Optimal. Leaf size=60 \[ \frac{2 i \sqrt{\frac{a+b \sinh (x)}{a-i b}} \text{EllipticF}\left (\frac{\pi }{4}-\frac{i x}{2},\frac{2 b}{b+i a}\right )}{\sqrt{a+b \sinh (x)}} \]
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Rubi [A] time = 0.0368365, antiderivative size = 60, 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 = {2663, 2661} \[ \frac{2 i \sqrt{\frac{a+b \sinh (x)}{a-i b}} F\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right )}{\sqrt{a+b \sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 2663
Rule 2661
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \sinh (x)}} \, dx &=\frac{\sqrt{\frac{a+b \sinh (x)}{a-i b}} \int \frac{1}{\sqrt{\frac{a}{a-i b}+\frac{b \sinh (x)}{a-i b}}} \, dx}{\sqrt{a+b \sinh (x)}}\\ &=\frac{2 i F\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right ) \sqrt{\frac{a+b \sinh (x)}{a-i b}}}{\sqrt{a+b \sinh (x)}}\\ \end{align*}
Mathematica [A] time = 0.185854, size = 60, normalized size = 1. \[ \frac{2 i \sqrt{\frac{a+b \sinh (x)}{a-i b}} \text{EllipticF}\left (\frac{1}{4} (\pi -2 i x),-\frac{2 i b}{a-i b}\right )}{\sqrt{a+b \sinh (x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 125, normalized size = 2.1 \begin{align*} -2\,{\frac{ib-a}{b\cosh \left ( x \right ) \sqrt{a+b\sinh \left ( x \right ) }}\sqrt{-{\frac{a+b\sinh \left ( x \right ) }{ib-a}}}\sqrt{{\frac{ \left ( i-\sinh \left ( x \right ) \right ) b}{ib+a}}}\sqrt{{\frac{ \left ( i+\sinh \left ( x \right ) \right ) b}{ib-a}}}{\it EllipticF} \left ( \sqrt{-{\frac{a+b\sinh \left ( x \right ) }{ib-a}}},\sqrt{-{\frac{ib-a}{ib+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{b \sinh \left (x\right ) + a}}\,{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{b \sinh \left (x\right ) + a}}, 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{a + b \sinh{\left (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{b \sinh \left (x\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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