Optimal. Leaf size=81 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {2 \sinh ^{\frac {3}{2}}(a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)} \]
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Rubi [A]
time = 0.06, 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 = {2646, 2654,
304, 209, 212} \begin {gather*} -\frac {\text {ArcTan}\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)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 304
Rule 2646
Rule 2654
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 \text {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 {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {\text {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 59, normalized size = 0.73 \begin {gather*} \frac {2 \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 ) \sinh ^{\frac {7}{2}}(a+b x)}{7 b \cosh ^{\frac {3}{2}}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.67, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 591 vs.
\(2 (67) = 134\).
time = 0.36, size = 591, normalized size = 7.30 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^{5/2}}{{\mathrm {cosh}\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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