Optimal. Leaf size=39 \[ \frac{\sinh ^4(a+b x)}{4 b}+\frac{\sinh ^2(a+b x)}{b}+\frac{\log (\sinh (a+b x))}{b} \]
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Rubi [A] time = 0.0308386, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2590, 266, 43} \[ \frac{\sinh ^4(a+b x)}{4 b}+\frac{\sinh ^2(a+b x)}{b}+\frac{\log (\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2590
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \cosh ^4(a+b x) \coth (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x} \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1-x)^2}{x} \, dx,x,-\sinh ^2(a+b x)\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2+\frac{1}{x}+x\right ) \, dx,x,-\sinh ^2(a+b x)\right )}{2 b}\\ &=\frac{\log (\sinh (a+b x))}{b}+\frac{\sinh ^2(a+b x)}{b}+\frac{\sinh ^4(a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0323827, size = 35, normalized size = 0.9 \[ \frac{\sinh ^4(a+b x)+4 \sinh ^2(a+b x)+4 \log (\sinh (a+b x))}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 39, normalized size = 1. \begin{align*}{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{4\,b}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2\,b}}+{\frac{\ln \left ( \sinh \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09916, size = 128, normalized size = 3.28 \begin{align*} \frac{{\left (12 \, e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} + \frac{b x + a}{b} + \frac{12 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} + \frac{\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac{\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.90217, size = 1270, normalized size = 32.56 \begin{align*} \frac{\cosh \left (b x + a\right )^{8} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + \sinh \left (b x + a\right )^{8} + 4 \,{\left (7 \, \cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )^{6} - 64 \, b x \cosh \left (b x + a\right )^{4} + 12 \, \cosh \left (b x + a\right )^{6} + 8 \,{\left (7 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 2 \,{\left (35 \, \cosh \left (b x + a\right )^{4} - 32 \, b x + 90 \, \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{4} + 8 \,{\left (7 \, \cosh \left (b x + a\right )^{5} - 32 \, b x \cosh \left (b x + a\right ) + 30 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{3} + 4 \,{\left (7 \, \cosh \left (b x + a\right )^{6} - 96 \, b x \cosh \left (b x + a\right )^{2} + 45 \, \cosh \left (b x + a\right )^{4} + 3\right )} \sinh \left (b x + a\right )^{2} + 12 \, \cosh \left (b x + a\right )^{2} + 64 \,{\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4}\right )} \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 8 \,{\left (\cosh \left (b x + a\right )^{7} - 32 \, b x \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right )^{5} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{64 \,{\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4}\right )}} \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 [B] time = 1.23364, size = 117, normalized size = 3. \begin{align*} -\frac{64 \, b x -{\left (48 \, e^{\left (4 \, b x + 4 \, a\right )} + 12 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )} -{\left (e^{\left (4 \, b x + 16 \, a\right )} + 12 \, e^{\left (2 \, b x + 14 \, a\right )}\right )} e^{\left (-12 \, a\right )} - 64 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{64 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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