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 = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, 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 203
Rule 206
Rule 298
Rule 2566
Rule 2575
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*}
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Mathematica [C] time = 0.05, 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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fricas [B] time = 0.46, size = 310, normalized size = 3.92 \[ \frac {2 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \arctan \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) - 4 \, \cosh \left (b x + a\right )^{2} - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \log \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) - 8 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - 4 \, \sinh \left (b x + a\right )^{2} - 4}{2 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (b x + a\right )^{\frac {3}{2}}}{\cosh \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{\frac {3}{2}}\left (b x +a \right )}{\cosh \left (b x +a \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (b x + a\right )^{\frac {3}{2}}}{\cosh \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^{3/2}}{{\mathrm {cosh}\left (a+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{\frac {3}{2}}{\left (a + b x \right )}}{\cosh ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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