Optimal. Leaf size=33 \[ -\frac{2}{3} \sinh (x) \sqrt{-\sinh (x) \tanh (x)}-\frac{8}{3} \text{csch}(x) \sqrt{-\sinh (x) \tanh (x)} \]
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Rubi [A] time = 0.0967737, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4397, 4400, 2598, 2589} \[ -\frac{2}{3} \sinh (x) \sqrt{-\sinh (x) \tanh (x)}-\frac{8}{3} \text{csch}(x) \sqrt{-\sinh (x) \tanh (x)} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 4400
Rule 2598
Rule 2589
Rubi steps
\begin{align*} \int (-\cosh (x)+\text{sech}(x))^{3/2} \, dx &=\int (-\sinh (x) \tanh (x))^{3/2} \, dx\\ &=\frac{\sqrt{-\sinh (x) \tanh (x)} \int (i \sinh (x))^{3/2} (i \tanh (x))^{3/2} \, dx}{\sqrt{i \sinh (x)} \sqrt{i \tanh (x)}}\\ &=-\frac{2}{3} \sinh (x) \sqrt{-\sinh (x) \tanh (x)}+\frac{\left (4 \sqrt{-\sinh (x) \tanh (x)}\right ) \int \frac{(i \tanh (x))^{3/2}}{\sqrt{i \sinh (x)}} \, dx}{3 \sqrt{i \sinh (x)} \sqrt{i \tanh (x)}}\\ &=-\frac{8}{3} \text{csch}(x) \sqrt{-\sinh (x) \tanh (x)}-\frac{2}{3} \sinh (x) \sqrt{-\sinh (x) \tanh (x)}\\ \end{align*}
Mathematica [A] time = 0.0831163, size = 24, normalized size = 0.73 \[ \frac{2}{3} \coth (x) \left (4 \text{csch}^2(x)+1\right ) (-\sinh (x) \tanh (x))^{3/2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.129, size = 0, normalized size = 0. \begin{align*} \int \left ( -\cosh \left ( x \right ) +{\rm sech} \left (x\right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.95826, size = 104, normalized size = 3.15 \begin{align*} -\frac{\sqrt{2} e^{\left (\frac{3}{2} \, x\right )}}{6 \,{\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}} - \frac{5 \, \sqrt{2} e^{\left (-\frac{1}{2} \, x\right )}}{2 \,{\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}} - \frac{5 \, \sqrt{2} e^{\left (-\frac{5}{2} \, x\right )}}{2 \,{\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}} - \frac{\sqrt{2} e^{\left (-\frac{9}{2} \, x\right )}}{6 \,{\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84747, size = 354, normalized size = 10.73 \begin{align*} -\frac{\sqrt{\frac{1}{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} + 7\right )} \sinh \left (x\right )^{2} + 14 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \sqrt{-\frac{1}{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )}}}{3 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-\cosh \left (x\right ) + \operatorname{sech}\left (x\right )\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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