Optimal. Leaf size=34 \[ \frac{1}{2} \tanh (x) \sqrt{-\tanh ^2(x)}-\sqrt{-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]
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Rubi [A] time = 0.0288396, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4121, 3658, 3473, 3475} \[ \frac{1}{2} \tanh (x) \sqrt{-\tanh ^2(x)}-\sqrt{-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 4121
Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \left (-1+\text{sech}^2(x)\right )^{3/2} \, dx &=\int \left (-\tanh ^2(x)\right )^{3/2} \, dx\\ &=-\left (\left (\coth (x) \sqrt{-\tanh ^2(x)}\right ) \int \tanh ^3(x) \, dx\right )\\ &=\frac{1}{2} \tanh (x) \sqrt{-\tanh ^2(x)}-\left (\coth (x) \sqrt{-\tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=-\coth (x) \log (\cosh (x)) \sqrt{-\tanh ^2(x)}+\frac{1}{2} \tanh (x) \sqrt{-\tanh ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0154117, size = 27, normalized size = 0.79 \[ -\frac{1}{2} \sqrt{-\tanh ^2(x)} (\text{csch}(x) \text{sech}(x)+2 \coth (x) \log (\cosh (x))) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.102, size = 123, normalized size = 3.6 \begin{align*}{\frac{ \left ({{\rm e}^{2\,x}}+1 \right ) x}{{{\rm e}^{2\,x}}-1}\sqrt{-{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}-2\,{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) \left ({{\rm e}^{2\,x}}+1 \right ) }\sqrt{-{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}-{\frac{ \left ({{\rm e}^{2\,x}}+1 \right ) \ln \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{2\,x}}-1}\sqrt{-{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.66048, size = 45, normalized size = 1.32 \begin{align*} i \, x + \frac{2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05415, size = 4, normalized size = 0.12 \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 (\operatorname{sech}^{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 = 1.13726, size = 112, normalized size = 3.29 \begin{align*} -i \, x \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + i \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - \frac{i \,{\left (3 \, e^{\left (4 \, x\right )} \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + 2 \, e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + 3 \, \mathrm{sgn}\left (-e^{\left (4 \, x\right )} + 1\right )\right )}}{2 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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