Integrand size = 18, antiderivative size = 64 \[ \int x \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {2 x \cosh ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {12 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{25 b^2}-\frac {4 \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{25 b^2} \]
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Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5481, 2715, 2719} \[ \int x \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {12 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{25 b^2}-\frac {4 \sinh (a+b x) \cosh ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \cosh ^{\frac {5}{2}}(a+b x)}{5 b} \]
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Rule 2715
Rule 2719
Rule 5481
Rubi steps \begin{align*} \text {integral}& = \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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 2.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.22 \[ \int x \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {e^{-3 (a+b x)} \left (\left (1+e^{2 (a+b x)}\right ) \left (2+5 b x+2 e^{2 (a+b x)} (-12+5 b x)+e^{4 (a+b x)} (-2+5 b x)\right )+48 e^{2 (a+b x)} \sqrt {1+e^{2 (a+b x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 (a+b x)}\right )\right )}{50 \sqrt {2} b^2 \sqrt {e^{-a-b x}+e^{a+b x}}} \]
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\[\int x \cosh \left (b x +a \right )^{\frac {3}{2}} \sinh \left (b x +a \right )d x\]
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Exception generated. \[ \int x \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\text {Timed out} \]
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\[ \int x \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\int { x \cosh \left (b x + a\right )^{\frac {3}{2}} \sinh \left (b x + a\right ) \,d x } \]
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\[ \int x \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\int { x \cosh \left (b x + a\right )^{\frac {3}{2}} \sinh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x) \, dx=\int x\,{\mathrm {cosh}\left (a+b\,x\right )}^{3/2}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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