Integrand size = 18, antiderivative size = 71 \[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\frac {2 x}{b \sqrt {\text {csch}(a+b x)}}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \]
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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, 2719} \[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\frac {2 x}{b \sqrt {\text {csch}(a+b x)}}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}} \]
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Rule 2719
Rule 3856
Rule 5553
Rubi steps \begin{align*} \text {integral}& = \frac {2 x}{b \sqrt {\text {csch}(a+b x)}}-\frac {2 \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx}{b} \\ & = \frac {2 x}{b \sqrt {\text {csch}(a+b x)}}-\frac {2 \int \sqrt {i \sinh (a+b x)} \, dx}{b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \\ & = \frac {2 x}{b \sqrt {\text {csch}(a+b x)}}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{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 = 0.90 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.41 \[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\frac {\sqrt {2} e^{-a-b x} \sqrt {\frac {e^{a+b x}}{-1+e^{2 (a+b x)}}} \left (\left (-1+e^{2 (a+b x)}\right ) (-2+b x)-4 \sqrt {1-e^{2 (a+b x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},e^{2 (a+b x)}\right )\right )}{b^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(228\) vs. \(2(92)=184\).
Time = 0.28 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.23
method | result | size |
risch | \(\frac {\left (b x -2\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{{\mathrm e}^{2 b x +2 a}-1}}\, {\mathrm e}^{-b x -a}}{b^{2}}+\frac {2 \left (\frac {2 \,{\mathrm e}^{2 b x +2 a}-2}{\sqrt {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}}-\frac {\sqrt {{\mathrm e}^{b x +a}+1}\, \sqrt {-2 \,{\mathrm e}^{b x +a}+2}\, \sqrt {-{\mathrm e}^{b x +a}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}-{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{{\mathrm e}^{2 b x +2 a}-1}}\, \sqrt {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}\, {\mathrm e}^{-b x -a}}{b^{2}}\) | \(229\) |
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Exception generated. \[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\int x \cosh {\left (a + b x \right )} \sqrt {\operatorname {csch}{\left (a + b x \right )}}\, dx \]
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\[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\int { x \cosh \left (b x + a\right ) \sqrt {\operatorname {csch}\left (b x + a\right )} \,d x } \]
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\[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\int { x \cosh \left (b x + a\right ) \sqrt {\operatorname {csch}\left (b x + a\right )} \,d x } \]
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Timed out. \[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,\sqrt {\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}} \,d x \]
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