Integrand size = 10, antiderivative size = 80 \[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 \cosh (a+b x)}{5 b \text {csch}^{\frac {3}{2}}(a+b x)}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{5 b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \]
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Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3854, 3856, 2719} \[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 \cosh (a+b x)}{5 b \text {csch}^{\frac {3}{2}}(a+b x)}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{5 b \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}} \]
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Rule 2719
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cosh (a+b x)}{5 b \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {3}{5} \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx \\ & = \frac {2 \cosh (a+b x)}{5 b \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {3 \int \sqrt {i \sinh (a+b x)} \, dx}{5 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \\ & = \frac {2 \cosh (a+b x)}{5 b \text {csch}^{\frac {3}{2}}(a+b x)}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{5 b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 \left (\cosh (a+b x)-3 \text {csch}^2(a+b x) E\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) \sqrt {i \sinh (a+b x)}\right )}{5 b \text {csch}^{\frac {3}{2}}(a+b x)} \]
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Time = 0.40 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.05
method | result | size |
default | \(\frac {-\frac {6 \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (b x +a \right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (b x +a \right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 \cosh \left (b x +a \right )^{4}}{5}-\frac {2 \cosh \left (b x +a \right )^{2}}{5}}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b}\) | \(164\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 370, normalized size of antiderivative = 4.62 \[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\frac {\sqrt {2} {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + {\left (15 \, \cosh \left (b x + a\right )^{2} + 11\right )} \sinh \left (b x + a\right )^{4} + 11 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 11 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (15 \, \cosh \left (b x + a\right )^{4} + 66 \, \cosh \left (b x + a\right )^{2} - 13\right )} \sinh \left (b x + a\right )^{2} - 13 \, \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{5} + 22 \, \cosh \left (b x + a\right )^{3} - 13 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}} + 24 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{3} + 3 \, \sqrt {2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sqrt {2} \sinh \left (b x + a\right )^{3}\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )}{20 \, {\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\right )}} \]
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\[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {1}{\operatorname {csch}^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {1}{{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{5/2}} \,d x \]
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