Optimal. Leaf size=35 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}\right )-\frac{1}{2} \tanh (x) \sqrt{-\text{sech}^2(x)} \]
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Rubi [A] time = 0.0235295, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3657, 4122, 195, 217, 206} \[ \frac{1}{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}\right )-\frac{1}{2} \tanh (x) \sqrt{-\text{sech}^2(x)} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \left (-1+\tanh ^2(x)\right )^{3/2} \, dx &=\int \left (-\text{sech}^2(x)\right )^{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} \sqrt{-\text{sech}^2(x)} \tanh (x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}\right )\\ &=\frac{1}{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}}\right )-\frac{1}{2} \sqrt{-\text{sech}^2(x)} \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0193486, size = 28, normalized size = 0.8 \[ -\frac{1}{2} \sqrt{-\text{sech}^2(x)} \left (\tanh (x)+2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 28, normalized size = 0.8 \begin{align*} -{\frac{\tanh \left ( x \right ) }{2}\sqrt{-1+ \left ( \tanh \left ( x \right ) \right ) ^{2}}}+{\frac{1}{2}\ln \left ( \tanh \left ( x \right ) +\sqrt{-1+ \left ( \tanh \left ( x \right ) \right ) ^{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.72371, size = 43, normalized size = 1.23 \begin{align*} \frac{-i \, e^{\left (3 \, x\right )} + i \, e^{x}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} - i \, \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35453, size = 4, normalized size = 0.11 \begin{align*} 0 \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 (\tanh ^{2}{\left (x \right )} - 1\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.20675, size = 84, normalized size = 2.4 \begin{align*} -\frac{i \, e^{\left (-x\right )} - i \, e^{x}}{{\left (-i \, e^{\left (-x\right )} + i \, e^{x}\right )}^{2} - 4} + \frac{1}{8} \, \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right ) - \frac{1}{4} \, \log \left (-i \, e^{\left (-x\right )} + i \, e^{x} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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