Optimal. Leaf size=50 \[ \frac{2}{5} \sinh ^2(x) \tanh (x) \sqrt{\sinh (x) \tanh (x)}+\frac{16}{15} \tanh (x) \sqrt{\sinh (x) \tanh (x)}-\frac{64}{15} \coth (x) \sqrt{\sinh (x) \tanh (x)} \]
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Rubi [A] time = 0.117818, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4398, 4400, 2598, 2594, 2589} \[ \frac{2}{5} \sinh ^2(x) \tanh (x) \sqrt{\sinh (x) \tanh (x)}+\frac{16}{15} \tanh (x) \sqrt{\sinh (x) \tanh (x)}-\frac{64}{15} \coth (x) \sqrt{\sinh (x) \tanh (x)} \]
Antiderivative was successfully verified.
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Rule 4398
Rule 4400
Rule 2598
Rule 2594
Rule 2589
Rubi steps
\begin{align*} \int (\sinh (x) \tanh (x))^{5/2} \, dx &=\frac{\sqrt{\sinh (x) \tanh (x)} \int (-\sinh (x) \tanh (x))^{5/2} \, dx}{\sqrt{-\sinh (x) \tanh (x)}}\\ &=\frac{\sqrt{\sinh (x) \tanh (x)} \int (i \sinh (x))^{5/2} (i \tanh (x))^{5/2} \, dx}{\sqrt{i \sinh (x)} \sqrt{i \tanh (x)}}\\ &=\frac{2}{5} \sinh ^2(x) \tanh (x) \sqrt{\sinh (x) \tanh (x)}+\frac{\left (8 \sqrt{\sinh (x) \tanh (x)}\right ) \int \sqrt{i \sinh (x)} (i \tanh (x))^{5/2} \, dx}{5 \sqrt{i \sinh (x)} \sqrt{i \tanh (x)}}\\ &=\frac{16}{15} \tanh (x) \sqrt{\sinh (x) \tanh (x)}+\frac{2}{5} \sinh ^2(x) \tanh (x) \sqrt{\sinh (x) \tanh (x)}-\frac{\left (32 \sqrt{\sinh (x) \tanh (x)}\right ) \int \sqrt{i \sinh (x)} \sqrt{i \tanh (x)} \, dx}{15 \sqrt{i \sinh (x)} \sqrt{i \tanh (x)}}\\ &=-\frac{64}{15} \coth (x) \sqrt{\sinh (x) \tanh (x)}+\frac{16}{15} \tanh (x) \sqrt{\sinh (x) \tanh (x)}+\frac{2}{5} \sinh ^2(x) \tanh (x) \sqrt{\sinh (x) \tanh (x)}\\ \end{align*}
Mathematica [A] time = 0.113306, size = 29, normalized size = 0.58 \[ -\frac{2}{15} \tanh (x) \sqrt{\sinh (x) \tanh (x)} \left (-3 \cosh ^2(x)+32 \coth ^2(x)-5\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.103, size = 0, normalized size = 0. \begin{align*} \int \left ( \sinh \left ( x \right ) \tanh \left ( x \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.75192, size = 139, normalized size = 2.78 \begin{align*} -\frac{\sqrt{2} e^{\left (\frac{5}{2} \, x\right )}}{20 \,{\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} + \frac{7 \, \sqrt{2} e^{\left (\frac{1}{2} \, x\right )}}{4 \,{\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} + \frac{41 \, \sqrt{2} e^{\left (-\frac{3}{2} \, x\right )}}{6 \,{\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} + \frac{41 \, \sqrt{2} e^{\left (-\frac{7}{2} \, x\right )}}{6 \,{\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} + \frac{7 \, \sqrt{2} e^{\left (-\frac{11}{2} \, x\right )}}{4 \,{\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} - \frac{\sqrt{2} e^{\left (-\frac{15}{2} \, x\right )}}{20 \,{\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.39637, size = 883, normalized size = 17.66 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (3 \, \cosh \left (x\right )^{8} + 24 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + 3 \, \sinh \left (x\right )^{8} + 12 \,{\left (7 \, \cosh \left (x\right )^{2} - 9\right )} \sinh \left (x\right )^{6} - 108 \, \cosh \left (x\right )^{6} + 24 \,{\left (7 \, \cosh \left (x\right )^{3} - 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \,{\left (105 \, \cosh \left (x\right )^{4} - 810 \, \cosh \left (x\right )^{2} - 151\right )} \sinh \left (x\right )^{4} - 302 \, \cosh \left (x\right )^{4} + 8 \,{\left (21 \, \cosh \left (x\right )^{5} - 270 \, \cosh \left (x\right )^{3} - 151 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 12 \,{\left (7 \, \cosh \left (x\right )^{6} - 135 \, \cosh \left (x\right )^{4} - 151 \, \cosh \left (x\right )^{2} - 9\right )} \sinh \left (x\right )^{2} - 108 \, \cosh \left (x\right )^{2} + 8 \,{\left (3 \, \cosh \left (x\right )^{7} - 81 \, \cosh \left (x\right )^{5} - 151 \, \cosh \left (x\right )^{3} - 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3\right )}}{30 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \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 )}} \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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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