Optimal. Leaf size=85 \[ -\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{x}-\frac {\log (x)}{3}+\log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6126, 94, 91} \[ -\frac {(1-x)^{2/3} \sqrt [3]{x+1}}{x}-\frac {\log (x)}{3}+\log \left (\sqrt [3]{1-x}-\sqrt [3]{x+1}\right )+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 91
Rule 94
Rule 6126
Rubi steps
\begin {align*} \int \frac {e^{\frac {2}{3} \tanh ^{-1}(x)}}{x^2} \, dx &=\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^2} \, dx\\ &=-\frac {(1-x)^{2/3} \sqrt [3]{1+x}}{x}+\frac {2}{3} \int \frac {1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx\\ &=-\frac {(1-x)^{2/3} \sqrt [3]{1+x}}{x}+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{\sqrt {3}}-\frac {\log (x)}{3}+\log \left (\sqrt [3]{1-x}-\sqrt [3]{1+x}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 45, normalized size = 0.53 \[ -\frac {(1-x)^{2/3} \left (x \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1-x}{x+1}\right )+x+1\right )}{x (x+1)^{2/3}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.52, size = 152, normalized size = 1.79 \[ -\frac {2 \, \sqrt {3} x \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 2 \, x \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - 1\right ) + x \log \left (\frac {{\left (x - 1\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + x - \sqrt {-x^{2} + 1} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - 1}{x - 1}\right ) - 3 \, {\left (x - 1\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}}}{3 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {2}{3}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {2}{3}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {2}{3}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{2/3}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {x + 1}{\sqrt {1 - x^{2}}}\right )^{\frac {2}{3}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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