Optimal. Leaf size=64 \[ \frac{12 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{25 b^2}-\frac{4 \sinh (a+b x) \cosh ^{\frac{3}{2}}(a+b x)}{25 b^2}+\frac{2 x \cosh ^{\frac{5}{2}}(a+b x)}{5 b} \]
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Rubi [A] time = 0.0426493, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5373, 2635, 2639} \[ \frac{12 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{25 b^2}-\frac{4 \sinh (a+b x) \cosh ^{\frac{3}{2}}(a+b x)}{25 b^2}+\frac{2 x \cosh ^{\frac{5}{2}}(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 5373
Rule 2635
Rule 2639
Rubi steps
\begin{align*} \int x \cosh ^{\frac{3}{2}}(a+b x) \sinh (a+b x) \, dx &=\frac{2 x \cosh ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{2 \int \cosh ^{\frac{5}{2}}(a+b x) \, dx}{5 b}\\ &=\frac{2 x \cosh ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{4 \cosh ^{\frac{3}{2}}(a+b x) \sinh (a+b x)}{25 b^2}-\frac{6 \int \sqrt{\cosh (a+b x)} \, dx}{25 b}\\ &=\frac{2 x \cosh ^{\frac{5}{2}}(a+b x)}{5 b}+\frac{12 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{25 b^2}-\frac{4 \cosh ^{\frac{3}{2}}(a+b x) \sinh (a+b x)}{25 b^2}\\ \end{align*}
Mathematica [C] time = 1.84358, size = 142, normalized size = 2.22 \[ \frac{e^{-3 (a+b x)} \left (48 e^{2 (a+b x)} \sqrt{e^{2 (a+b x)}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 (a+b x)}\right )+\left (e^{2 (a+b x)}+1\right ) \left (2 (5 b x-12) e^{2 (a+b x)}+(5 b x-2) e^{4 (a+b x)}+5 b x+2\right )\right )}{50 \sqrt{2} b^2 \sqrt{e^{-a-b x}+e^{a+b x}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int x \left ( \cosh \left ( bx+a \right ) \right ) ^{{\frac{3}{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 \cosh \left (b x + a\right )^{\frac{3}{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 \cosh \left (b x + a\right )^{\frac{3}{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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