Optimal. Leaf size=107 \[ \frac{20 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{147 b^2}-\frac{4 \sinh (a+b x)}{49 b^2 \text{sech}^{\frac{5}{2}}(a+b x)}-\frac{20 \sinh (a+b x)}{147 b^2 \sqrt{\text{sech}(a+b x)}}+\frac{2 x}{7 b \text{sech}^{\frac{7}{2}}(a+b x)} \]
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Rubi [A] time = 0.069102, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5444, 3769, 3771, 2641} \[ -\frac{4 \sinh (a+b x)}{49 b^2 \text{sech}^{\frac{5}{2}}(a+b x)}-\frac{20 \sinh (a+b x)}{147 b^2 \sqrt{\text{sech}(a+b x)}}+\frac{20 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{147 b^2}+\frac{2 x}{7 b \text{sech}^{\frac{7}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 5444
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{x \sinh (a+b x)}{\text{sech}^{\frac{5}{2}}(a+b x)} \, dx &=\frac{2 x}{7 b \text{sech}^{\frac{7}{2}}(a+b x)}-\frac{2 \int \frac{1}{\text{sech}^{\frac{7}{2}}(a+b x)} \, dx}{7 b}\\ &=\frac{2 x}{7 b \text{sech}^{\frac{7}{2}}(a+b x)}-\frac{4 \sinh (a+b x)}{49 b^2 \text{sech}^{\frac{5}{2}}(a+b x)}-\frac{10 \int \frac{1}{\text{sech}^{\frac{3}{2}}(a+b x)} \, dx}{49 b}\\ &=\frac{2 x}{7 b \text{sech}^{\frac{7}{2}}(a+b x)}-\frac{4 \sinh (a+b x)}{49 b^2 \text{sech}^{\frac{5}{2}}(a+b x)}-\frac{20 \sinh (a+b x)}{147 b^2 \sqrt{\text{sech}(a+b x)}}-\frac{10 \int \sqrt{\text{sech}(a+b x)} \, dx}{147 b}\\ &=\frac{2 x}{7 b \text{sech}^{\frac{7}{2}}(a+b x)}-\frac{4 \sinh (a+b x)}{49 b^2 \text{sech}^{\frac{5}{2}}(a+b x)}-\frac{20 \sinh (a+b x)}{147 b^2 \sqrt{\text{sech}(a+b x)}}-\frac{\left (10 \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)}\right ) \int \frac{1}{\sqrt{\cosh (a+b x)}} \, dx}{147 b}\\ &=\frac{2 x}{7 b \text{sech}^{\frac{7}{2}}(a+b x)}+\frac{20 i \sqrt{\cosh (a+b x)} F\left (\left .\frac{1}{2} i (a+b x)\right |2\right ) \sqrt{\text{sech}(a+b x)}}{147 b^2}-\frac{4 \sinh (a+b x)}{49 b^2 \text{sech}^{\frac{5}{2}}(a+b x)}-\frac{20 \sinh (a+b x)}{147 b^2 \sqrt{\text{sech}(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.273311, size = 93, normalized size = 0.87 \[ \frac{\sqrt{\text{sech}(a+b x)} \left (80 i \sqrt{\cosh (a+b x)} \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )-52 \sinh (2 (a+b x))-6 \sinh (4 (a+b x))+84 b x \cosh (2 (a+b x))+21 b x \cosh (4 (a+b x))+63 b x\right )}{588 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.032, size = 0, normalized size = 0. \begin{align*} \int{x\sinh \left ( bx+a \right ) \left ({\rm sech} \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 \sinh \left (b x + a\right )}{\operatorname{sech}\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 \sinh \left (b x + a\right )}{\operatorname{sech}\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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