Optimal. Leaf size=54 \[ -\frac{2 i \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{b} \]
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Rubi [A] time = 0.0197578, 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, 2641} \[ -\frac{2 i \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b} \]
Antiderivative was successfully verified.
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Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\text{csch}(a+b x)} \, dx &=\left (\sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}\right ) \int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx\\ &=-\frac{2 i \sqrt{\text{csch}(a+b x)} F\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right ) \sqrt{i \sinh (a+b x)}}{b}\\ \end{align*}
Mathematica [A] time = 0.187321, size = 48, normalized size = 0.89 \[ \frac{2 (i \sinh (a+b x))^{3/2} \text{csch}^{\frac{3}{2}}(a+b x) \text{EllipticF}\left (\frac{1}{4} (-2 i a-2 i b x+\pi ),2\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.204, size = 87, normalized size = 1.6 \begin{align*}{\frac{i\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 ) }{\it EllipticF} \left ( \sqrt{-i \left ( i+\sinh \left ( bx+a \right ) \right ) },{\frac{\sqrt{2}}{2}} \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 \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 (\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 \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 \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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