Optimal. Leaf size=20 \[ 4 \sqrt{\sinh (x)}-\frac{2 x \cosh (x)}{\sqrt{\sinh (x)}} \]
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Rubi [A] time = 0.0583554, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {3315} \[ 4 \sqrt{\sinh (x)}-\frac{2 x \cosh (x)}{\sqrt{\sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin{align*} \int \left (\frac{x}{\sinh ^{\frac{3}{2}}(x)}-x \sqrt{\sinh (x)}\right ) \, dx &=\int \frac{x}{\sinh ^{\frac{3}{2}}(x)} \, dx-\int x \sqrt{\sinh (x)} \, dx\\ &=-\frac{2 x \cosh (x)}{\sqrt{\sinh (x)}}+4 \sqrt{\sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.113241, size = 17, normalized size = 0.85 \[ \frac{4 \sinh (x)-2 x \cosh (x)}{\sqrt{\sinh (x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.075, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \sinh \left ( x \right ) \right ) ^{-{\frac{3}{2}}}}-x\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 -x \sqrt{\sinh \left (x\right )} + \frac{x}{\sinh \left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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*} - \int - \frac{x}{\sinh ^{\frac{3}{2}}{\left (x \right )}}\, dx - \int x \sqrt{\sinh{\left (x \right )}}\, dx \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 -x \sqrt{\sinh \left (x\right )} + \frac{x}{\sinh \left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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