Optimal. Leaf size=38 \[ \frac{\cosh ^3(a+b x)}{3 b}+\frac{\cosh (a+b x)}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]
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Rubi [A] time = 0.0300179, antiderivative size = 38, 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 = {2592, 302, 206} \[ \frac{\cosh ^3(a+b x)}{3 b}+\frac{\cosh (a+b x)}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2592
Rule 302
Rule 206
Rubi steps
\begin{align*} \int \cosh ^3(a+b x) \coth (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^4}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-1-x^2+\frac{1}{1-x^2}\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac{\cosh (a+b x)}{b}+\frac{\cosh ^3(a+b x)}{3 b}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac{\tanh ^{-1}(\cosh (a+b x))}{b}+\frac{\cosh (a+b x)}{b}+\frac{\cosh ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0293825, size = 44, normalized size = 1.16 \[ \frac{5 \cosh (a+b x)}{4 b}+\frac{\cosh (3 (a+b x))}{12 b}+\frac{\log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 31, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{3}}+\cosh \left ( bx+a \right ) -2\,{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04711, size = 117, normalized size = 3.08 \begin{align*} \frac{{\left (15 \, e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} + \frac{15 \, e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, 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 = 1.89726, size = 1037, normalized size = 27.29 \begin{align*} \frac{\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + 15 \,{\left (\cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{4} + 15 \, \cosh \left (b x + a\right )^{4} + 20 \,{\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 15 \,{\left (\cosh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 15 \, \cosh \left (b x + a\right )^{2} - 24 \,{\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 24 \,{\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 6 \,{\left (\cosh \left (b x + a\right )^{5} + 10 \, \cosh \left (b x + a\right )^{3} + 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{24 \,{\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh ^{3}{\left (a + b x \right )} \coth{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19812, size = 104, normalized size = 2.74 \begin{align*} \frac{{\left (15 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} +{\left (e^{\left (3 \, b x + 18 \, a\right )} + 15 \, e^{\left (b x + 16 \, a\right )}\right )} e^{\left (-15 \, a\right )} - 24 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 24 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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