Optimal. Leaf size=79 \[ -\frac{2 \sqrt{\sinh (a+b x)}}{b \sqrt{\cosh (a+b x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b} \]
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Rubi [A] time = 0.0733108, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2566, 2575, 298, 203, 206} \[ -\frac{2 \sqrt{\sinh (a+b x)}}{b \sqrt{\cosh (a+b x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2566
Rule 2575
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\sinh ^{\frac{3}{2}}(a+b x)}{\cosh ^{\frac{3}{2}}(a+b x)} \, dx &=-\frac{2 \sqrt{\sinh (a+b x)}}{b \sqrt{\cosh (a+b x)}}+\int \frac{\sqrt{\cosh (a+b x)}}{\sqrt{\sinh (a+b x)}} \, dx\\ &=-\frac{2 \sqrt{\sinh (a+b x)}}{b \sqrt{\cosh (a+b x)}}+\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{2 \sqrt{\sinh (a+b x)}}{b \sqrt{\cosh (a+b x)}}+\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}-\frac{2 \sqrt{\sinh (a+b x)}}{b \sqrt{\cosh (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.0512258, size = 59, normalized size = 0.75 \[ \frac{2 \sinh ^{\frac{5}{2}}(a+b x) \sqrt [4]{\cosh ^2(a+b x)} \, _2F_1\left (\frac{5}{4},\frac{5}{4};\frac{9}{4};-\sinh ^2(a+b x)\right )}{5 b \sqrt{\cosh (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sinh \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}} \left ( \cosh \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}}\, 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{\sinh \left (b x + a\right )^{\frac{3}{2}}}{\cosh \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.55228, size = 923, normalized size = 11.68 \begin{align*} \frac{2 \,{\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \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 ) - 4 \, \cosh \left (b x + a\right )^{2} -{\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\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 ) - 8 \,{\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 )} - 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - 4 \, \sinh \left (b x + a\right )^{2} - 4}{2 \,{\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} + b\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{\sinh ^{\frac{3}{2}}{\left (a + b x \right )}}{\cosh ^{\frac{3}{2}}{\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{\sinh \left (b x + a\right )^{\frac{3}{2}}}{\cosh \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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