Optimal. Leaf size=85 \[ -\frac {(1-x)^{2/3} \sqrt [3]{1+x}}{x}+\frac {2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{\sqrt {3}}-\frac {\log (x)}{3}+\log \left (\sqrt [3]{1-x}-\sqrt [3]{1+x}\right ) \]
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Rubi [A]
time = 0.02, 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 = {6261, 96, 93}
\begin {gather*} \frac {2 \text {ArcTan}\left (\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}}-\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 96
Rule 6261
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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 45, normalized size = 0.53 \begin {gather*} -\frac {(1-x)^{2/3} \left (1+x+x \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1-x}{1+x}\right )\right )}{x (1+x)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 152 vs.
\(2 (67) = 134\).
time = 0.34, size = 152, normalized size = 1.79 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \int \frac {\left (\frac {x + 1}{\sqrt {1 - x^{2}}}\right )^{\frac {2}{3}}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{2/3}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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