Optimal. Leaf size=81 \[ -\frac{2 \sinh ^{\frac{3}{2}}(a+b x)}{3 b \cosh ^{\frac{3}{2}}(a+b x)}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sinh (a+b x)}}{\sqrt{\cosh (a+b x)}}\right )}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\sinh (a+b x)}}{\sqrt{\cosh (a+b x)}}\right )}{b} \]
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Rubi [A] time = 0.0811908, antiderivative size = 81, 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, 2574, 298, 203, 206} \[ -\frac{2 \sinh ^{\frac{3}{2}}(a+b x)}{3 b \cosh ^{\frac{3}{2}}(a+b x)}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sinh (a+b x)}}{\sqrt{\cosh (a+b x)}}\right )}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\sinh (a+b x)}}{\sqrt{\cosh (a+b x)}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2566
Rule 2574
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\sinh ^{\frac{5}{2}}(a+b x)}{\cosh ^{\frac{5}{2}}(a+b x)} \, dx &=-\frac{2 \sinh ^{\frac{3}{2}}(a+b x)}{3 b \cosh ^{\frac{3}{2}}(a+b x)}+\int \frac{\sqrt{\sinh (a+b x)}}{\sqrt{\cosh (a+b x)}} \, dx\\ &=-\frac{2 \sinh ^{\frac{3}{2}}(a+b x)}{3 b \cosh ^{\frac{3}{2}}(a+b x)}-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\frac{\sqrt{\sinh (a+b x)}}{\sqrt{\cosh (a+b x)}}\right )}{b}\\ &=-\frac{2 \sinh ^{\frac{3}{2}}(a+b x)}{3 b \cosh ^{\frac{3}{2}}(a+b x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt{\sinh (a+b x)}}{\sqrt{\cosh (a+b x)}}\right )}{b}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt{\sinh (a+b x)}}{\sqrt{\cosh (a+b x)}}\right )}{b}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{\sinh (a+b x)}}{\sqrt{\cosh (a+b x)}}\right )}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\sinh (a+b x)}}{\sqrt{\cosh (a+b x)}}\right )}{b}-\frac{2 \sinh ^{\frac{3}{2}}(a+b x)}{3 b \cosh ^{\frac{3}{2}}(a+b x)}\\ \end{align*}
Mathematica [C] time = 0.0479155, size = 59, normalized size = 0.73 \[ \frac{2 \sinh ^{\frac{7}{2}}(a+b x) \cosh ^2(a+b x)^{3/4} \, _2F_1\left (\frac{7}{4},\frac{7}{4};\frac{11}{4};-\sinh ^2(a+b x)\right )}{7 b \cosh ^{\frac{3}{2}}(a+b x)} \]
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{5}{2}}} \left ( \cosh \left ( bx+a \right ) \right ) ^{-{\frac{5}{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{5}{2}}}{\cosh \left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.68691, size = 1705, normalized size = 21.05 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{5}{2}}}{\cosh \left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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