Optimal. Leaf size=112 \[ \frac{2 (a \sinh (x)+b \cosh (x))}{\left (a^2-b^2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}+\frac{2 i \sqrt{a \cosh (x)+b \sinh (x)} E\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\left (a^2-b^2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}} \]
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Rubi [A] time = 0.0505367, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3076, 3078, 2639} \[ \frac{2 (a \sinh (x)+b \cosh (x))}{\left (a^2-b^2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}+\frac{2 i \sqrt{a \cosh (x)+b \sinh (x)} E\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\left (a^2-b^2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}} \]
Antiderivative was successfully verified.
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Rule 3076
Rule 3078
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx &=\frac{2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}+\frac{\int \sqrt{a \cosh (x)+b \sinh (x)} \, dx}{-a^2+b^2}\\ &=\frac{2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}+\frac{\sqrt{a \cosh (x)+b \sinh (x)} \int \sqrt{\cosh \left (x+i \tan ^{-1}(a,-i b)\right )} \, dx}{\left (-a^2+b^2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}}\\ &=\frac{2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}+\frac{2 i E\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt{a \cosh (x)+b \sinh (x)}}{\left (a^2-b^2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}}\\ \end{align*}
Mathematica [C] time = 0.446826, size = 148, normalized size = 1.32 \[ \frac{b \sinh \left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right ) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cosh ^2\left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )\right )-\sqrt{-\sinh ^2\left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )} \left (2 a \sqrt{1-\frac{b^2}{a^2}} \cosh (x)+b \sinh \left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )-2 a \cosh \left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )\right )}{a b \sqrt{1-\frac{b^2}{a^2}} \sqrt{-\sinh ^2\left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )} \sqrt{a \cosh (x)+b \sinh (x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.12, size = 33, normalized size = 0.3 \begin{align*}{{\it Artanh} \left ( \cosh \left ( x \right ) \right ){\frac{1}{\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}{{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac{3}{2}}}\,{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{\sqrt{a \cosh \left (x\right ) + b \sinh \left (x\right )}}{a^{2} \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2}}, 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}{\left (a \cosh{\left (x \right )} + b \sinh{\left (x \right )}\right )^{\frac{3}{2}}}\, 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}{{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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