Optimal. Leaf size=22 \[ \frac{1}{2} \sin ^{-1}(\tanh (x))+\frac{1}{2} \tanh (x) \sqrt{\text{sech}^2(x)} \]
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Rubi [A] time = 0.0199402, antiderivative size = 22, 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 = {3657, 4122, 195, 216} \[ \frac{1}{2} \sin ^{-1}(\tanh (x))+\frac{1}{2} \tanh (x) \sqrt{\text{sech}^2(x)} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 195
Rule 216
Rubi steps
\begin{align*} \int \left (1-\tanh ^2(x)\right )^{3/2} \, dx &=\int \text{sech}^2(x)^{3/2} \, dx\\ &=\operatorname{Subst}\left (\int \sqrt{1-x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{2} \sqrt{\text{sech}^2(x)} \tanh (x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{2} \sin ^{-1}(\tanh (x))+\frac{1}{2} \sqrt{\text{sech}^2(x)} \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0181831, size = 29, normalized size = 1.32 \[ \frac{\text{sech}(x) \left (2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+\tanh (x) \text{sech}(x)\right )}{2 \sqrt{\text{sech}^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 21, normalized size = 1. \begin{align*}{\frac{\tanh \left ( x \right ) }{2}\sqrt{1- \left ( \tanh \left ( x \right ) \right ) ^{2}}}+{\frac{\arcsin \left ( \tanh \left ( x \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68189, size = 38, normalized size = 1.73 \begin{align*} \frac{e^{\left (3 \, x\right )} - e^{x}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} + \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14712, size = 504, normalized size = 22.91 \begin{align*} \frac{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) +{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (1 - \tanh ^{2}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29679, size = 61, normalized size = 2.77 \begin{align*} \frac{1}{4} \, \pi - \frac{e^{\left (-x\right )} - e^{x}}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4} + \frac{1}{2} \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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