Optimal. Leaf size=65 \[ -\frac{2 i \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}} \text{EllipticF}\left (\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right )}{\sqrt{a \cosh (x)+b \sinh (x)}} \]
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Rubi [A] time = 0.0268969, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3078, 2641} \[ -\frac{2 i \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}} F\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt{a \cosh (x)+b \sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 3078
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \cosh (x)+b \sinh (x)}} \, dx &=\frac{\sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}} \int \frac{1}{\sqrt{\cosh \left (x+i \tan ^{-1}(a,-i b)\right )}} \, dx}{\sqrt{a \cosh (x)+b \sinh (x)}}\\ &=-\frac{2 i F\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}}{\sqrt{a \cosh (x)+b \sinh (x)}}\\ \end{align*}
Mathematica [C] time = 0.0926322, size = 81, normalized size = 1.25 \[ \frac{2 \sqrt{a \cosh (x)+b \sinh (x)} \sqrt{\cosh ^2\left (\tanh ^{-1}\left (\frac{a}{b}\right )+x\right )} \text{sech}\left (\tanh ^{-1}\left (\frac{a}{b}\right )+x\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},-\sinh ^2\left (\tanh ^{-1}\left (\frac{a}{b}\right )+x\right )\right )}{b \sqrt{1-\frac{a^2}{b^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.19, size = 97, normalized size = 1.5 \begin{align*}{\frac{1}{\sinh \left ( x \right ) }\sqrt{-\sqrt{{a}^{2}-{b}^{2}} \left ( \sinh \left ( x \right ) \right ) ^{3}}\arctan \left ({\cosh \left ( x \right ) \sqrt{\sinh \left ( x \right ) \sqrt{{a}^{2}-{b}^{2}}}{\frac{1}{\sqrt{-\sqrt{{a}^{2}-{b}^{2}} \left ( \sinh \left ( x \right ) \right ) ^{3}}}}} \right ){\frac{1}{\sqrt{\sinh \left ( x \right ) \sqrt{{a}^{2}-{b}^{2}}}}}{\frac{1}{\sqrt{-\sinh \left ( x \right ) \sqrt{{a}^{2}-{b}^{2}}}}}} \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{a \cosh \left (x\right ) + b \sinh \left (x\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{a \cosh \left (x\right ) + b \sinh \left (x\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{a \cosh{\left (x \right )} + 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{a \cosh \left (x\right ) + b \sinh \left (x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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