Optimal. Leaf size=135 \[ -\frac {\log (x)}{2}+\frac {1}{2} \log (x+1)+\frac {3}{2} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+1\right )+\frac {3}{2} \log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6126, 105, 60, 91} \[ -\frac {\log (x)}{2}+\frac {1}{2} \log (x+1)+\frac {3}{2} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+1\right )+\frac {3}{2} \log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 60
Rule 91
Rule 105
Rule 6126
Rubi steps
\begin {align*} \int \frac {e^{\frac {2}{3} \tanh ^{-1}(x)}}{x} \, dx &=\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x} \, dx\\ &=\int \frac {1}{\sqrt [3]{1-x} (1+x)^{2/3}} \, dx+\int \frac {1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx\\ &=\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )-\frac {\log (x)}{2}+\frac {1}{2} \log (1+x)+\frac {3}{2} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )+\frac {3}{2} \log \left (\sqrt [3]{1-x}-\sqrt [3]{1+x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 74, normalized size = 0.55 \[ -\frac {3 (1-x)^{2/3} \left (\sqrt [3]{2} (x+1)^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {1-x}{2}\right )+2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1-x}{x+1}\right )\right )}{4 (x+1)^{2/3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.29, size = 158, normalized size = 1.17 \[ -\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (x - 1\right )} - 2 \, \sqrt {3} \sqrt {-x^{2} + 1} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}}{3 \, {\left (x - 1\right )}}\right ) - \frac {1}{2} \, \log \left (-\frac {{\left (x + 1\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - x + \sqrt {-x^{2} + 1} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1}{x - 1}\right ) + \log \left (-\frac {x + \sqrt {-x^{2} + 1} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - 1}{x - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {2}{3}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {2}{3}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {2}{3}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{2/3}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {x + 1}{\sqrt {1 - x^{2}}}\right )^{\frac {2}{3}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________