Optimal. Leaf size=46 \[ -\frac{2 i \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )}{\sqrt{a+b \cosh (x)}} \]
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Rubi [A] time = 0.0341934, antiderivative size = 46, 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} \[ -\frac{2 i \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{\sqrt{a+b \cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 2663
Rule 2661
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \cosh (x)}} \, dx &=\frac{\sqrt{\frac{a+b \cosh (x)}{a+b}} \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}}} \, dx}{\sqrt{a+b \cosh (x)}}\\ &=-\frac{2 i \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{\sqrt{a+b \cosh (x)}}\\ \end{align*}
Mathematica [A] time = 0.036856, size = 46, normalized size = 1. \[ -\frac{2 i \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )}{\sqrt{a+b \cosh (x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 146, normalized size = 3.2 \begin{align*} 2\,{\frac{\sqrt{ \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}b+a-b \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}}{\sqrt{2\,b \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}\sinh \left ( x/2 \right ) \sqrt{2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}b+a+b}}\sqrt{{\frac{2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}b+a-b}{a-b}}}{\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ){\frac{1}{\sqrt{-2\,{\frac{b}{a-b}}}}}} \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 \cosh \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 \cosh \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 \cosh{\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 \cosh \left (x\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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