ 2^{2/3} \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{11}}{{x}^{3}+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.4344, size = 161, normalized size = 1.26 \begin{align*} -\frac{1}{8} \,{\left (-x^{3} + 1\right )}^{\frac{8}{3}} - \frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) + \frac{1}{5} \,{\left (-x^{3} + 1\right )}^{\frac{5}{3}} + \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) - \frac{1}{6} \cdot 2^{\frac{2}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) - \frac{1}{2} \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.39276, size = 448, normalized size = 3.5 \begin{align*} -\frac{1}{6} \, \sqrt{6} 2^{\frac{1}{6}} \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{1}{6} \cdot 2^{\frac{1}{6}}{\left (2 \, \sqrt{6} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \sqrt{6} 2^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{6} \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) - \frac{1}{40} \,{\left (5 \, x^{6} - 2 \, x^{3} + 17\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{11}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]