Optimal. Leaf size=98 \[ -\frac{4 \cosh (a+b x)}{25 b^2 \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{12 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}}+\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)} \]
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Rubi [A] time = 0.0527631, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5445, 3769, 3771, 2639} \[ -\frac{4 \cosh (a+b x)}{25 b^2 \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{12 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}}+\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 5445
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{x \cosh (a+b x)}{\text{csch}^{\frac{3}{2}}(a+b x)} \, dx &=\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)}-\frac{2 \int \frac{1}{\text{csch}^{\frac{5}{2}}(a+b x)} \, dx}{5 b}\\ &=\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{25 b^2 \text{csch}^{\frac{3}{2}}(a+b x)}+\frac{6 \int \frac{1}{\sqrt{\text{csch}(a+b x)}} \, dx}{25 b}\\ &=\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{25 b^2 \text{csch}^{\frac{3}{2}}(a+b x)}+\frac{6 \int \sqrt{i \sinh (a+b x)} \, dx}{25 b \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ &=\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{25 b^2 \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{12 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right )}{25 b^2 \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ \end{align*}
Mathematica [C] time = 1.91286, size = 111, normalized size = 1.13 \[ \frac{e^{-2 (a+b x)} \left (-\frac{48 e^{2 (a+b x)} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};e^{2 (a+b x)}\right )}{\sqrt{1-e^{2 (a+b x)}}}+(24-10 b x) e^{2 (a+b x)}+(5 b x-2) e^{4 (a+b x)}+5 b x+2\right )}{50 b^2 \sqrt{\text{csch}(a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{x\cosh \left ( bx+a \right ) \left ({\rm csch} \left (bx+a\right ) \right ) ^{-{\frac{3}{2}}}}\, 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 \cosh \left (b x + a\right )}{\operatorname{csch}\left (b x + a\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(-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{x \cosh \left (b x + a\right )}{\operatorname{csch}\left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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