Optimal. Leaf size=16 \[ \frac{3 \sinh ^{\frac{5}{3}}(x)}{5 \cosh ^{\frac{5}{3}}(x)} \]
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Rubi [A] time = 0.0296269, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2563} \[ \frac{3 \sinh ^{\frac{5}{3}}(x)}{5 \cosh ^{\frac{5}{3}}(x)} \]
Antiderivative was successfully verified.
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Rule 2563
Rubi steps
\begin{align*} \int \frac{\sinh ^{\frac{2}{3}}(x)}{\cosh ^{\frac{8}{3}}(x)} \, dx &=\frac{3 \sinh ^{\frac{5}{3}}(x)}{5 \cosh ^{\frac{5}{3}}(x)}\\ \end{align*}
Mathematica [A] time = 0.0159079, size = 16, normalized size = 1. \[ \frac{3 \sinh ^{\frac{5}{3}}(x)}{5 \cosh ^{\frac{5}{3}}(x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sinh \left ( x \right ) \right ) ^{{\frac{2}{3}}} \left ( \cosh \left ( x \right ) \right ) ^{-{\frac{8}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.63048, size = 82, normalized size = 5.12 \begin{align*} -\frac{3 \,{\left (e^{\left (-x\right )} + 1\right )}^{\frac{2}{3}}{\left (-e^{\left (-x\right )} + 1\right )}^{\frac{2}{3}} e^{\left (-4 \, x\right )}}{5 \,{\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac{8}{3}}} + \frac{3 \,{\left (e^{\left (-x\right )} + 1\right )}^{\frac{2}{3}}{\left (-e^{\left (-x\right )} + 1\right )}^{\frac{2}{3}}}{5 \,{\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac{8}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84881, size = 333, normalized size = 20.81 \begin{align*} \frac{6 \,{\left (\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 )\right )} \cosh \left (x\right )^{\frac{1}{3}} \sinh \left (x\right )^{\frac{2}{3}}}{5 \,{\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 )}} \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 \frac{\sinh \left (x\right )^{\frac{2}{3}}}{\cosh \left (x\right )^{\frac{8}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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