Optimal. Leaf size=87 \[ \frac{12 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{35 b^2}+\frac{4 \sinh (a+b x)}{35 b^2 \cosh ^{\frac{5}{2}}(a+b x)}+\frac{12 \sinh (a+b x)}{35 b^2 \sqrt{\cosh (a+b x)}}-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)} \]
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Rubi [A] time = 0.0534982, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5373, 2636, 2639} \[ \frac{12 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{35 b^2}+\frac{4 \sinh (a+b x)}{35 b^2 \cosh ^{\frac{5}{2}}(a+b x)}+\frac{12 \sinh (a+b x)}{35 b^2 \sqrt{\cosh (a+b x)}}-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 5373
Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{x \sinh (a+b x)}{\cosh ^{\frac{9}{2}}(a+b x)} \, dx &=-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)}+\frac{2 \int \frac{1}{\cosh ^{\frac{7}{2}}(a+b x)} \, dx}{7 b}\\ &=-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)}+\frac{4 \sinh (a+b x)}{35 b^2 \cosh ^{\frac{5}{2}}(a+b x)}+\frac{6 \int \frac{1}{\cosh ^{\frac{3}{2}}(a+b x)} \, dx}{35 b}\\ &=-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)}+\frac{4 \sinh (a+b x)}{35 b^2 \cosh ^{\frac{5}{2}}(a+b x)}+\frac{12 \sinh (a+b x)}{35 b^2 \sqrt{\cosh (a+b x)}}-\frac{6 \int \sqrt{\cosh (a+b x)} \, dx}{35 b}\\ &=-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)}+\frac{12 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{35 b^2}+\frac{4 \sinh (a+b x)}{35 b^2 \cosh ^{\frac{5}{2}}(a+b x)}+\frac{12 \sinh (a+b x)}{35 b^2 \sqrt{\cosh (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.279004, size = 69, normalized size = 0.79 \[ \frac{10 \sinh (2 (a+b x))+3 \sinh (4 (a+b x))+24 i \cosh ^{\frac{7}{2}}(a+b x) E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )-20 b x}{70 b^2 \cosh ^{\frac{7}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{x\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{-{\frac{9}{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 )}{\cosh \left (b x + a\right )^{\frac{9}{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 )}{\cosh \left (b x + a\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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