Optimal. Leaf size=98 \[ -\frac{4 \cosh (a+b x)}{3 b^2 \sqrt{\sinh (a+b x)}}-\frac{4 i \sqrt{\sinh (a+b x)} E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{3 b^2 \sqrt{i \sinh (a+b x)}}-\frac{2 x}{3 b \sinh ^{\frac{3}{2}}(a+b x)} \]
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Rubi [A] time = 0.0516689, 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 = {5372, 2636, 2640, 2639} \[ -\frac{4 \cosh (a+b x)}{3 b^2 \sqrt{\sinh (a+b x)}}-\frac{4 i \sqrt{\sinh (a+b x)} E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{3 b^2 \sqrt{i \sinh (a+b x)}}-\frac{2 x}{3 b \sinh ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 5372
Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{x \cosh (a+b x)}{\sinh ^{\frac{5}{2}}(a+b x)} \, dx &=-\frac{2 x}{3 b \sinh ^{\frac{3}{2}}(a+b x)}+\frac{2 \int \frac{1}{\sinh ^{\frac{3}{2}}(a+b x)} \, dx}{3 b}\\ &=-\frac{2 x}{3 b \sinh ^{\frac{3}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{3 b^2 \sqrt{\sinh (a+b x)}}+\frac{2 \int \sqrt{\sinh (a+b x)} \, dx}{3 b}\\ &=-\frac{2 x}{3 b \sinh ^{\frac{3}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{3 b^2 \sqrt{\sinh (a+b x)}}+\frac{\left (2 \sqrt{\sinh (a+b x)}\right ) \int \sqrt{i \sinh (a+b x)} \, dx}{3 b \sqrt{i \sinh (a+b x)}}\\ &=-\frac{2 x}{3 b \sinh ^{\frac{3}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{3 b^2 \sqrt{\sinh (a+b x)}}-\frac{4 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right ) \sqrt{\sinh (a+b x)}}{3 b^2 \sqrt{i \sinh (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.203667, size = 66, normalized size = 0.67 \[ -\frac{2 \left (\sinh (2 (a+b x))+2 i (i \sinh (a+b x))^{3/2} E\left (\left .\frac{1}{4} (-2 i a-2 i b x+\pi )\right |2\right )+b x\right )}{3 b^2 \sinh ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{x\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{-{\frac{5}{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 )}{\sinh \left (b x + a\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(-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 )}{\sinh \left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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