Optimal. Leaf size=24 \[ \frac{2 x \cosh (x)}{5 \text{csch}^{\frac{3}{2}}(x)}-\frac{4}{25 \text{csch}^{\frac{5}{2}}(x)} \]
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Rubi [A] time = 0.113536, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4187, 4189} \[ \frac{2 x \cosh (x)}{5 \text{csch}^{\frac{3}{2}}(x)}-\frac{4}{25 \text{csch}^{\frac{5}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 4187
Rule 4189
Rubi steps
\begin{align*} \int \left (\frac{x}{\text{csch}^{\frac{5}{2}}(x)}+\frac{3 x}{5 \sqrt{\text{csch}(x)}}\right ) \, dx &=\frac{3}{5} \int \frac{x}{\sqrt{\text{csch}(x)}} \, dx+\int \frac{x}{\text{csch}^{\frac{5}{2}}(x)} \, dx\\ &=-\frac{4}{25 \text{csch}^{\frac{5}{2}}(x)}+\frac{2 x \cosh (x)}{5 \text{csch}^{\frac{3}{2}}(x)}-\frac{3}{5} \int \frac{x}{\sqrt{\text{csch}(x)}} \, dx+\frac{3 \int x \sqrt{-\sinh (x)} \, dx}{5 \sqrt{\text{csch}(x)} \sqrt{-\sinh (x)}}\\ &=-\frac{4}{25 \text{csch}^{\frac{5}{2}}(x)}+\frac{2 x \cosh (x)}{5 \text{csch}^{\frac{3}{2}}(x)}\\ \end{align*}
Mathematica [A] time = 0.101049, size = 17, normalized size = 0.71 \[ \frac{2 (5 x \coth (x)-2)}{25 \text{csch}^{\frac{5}{2}}(x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{x \left ({\rm csch} \left (x\right ) \right ) ^{-{\frac{5}{2}}}}+{\frac{3\,x}{5}{\frac{1}{\sqrt{{\rm csch} \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{3 \, x}{5 \, \sqrt{\operatorname{csch}\left (x\right )}} + \frac{x}{\operatorname{csch}\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 [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*} \frac{\int \frac{5 x}{\operatorname{csch}^{\frac{5}{2}}{\left (x \right )}}\, dx + \int \frac{3 x}{\sqrt{\operatorname{csch}{\left (x \right )}}}\, dx}{5} \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{3 \, x}{5 \, \sqrt{\operatorname{csch}\left (x\right )}} + \frac{x}{\operatorname{csch}\left (x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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