Optimal. Leaf size=29 \[ \sqrt{\tanh ^2(x)} \coth (x) \log (\cosh (x))-\frac{1}{2} \tanh ^2(x)^{3/2} \coth (x) \]
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Rubi [A] time = 0.0310718, antiderivative size = 29, 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 = {4121, 3658, 3473, 3475} \[ \sqrt{\tanh ^2(x)} \coth (x) \log (\cosh (x))-\frac{1}{2} \tanh ^2(x)^{3/2} \coth (x) \]
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 \tanh ^2(x)^{3/2} \, dx\\ &=\left (\coth (x) \sqrt{\tanh ^2(x)}\right ) \int \tanh ^3(x) \, dx\\ &=-\frac{1}{2} \coth (x) \tanh ^2(x)^{3/2}+\left (\coth (x) \sqrt{\tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=\coth (x) \log (\cosh (x)) \sqrt{\tanh ^2(x)}-\frac{1}{2} \coth (x) \tanh ^2(x)^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0225163, size = 25, normalized size = 0.86 \[ \frac{1}{2} \sqrt{\tanh ^2(x)} (\text{csch}(x) \text{sech}(x)+2 \coth (x) \log (\cosh (x))) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.108, size = 120, normalized size = 4.1 \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 [A] time = 1.64972, size = 45, normalized size = 1.55 \begin{align*} -x - \frac{2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} - \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 [B] time = 1.95346, size = 620, normalized size = 21.38 \begin{align*} -\frac{x \cosh \left (x\right )^{4} + 4 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + x \sinh \left (x\right )^{4} + 2 \,{\left (x - 1\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, x \cosh \left (x\right )^{2} + x - 1\right )} \sinh \left (x\right )^{2} -{\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 )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \,{\left (x \cosh \left (x\right )^{3} +{\left (x - 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + x}{\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 - \operatorname{sech}^{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 = 1.12704, size = 97, normalized size = 3.34 \begin{align*} -x \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac{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 )}{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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