Optimal. Leaf size=24 \[ -\frac{4}{3 \sqrt{\sinh (x)}}-\frac{2 x \cosh (x)}{3 \sinh ^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.0602881, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {3315} \[ -\frac{4}{3 \sqrt{\sinh (x)}}-\frac{2 x \cosh (x)}{3 \sinh ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin{align*} \int \left (\frac{x}{\sinh ^{\frac{5}{2}}(x)}+\frac{x}{3 \sqrt{\sinh (x)}}\right ) \, dx &=\frac{1}{3} \int \frac{x}{\sqrt{\sinh (x)}} \, dx+\int \frac{x}{\sinh ^{\frac{5}{2}}(x)} \, dx\\ &=-\frac{2 x \cosh (x)}{3 \sinh ^{\frac{3}{2}}(x)}-\frac{4}{3 \sqrt{\sinh (x)}}\\ \end{align*}
Mathematica [A] time = 0.0705882, size = 22, normalized size = 0.92 \[ \frac{1}{6} \sqrt{\sinh (x)} (-8 \text{csch}(x)-4 x \coth (x) \text{csch}(x)) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \sinh \left ( x \right ) \right ) ^{-{\frac{5}{2}}}}+{\frac{x}{3}{\frac{1}{\sqrt{\sinh \left ( x \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{3 \, \sqrt{\sinh \left (x\right )}} + \frac{x}{\sinh \left (x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.621, size = 375, normalized size = 15.62 \begin{align*} -\frac{4 \,{\left ({\left (x + 2\right )} \cosh \left (x\right )^{3} + 3 \,{\left (x + 2\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} +{\left (x + 2\right )} \sinh \left (x\right )^{3} +{\left (x - 2\right )} \cosh \left (x\right ) +{\left (3 \,{\left (x + 2\right )} \cosh \left (x\right )^{2} + x - 2\right )} \sinh \left (x\right )\right )} \sqrt{\sinh \left (x\right )}}{3 \,{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{3 x}{\sinh ^{\frac{5}{2}}{\left (x \right )}}\, dx + \int \frac{x}{\sqrt{\sinh{\left (x \right )}}}\, dx}{3} \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{x}{3 \, \sqrt{\sinh \left (x\right )}} + \frac{x}{\sinh \left (x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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