Optimal. Leaf size=37 \[ \frac{\sinh (a+b x)}{b}-\frac{\text{csch}^3(a+b x)}{3 b}-\frac{2 \text{csch}(a+b x)}{b} \]
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Rubi [A] time = 0.0250251, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2590, 270} \[ \frac{\sinh (a+b x)}{b}-\frac{\text{csch}^3(a+b x)}{3 b}-\frac{2 \text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2590
Rule 270
Rubi steps
\begin{align*} \int \cosh (a+b x) \coth ^4(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^4} \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=\frac{i \operatorname{Subst}\left (\int \left (1+\frac{1}{x^4}-\frac{2}{x^2}\right ) \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=-\frac{2 \text{csch}(a+b x)}{b}-\frac{\text{csch}^3(a+b x)}{3 b}+\frac{\sinh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0213773, size = 37, normalized size = 1. \[ \frac{\sinh (a+b x)}{b}-\frac{\text{csch}^3(a+b x)}{3 b}-\frac{2 \text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 67, normalized size = 1.8 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{ \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}}-{\frac{4\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3\, \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}}-{\frac{8\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3\,\sinh \left ( bx+a \right ) }}+{\frac{8\,\sinh \left ( bx+a \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15323, size = 135, normalized size = 3.65 \begin{align*} -\frac{e^{\left (-b x - a\right )}}{2 \, b} - \frac{33 \, e^{\left (-2 \, b x - 2 \, a\right )} - 41 \, e^{\left (-4 \, b x - 4 \, a\right )} + 27 \, e^{\left (-6 \, b x - 6 \, a\right )} - 3}{6 \, b{\left (e^{\left (-b x - a\right )} - 3 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )} - e^{\left (-7 \, b x - 7 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82168, size = 238, normalized size = 6.43 \begin{align*} \frac{3 \, \cosh \left (b x + a\right )^{4} + 3 \, \sinh \left (b x + a\right )^{4} + 18 \,{\left (\cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )^{2} - 36 \, \cosh \left (b x + a\right )^{2} + 25}{6 \,{\left (b \sinh \left (b x + a\right )^{3} + 3 \,{\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )\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 = 1.25634, size = 96, normalized size = 2.59 \begin{align*} -\frac{\frac{8 \,{\left (3 \, e^{\left (5 \, b x + 5 \, a\right )} - 4 \, e^{\left (3 \, b x + 3 \, a\right )} + 3 \, e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} - 3 \, e^{\left (b x + a\right )} + 3 \, e^{\left (-b x - a\right )}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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