Integrand size = 10, antiderivative size = 54 \[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \]
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Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3856, 2719} \[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}} \]
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Rule 2719
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sqrt {i \sinh (a+b x)} \, dx}{\sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \\ & = -\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=\frac {2 \sqrt {\text {csch}(a+b x)} E\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-i (a+b x)\right )\right |2\right ) \sqrt {i \sinh (a+b x)}}{b} \]
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Time = 0.47 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.00
method | result | size |
default | \(\frac {\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b}\) | \(108\) |
risch | \(\frac {\sqrt {2}}{b \sqrt {\frac {{\mathrm e}^{b x +a}}{{\mathrm e}^{2 b x +2 a}-1}}}-\frac {\left (\frac {2 \,{\mathrm e}^{2 b x +2 a}-2}{\sqrt {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}}-\frac {\sqrt {{\mathrm e}^{b x +a}+1}\, \sqrt {-2 \,{\mathrm e}^{b x +a}+2}\, \sqrt {-{\mathrm e}^{b x +a}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}-{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}}{b \sqrt {\frac {{\mathrm e}^{b x +a}}{{\mathrm e}^{2 b x +2 a}-1}}\, \left ({\mathrm e}^{2 b x +2 a}-1\right )}\) | \(210\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.78 \[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=-\frac {\sqrt {2} {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}} + 2 \, {\left (\sqrt {2} \cosh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )}{b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )} \]
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\[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=\int \frac {1}{\sqrt {\operatorname {csch}{\left (a + b x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=\int { \frac {1}{\sqrt {\operatorname {csch}\left (b x + a\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=\int { \frac {1}{\sqrt {\operatorname {csch}\left (b x + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=\int \frac {1}{\sqrt {\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}}} \,d x \]
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