Integrand size = 18, antiderivative size = 98 \[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {12 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{25 b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \]
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Time = 0.04 (sec) , antiderivative size = 98, 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 = {5553, 3854, 3856, 2719} \[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=-\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {12 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}+\frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)} \]
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Rule 2719
Rule 3854
Rule 3856
Rule 5553
Rubi steps \begin{align*} \text {integral}& = \frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)}-\frac {2 \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx}{5 b} \\ & = \frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(a+b x)}+\frac {6 \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx}{25 b} \\ & = \frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(a+b x)}+\frac {6 \int \sqrt {i \sinh (a+b x)} \, dx}{25 b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \\ & = \frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(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 )}{25 b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.13 \[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\frac {e^{-2 (a+b x)} \left (2+5 b x+e^{2 (a+b x)} (24-10 b x)+e^{4 (a+b x)} (-2+5 b x)-\frac {48 e^{2 (a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},e^{2 (a+b x)}\right )}{\sqrt {1-e^{2 (a+b x)}}}\right )}{50 b^2 \sqrt {\text {csch}(a+b x)}} \]
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\[\int \frac {x \cosh \left (b x +a \right )}{\operatorname {csch}\left (b x +a \right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x \cosh {\left (a + b x \right )}}{\operatorname {csch}^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {x \cosh \left (b x + a\right )}{\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {x \cosh \left (b x + a\right )}{\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{3/2}} \,d x \]
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