Integrand size = 18, antiderivative size = 121 \[ \int \frac {x \cosh (a+b x)}{\sinh ^{\frac {9}{2}}(a+b x)} \, dx=-\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}+\frac {12 \cosh (a+b x)}{35 b^2 \sqrt {\sinh (a+b x)}}+\frac {12 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{35 b^2 \sqrt {i \sinh (a+b x)}} \]
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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 = {5480, 2716, 2721, 2719} \[ \int \frac {x \cosh (a+b x)}{\sinh ^{\frac {9}{2}}(a+b x)} \, dx=-\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}+\frac {12 \cosh (a+b x)}{35 b^2 \sqrt {\sinh (a+b x)}}+\frac {12 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{35 b^2 \sqrt {i \sinh (a+b x)}}-\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)} \]
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Rule 2716
Rule 2719
Rule 2721
Rule 5480
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}+\frac {2 \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx}{7 b} \\ & = -\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}-\frac {6 \int \frac {1}{\sinh ^{\frac {3}{2}}(a+b x)} \, dx}{35 b} \\ & = -\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}+\frac {12 \cosh (a+b x)}{35 b^2 \sqrt {\sinh (a+b x)}}-\frac {6 \int \sqrt {\sinh (a+b x)} \, dx}{35 b} \\ & = -\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}+\frac {12 \cosh (a+b x)}{35 b^2 \sqrt {\sinh (a+b x)}}-\frac {\left (6 \sqrt {\sinh (a+b x)}\right ) \int \sqrt {i \sinh (a+b x)} \, dx}{35 b \sqrt {i \sinh (a+b x)}} \\ & = -\frac {2 x}{7 b \sinh ^{\frac {7}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{35 b^2 \sinh ^{\frac {5}{2}}(a+b x)}+\frac {12 \cosh (a+b x)}{35 b^2 \sqrt {\sinh (a+b x)}}+\frac {12 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{35 b^2 \sqrt {i \sinh (a+b x)}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.74 \[ \int \frac {x \cosh (a+b x)}{\sinh ^{\frac {9}{2}}(a+b x)} \, dx=-\frac {2 \left (5 b x-6 \cosh (a+b x) \sinh ^3(a+b x)+6 E\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) \sqrt {i \sinh (a+b x)} \sinh ^3(a+b x)+\sinh (2 (a+b x))\right )}{35 b^2 \sinh ^{\frac {7}{2}}(a+b x)} \]
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\[\int \frac {x \cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{\frac {9}{2}}}d x\]
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Exception generated. \[ \int \frac {x \cosh (a+b x)}{\sinh ^{\frac {9}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x \cosh (a+b x)}{\sinh ^{\frac {9}{2}}(a+b x)} \, dx=\text {Timed out} \]
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\[ \int \frac {x \cosh (a+b x)}{\sinh ^{\frac {9}{2}}(a+b x)} \, dx=\int { \frac {x \cosh \left (b x + a\right )}{\sinh \left (b x + a\right )^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {x \cosh (a+b x)}{\sinh ^{\frac {9}{2}}(a+b x)} \, dx=\int { \frac {x \cosh \left (b x + a\right )}{\sinh \left (b x + a\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \cosh (a+b x)}{\sinh ^{\frac {9}{2}}(a+b x)} \, dx=\int \frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{{\mathrm {sinh}\left (a+b\,x\right )}^{9/2}} \,d x \]
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