Optimal. Leaf size=23 \[ \frac{\cosh (a+b x)}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]
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Rubi [A] time = 0.0180681, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2592, 321, 206} \[ \frac{\cosh (a+b x)}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2592
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \cosh (a+b x) \coth (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac{\cosh (a+b x)}{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}\\ \end{align*}
Mathematica [A] time = 0.0257619, size = 26, normalized size = 1.13 \[ \frac{\cosh (a+b x)}{b}+\frac{\log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 21, normalized size = 0.9 \begin{align*}{\frac{\cosh \left ( bx+a \right ) -2\,{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20723, size = 80, normalized size = 3.48 \begin{align*} \frac{e^{\left (b x + a\right )}}{2 \, b} + \frac{e^{\left (-b x - a\right )}}{2 \, 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 = 1.84889, size = 355, normalized size = 15.43 \begin{align*} \frac{\cosh \left (b x + a\right )^{2} - 2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}{2 \,{\left (b \cosh \left (b x + a\right ) + b \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\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 [A] time = 1.18726, size = 59, normalized size = 2.57 \begin{align*} \frac{e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, \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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