Optimal. Leaf size=116 \[ -\frac{(1-x)^{2/3} (x+1)^{4/3}}{2 x^2}-\frac{(1-x)^{2/3} \sqrt [3]{x+1}}{3 x}-\frac{\log (x)}{9}+\frac{1}{3} \log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{x+1}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0377013, antiderivative size = 116, normalized size of antiderivative = 1., 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, 96, 94, 91} \[ -\frac{(1-x)^{2/3} (x+1)^{4/3}}{2 x^2}-\frac{(1-x)^{2/3} \sqrt [3]{x+1}}{3 x}-\frac{\log (x)}{9}+\frac{1}{3} \log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{x+1}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 96
Rule 94
Rule 91
Rubi steps
\begin{align*} \int \frac{e^{\frac{2}{3} \tanh ^{-1}(x)}}{x^3} \, dx &=\int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^3} \, dx\\ &=-\frac{(1-x)^{2/3} (1+x)^{4/3}}{2 x^2}+\frac{1}{3} \int \frac{\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^2} \, dx\\ &=-\frac{(1-x)^{2/3} \sqrt [3]{1+x}}{3 x}-\frac{(1-x)^{2/3} (1+x)^{4/3}}{2 x^2}+\frac{2}{9} \int \frac{1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx\\ &=-\frac{(1-x)^{2/3} \sqrt [3]{1+x}}{3 x}-\frac{(1-x)^{2/3} (1+x)^{4/3}}{2 x^2}+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{1+x}}\right )}{3 \sqrt{3}}-\frac{\log (x)}{9}+\frac{1}{3} \log \left (\sqrt [3]{1-x}-\sqrt [3]{1+x}\right )\\ \end{align*}
Mathematica [C] time = 0.0128138, size = 57, normalized size = 0.49 \[ -\frac{(1-x)^{2/3} \left (2 x^2 \text{Hypergeometric2F1}\left (\frac{2}{3},1,\frac{5}{3},\frac{1-x}{x+1}\right )+5 x^2+8 x+3\right )}{6 x^2 (x+1)^{2/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ({(1+x){\frac{1}{\sqrt{-{x}^{2}+1}}}} \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{x + 1}{\sqrt{-x^{2} + 1}}\right )^{\frac{2}{3}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72863, size = 427, normalized size = 3.68 \begin{align*} -\frac{4 \, \sqrt{3} x^{2} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - 4 \, x^{2} \log \left (\left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}} - 1\right ) + 2 \, x^{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 ) - 3 \,{\left (5 \, x^{2} - 2 \, x - 3\right )} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{2}{3}}}{18 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{x + 1}{\sqrt{-x^{2} + 1}}\right )^{\frac{2}{3}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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