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 = 0.08, antiderivative size = 81, 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, 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 203
Rule 206
Rule 298
Rule 2566
Rule 2574
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*}
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Mathematica [C] time = 0.05, 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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fricas [B] time = 0.45, size = 591, normalized size = 7.30 \[ -\frac {4 \, \cosh \left (b x + a\right )^{4} + 16 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 4 \, \sinh \left (b x + a\right )^{4} + 8 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 6 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 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 ) + 8 \, \cosh \left (b x + a\right )^{2} + 3 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 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 )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} + 16 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 4}{6 \, {\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 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 \[ \int \frac {\sinh \left (b x + a\right )^{\frac {5}{2}}}{\cosh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{\frac {5}{2}}\left (b x +a \right )}{\cosh \left (b x +a \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {\sinh \left (b x + a\right )^{\frac {5}{2}}}{\cosh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^{5/2}}{{\mathrm {cosh}\left (a+b\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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