Optimal. Leaf size=49 \[ \frac{3 \cosh (a+b x)}{2 b}-\frac{3 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac{\cosh (a+b x) \coth ^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0359131, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2592, 288, 321, 206} \[ \frac{3 \cosh (a+b x)}{2 b}-\frac{3 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac{\cosh (a+b x) \coth ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2592
Rule 288
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \cosh (a+b x) \coth ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^2} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac{\cosh (a+b x) \coth ^2(a+b x)}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{2 b}\\ &=\frac{3 \cosh (a+b x)}{2 b}-\frac{\cosh (a+b x) \coth ^2(a+b x)}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{2 b}\\ &=-\frac{3 \tanh ^{-1}(\cosh (a+b x))}{2 b}+\frac{3 \cosh (a+b x)}{2 b}-\frac{\cosh (a+b x) \coth ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0319093, size = 67, normalized size = 1.37 \[ \frac{\cosh (a+b x)}{b}-\frac{\text{csch}^2\left (\frac{1}{2} (a+b x)\right )}{8 b}-\frac{\text{sech}^2\left (\frac{1}{2} (a+b x)\right )}{8 b}+\frac{3 \log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 62, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}}-3\,{\frac{\cosh \left ( bx+a \right ) }{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}}+{\frac{3\,{\rm csch} \left (bx+a\right ){\rm coth} \left (bx+a\right )}{2}}-3\,{\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 = 1.21084, size = 146, normalized size = 2.98 \begin{align*} \frac{e^{\left (-b x - a\right )}}{2 \, b} - \frac{3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{2 \, b} + \frac{3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{2 \, b} - \frac{4 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} - 1}{2 \, b{\left (e^{\left (-b x - a\right )} - 2 \, e^{\left (-3 \, b x - 3 \, a\right )} + e^{\left (-5 \, b x - 5 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02582, size = 1710, normalized size = 34.9 \begin{align*} \text{result too large to display} \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{\left (a + b x \right )} \coth ^{3}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28521, size = 107, normalized size = 2.18 \begin{align*} -\frac{\frac{2 \,{\left (e^{\left (3 \, b x + 3 \, a\right )} + e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} - e^{\left (b x + a\right )} - e^{\left (-b x - a\right )} + 3 \, \log \left (e^{\left (b x + a\right )} + 1\right ) - 3 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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