Optimal. Leaf size=37 \[ \frac{e^{2 a+2 b x}}{2 b}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A] time = 0.032322, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2282, 444, 43} \[ \frac{e^{2 a+2 b x}}{2 b}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 444
Rule 43
Rubi steps
\begin{align*} \int e^{2 (a+b x)} \coth (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x \left (-1-x^2\right )}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{-1-x}{1-x} \, dx,x,e^{2 a+2 b x}\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{2}{-1+x}\right ) \, dx,x,e^{2 a+2 b x}\right )}{2 b}\\ &=\frac{e^{2 a+2 b x}}{2 b}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0245513, size = 35, normalized size = 0.95 \[ \frac{e^{2 a+2 b x}+2 \log \left (1-e^{2 a+2 b x}\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 38, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{2\,bx+2\,a}}}{2\,b}}-2\,{\frac{a}{b}}+{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0186, size = 77, normalized size = 2.08 \begin{align*} \frac{2 \,{\left (b x + a\right )}}{b} + \frac{e^{\left (2 \, b x + 2 \, 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 [A] time = 1.85139, size = 178, normalized size = 4.81 \begin{align*} \frac{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 2 \, \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{2 \, b} \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 [A] time = 1.16356, size = 41, normalized size = 1.11 \begin{align*} \frac{e^{\left (2 \, b x + 2 \, a\right )} + 2 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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