Optimal. Leaf size=54 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b}-\frac{\tan ^{-1}\left (\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b} \]
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Rubi [A] time = 0.0381403, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2575, 298, 203, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b}-\frac{\tan ^{-1}\left (\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2575
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{1-x^4} \, dx,x,\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b}\\ \end{align*}
Mathematica [C] time = 0.0231303, size = 57, normalized size = 1.06 \[ \frac{2 \sqrt{\sinh (a+b x)} \sqrt [4]{\cosh ^2(a+b x)} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\sinh ^2(a+b x)\right )}{b \sqrt{\cosh (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.09, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{\cosh \left ( bx+a \right ) }{\frac{1}{\sqrt{\sinh \left ( bx+a \right ) }}}}\, dx \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{\sqrt{\cosh \left (b x + a\right )}}{\sqrt{\sinh \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.47819, size = 421, normalized size = 7.8 \begin{align*} \frac{2 \, \arctan \left (-\cosh \left (b x + a\right )^{2} + 2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt{\cosh \left (b x + a\right )} \sqrt{\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) - \log \left (-\cosh \left (b x + a\right )^{2} + 2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt{\cosh \left (b x + a\right )} \sqrt{\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right )}{2 \, b} \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{\sqrt{\cosh{\left (a + b x \right )}}}{\sqrt{\sinh{\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 \frac{\sqrt{\cosh \left (b x + a\right )}}{\sqrt{\sinh \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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