Optimal. Leaf size=31 \[ \frac{2}{3} \sinh (x) \sqrt{\sinh (x) \tanh (x)}+\frac{8}{3} \text{csch}(x) \sqrt{\sinh (x) \tanh (x)} \]
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Rubi [A] time = 0.08793, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {4398, 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 4398
Rule 4400
Rule 2598
Rule 2589
Rubi steps
\begin{align*} \int (\sinh (x) \tanh (x))^{3/2} \, dx &=-\frac{\sqrt{\sinh (x) \tanh (x)} \int (-\sinh (x) \tanh (x))^{3/2} \, dx}{\sqrt{-\sinh (x) \tanh (x)}}\\ &=-\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 = 0.0673715, size = 23, normalized size = 0.74 \[ \frac{2}{3} \sinh (x) \left (4 \text{csch}^2(x)+1\right ) \sqrt{\sinh (x) \tanh (x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.105, size = 0, normalized size = 0. \begin{align*} \int \left ( \sinh \left ( x \right ) \tanh \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 = 1.62995, size = 93, normalized size = 3. \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 = 2.30302, size = 348, normalized size = 11.23 \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 )}}{3 \, \sqrt{\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 )}{\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 (\sinh \left (x\right ) \tanh \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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