Optimal. Leaf size=158 \[ -\frac{\sqrt [3]{1-x^3}}{3 x^3}-\frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}-\frac{1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\log (x)}{6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.102788, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {446, 103, 156, 57, 618, 204, 31, 617} \[ -\frac{\sqrt [3]{1-x^3}}{3 x^3}-\frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}-\frac{1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\log (x)}{6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 103
Rule 156
Rule 57
Rule 618
Rule 204
Rule 31
Rule 617
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{2/3} x^2 (1+x)} \, dx,x,x^3\right )\\ &=-\frac{\sqrt [3]{1-x^3}}{3 x^3}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\frac{1}{3}-\frac{2 x}{3}}{(1-x)^{2/3} x (1+x)} \, dx,x,x^3\right )\\ &=-\frac{\sqrt [3]{1-x^3}}{3 x^3}-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{2/3} x} \, dx,x,x^3\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{2/3} (1+x)} \, dx,x,x^3\right )\\ &=-\frac{\sqrt [3]{1-x^3}}{3 x^3}+\frac{\log (x)}{6}-\frac{\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+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\ 2^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=-\frac{\sqrt [3]{1-x^3}}{3 x^3}+\frac{\log (x)}{6}-\frac{\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac{1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^3}\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{2^{2/3}}\\ &=-\frac{\sqrt [3]{1-x^3}}{3 x^3}+\frac{\tan ^{-1}\left (\frac{1+2 \sqrt [3]{1-x^3}}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\log (x)}{6}-\frac{\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac{1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.066116, size = 196, normalized size = 1.24 \[ \frac{1}{36} \left (-\frac{12 \sqrt [3]{1-x^3}}{x^3}-4 \log \left (1-\sqrt [3]{1-x^3}\right )+6 \sqrt [3]{2} \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )-3 \sqrt [3]{2} \log \left (\left (1-x^3\right )^{2/3}+\sqrt [3]{2-2 x^3}+2^{2/3}\right )+2 \log \left (\left (1-x^3\right )^{2/3}+\sqrt [3]{1-x^3}+1\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )-6 \sqrt [3]{2} \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ({x}^{3}+1 \right ) } \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.59326, size = 560, normalized size = 3.54 \begin{align*} -\frac{12 \cdot 4^{\frac{1}{6}} \sqrt{3} x^{3} \arctan \left (\frac{1}{6} \cdot 4^{\frac{1}{6}}{\left (4^{\frac{2}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}} \sqrt{3}\right )}\right ) + 3 \cdot 4^{\frac{2}{3}} x^{3} \log \left (4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 2 \cdot 4^{\frac{1}{3}}\right ) - 6 \cdot 4^{\frac{2}{3}} x^{3} \log \left (-4^{\frac{2}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) - 8 \, \sqrt{3} x^{3} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - 4 \, x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + 8 \, x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) + 24 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{72 \, x^{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{1}{x^{4} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{2}{3}} \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]