Optimal. Leaf size=54 \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}} \]
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Rubi [A] time = 0.0201483, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3771, 2639} \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\text{csch}(a+b x)}} \, dx &=\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*}
Mathematica [A] time = 0.0339411, size = 50, normalized size = 0.93 \[ \frac{2 \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)} E\left (\left .\frac{1}{2} \left (\frac{\pi }{2}-i (a+b x)\right )\right |2\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.152, size = 108, normalized size = 2. \begin{align*}{\frac{\sqrt{2}}{\cosh \left ( bx+a \right ) b}\sqrt{-i \left ( i+\sinh \left ( bx+a \right ) \right ) }\sqrt{-i \left ( -\sinh \left ( bx+a \right ) +i \right ) }\sqrt{i\sinh \left ( bx+a \right ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{\sinh \left ( bx+a \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\operatorname{csch}\left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{\operatorname{csch}\left (b x + a\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\operatorname{csch}{\left (a + b x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\operatorname{csch}\left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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