Optimal. Leaf size=58 \[ \frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )-\frac {\log (x)}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 57, 618, 204, 31} \[ \frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )-\frac {\log (x)}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 57
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {1}{x \left (1-x^2\right )^{2/3}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1-x)^{2/3} x} \, dx,x,x^2\right )\\ &=-\frac {\log (x)}{2}-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^2}\right )-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )\\ &=-\frac {\log (x)}{2}+\frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^2}\right )\\ &=-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )-\frac {\log (x)}{2}+\frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 81, normalized size = 1.40 \[ \frac {1}{2} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {1}{4} \log \left (\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 64, normalized size = 1.10 \[ -\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.09, size = 64, normalized size = 1.10 \[ -\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left (-{\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.10, size = 48, normalized size = 0.83 \[ \frac {\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{2} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{2}\right )}{3}+\left (2 \ln \relax (x )+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{2 \Gamma \left (\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.30, size = 62, normalized size = 1.07 \[ -\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.46, size = 76, normalized size = 1.31 \[ \frac {\ln \left (\frac {9\,{\left (1-x^2\right )}^{1/3}}{4}-\frac {9}{4}\right )}{2}+\ln \left (\frac {9\,{\left (1-x^2\right )}^{1/3}}{2}+\frac {9}{4}-\frac {\sqrt {3}\,9{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )-\ln \left (\frac {9\,{\left (1-x^2\right )}^{1/3}}{2}+\frac {9}{4}+\frac {\sqrt {3}\,9{}\mathrm {i}}{4}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.98, size = 37, normalized size = 0.64 \[ - \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {1}{x^{2}}} \right )}}{2 x^{\frac {4}{3}} \Gamma \left (\frac {5}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________