Integrand size = 18, antiderivative size = 121 \[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\frac {12 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{35 b^2}-\frac {4 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {12 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{35 b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \]
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Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5553, 3853, 3856, 2719} \[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=-\frac {4 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{35 b^2}+\frac {12 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{35 b^2}+\frac {12 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{35 b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}-\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b} \]
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Rule 2719
Rule 3853
Rule 3856
Rule 5553
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \int \text {csch}^{\frac {7}{2}}(a+b x) \, dx}{7 b} \\ & = -\frac {4 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}-\frac {6 \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx}{35 b} \\ & = \frac {12 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{35 b^2}-\frac {4 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}-\frac {6 \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx}{35 b} \\ & = \frac {12 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{35 b^2}-\frac {4 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}-\frac {6 \int \sqrt {i \sinh (a+b x)} \, dx}{35 b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \\ & = \frac {12 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{35 b^2}-\frac {4 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {12 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{35 b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.69 \[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=-\frac {2 \sqrt {\text {csch}(a+b x)} \left (-6 \cosh (a+b x)+6 E\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) \sqrt {i \sinh (a+b x)}+\text {csch}^3(a+b x) (5 b x+\sinh (2 (a+b x)))\right )}{35 b^2} \]
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\[\int x \cosh \left (b x +a \right ) \operatorname {csch}\left (b x +a \right )^{\frac {9}{2}}d x\]
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Exception generated. \[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\text {Timed out} \]
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\[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{\frac {9}{2}} \,d x } \]
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\[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{\frac {9}{2}} \,d x } \]
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Timed out. \[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{9/2} \,d x \]
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