Integrand size = 18, antiderivative size = 71 \[ \int x \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 x \sqrt {\text {csch}(a+b x)}}{b}-\frac {4 i \sqrt {\text {csch}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{b^2} \]
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Time = 0.03 (sec) , antiderivative size = 71, 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 = {5553, 3856, 2720} \[ \int x \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 x \sqrt {\text {csch}(a+b x)}}{b}-\frac {4 i \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right ),2\right )}{b^2} \]
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Rule 2720
Rule 3856
Rule 5553
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \sqrt {\text {csch}(a+b x)}}{b}+\frac {2 \int \sqrt {\text {csch}(a+b x)} \, dx}{b} \\ & = -\frac {2 x \sqrt {\text {csch}(a+b x)}}{b}+\frac {\left (2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}\right ) \int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx}{b} \\ & = -\frac {2 x \sqrt {\text {csch}(a+b x)}}{b}-\frac {4 i \sqrt {\text {csch}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{b^2} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.79 \[ \int x \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \sqrt {\text {csch}(a+b x)} \left (b x-2 i \operatorname {EllipticF}\left (\frac {1}{4} (-2 i a+\pi -2 i b x),2\right ) \sqrt {i \sinh (a+b x)}\right )}{b^2} \]
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\[\int x \cosh \left (b x +a \right ) \operatorname {csch}\left (b x +a \right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int x \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x) \, dx=\text {Timed out} \]
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\[ \int x \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{\frac {3}{2}} \,d x } \]
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\[ \int x \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int x \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x) \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{3/2} \,d x \]
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