Integrand size = 18, antiderivative size = 84 \[ \int x \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {4 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{3 b^2}-\frac {2 x \text {sech}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 \sqrt {\text {sech}(a+b x)} \sinh (a+b x)}{3 b^2} \]
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Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5552, 3853, 3856, 2719} \[ \int x \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {4 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{3 b^2}+\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b^2}-\frac {2 x \text {sech}^{\frac {3}{2}}(a+b x)}{3 b} \]
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Rule 2719
Rule 3853
Rule 3856
Rule 5552
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.68 \[ \int x \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {2 \text {sech}^{\frac {3}{2}}(a+b x) \left (-b x+2 i \cosh ^{\frac {3}{2}}(a+b x) E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )+\sinh (2 (a+b x))\right )}{3 b^2} \]
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\[\int x \operatorname {sech}\left (b x +a \right )^{\frac {5}{2}} \sinh \left (b x +a \right )d x\]
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Exception generated. \[ \int x \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\text {Timed out} \]
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\[ \int x \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\int { x \operatorname {sech}\left (b x + a\right )^{\frac {5}{2}} \sinh \left (b x + a\right ) \,d x } \]
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\[ \int x \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\int { x \operatorname {sech}\left (b x + a\right )^{\frac {5}{2}} \sinh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\int x\,\mathrm {sinh}\left (a+b\,x\right )\,{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{5/2} \,d x \]
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