Optimal. Leaf size=84 \[ \frac{4 \sinh (a+b x) \sqrt{\text{sech}(a+b x)}}{3 b^2}+\frac{4 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{3 b^2}-\frac{2 x \text{sech}^{\frac{3}{2}}(a+b x)}{3 b} \]
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Rubi [A] time = 0.0517026, antiderivative size = 84, 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 = {5444, 3768, 3771, 2639} \[ \frac{4 \sinh (a+b x) \sqrt{\text{sech}(a+b x)}}{3 b^2}+\frac{4 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{3 b^2}-\frac{2 x \text{sech}^{\frac{3}{2}}(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 5444
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int x \text{sech}^{\frac{5}{2}}(a+b x) \sinh (a+b x) \, dx &=-\frac{2 x \text{sech}^{\frac{3}{2}}(a+b x)}{3 b}+\frac{2 \int \text{sech}^{\frac{3}{2}}(a+b x) \, dx}{3 b}\\ &=-\frac{2 x \text{sech}^{\frac{3}{2}}(a+b x)}{3 b}+\frac{4 \sqrt{\text{sech}(a+b x)} \sinh (a+b x)}{3 b^2}-\frac{2 \int \frac{1}{\sqrt{\text{sech}(a+b x)}} \, dx}{3 b}\\ &=-\frac{2 x \text{sech}^{\frac{3}{2}}(a+b x)}{3 b}+\frac{4 \sqrt{\text{sech}(a+b x)} \sinh (a+b x)}{3 b^2}-\frac{\left (2 \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)}\right ) \int \sqrt{\cosh (a+b x)} \, dx}{3 b}\\ &=\frac{4 i \sqrt{\cosh (a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right ) \sqrt{\text{sech}(a+b x)}}{3 b^2}-\frac{2 x \text{sech}^{\frac{3}{2}}(a+b x)}{3 b}+\frac{4 \sqrt{\text{sech}(a+b x)} \sinh (a+b x)}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.195932, size = 57, normalized size = 0.68 \[ \frac{2 \text{sech}^{\frac{3}{2}}(a+b x) \left (\sinh (2 (a+b x))+2 i \cosh ^{\frac{3}{2}}(a+b x) E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )-b x\right )}{3 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.034, size = 0, normalized size = 0. \begin{align*} \int x \left ({\rm sech} \left (bx+a\right ) \right ) ^{{\frac{5}{2}}}\sinh \left ( bx+a \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 \operatorname{sech}\left (b x + a\right )^{\frac{5}{2}} \sinh \left (b x + a\right )\,{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 x \operatorname{sech}\left (b x + a\right )^{\frac{5}{2}} \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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