Optimal. Leaf size=41 \[ \frac{2}{b \left (1-e^{2 a+2 b x}\right )}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A] time = 0.0448145, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2282, 12, 444, 43} \[ \frac{2}{b \left (1-e^{2 a+2 b x}\right )}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 444
Rule 43
Rubi steps
\begin{align*} \int e^{a+b x} \coth (a+b x) \text{csch}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{2 x \left (1+x^2\right )}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x \left (1+x^2\right )}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1+x}{(1-x)^2} \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{2}{(-1+x)^2}+\frac{1}{-1+x}\right ) \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=\frac{2}{b \left (1-e^{2 a+2 b x}\right )}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0466463, size = 34, normalized size = 0.83 \[ \frac{\log \left (1-e^{2 (a+b x)}\right )-\frac{2}{e^{2 (a+b x)}-1}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 30, normalized size = 0.7 \begin{align*} x+{\frac{\ln \left ( \sinh \left ( bx+a \right ) \right ) }{b}}-{\frac{{\rm coth} \left (bx+a\right )}{b}}+{\frac{a}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02518, size = 61, normalized size = 1.49 \begin{align*} \frac{\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac{\log \left (e^{\left (b x + a\right )} - 1\right )}{b} - \frac{2}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.47199, size = 284, normalized size = 6.93 \begin{align*} \frac{{\left (\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} - 1\right )} \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 \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\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 = 1.2552, size = 63, normalized size = 1.54 \begin{align*} \frac{\log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} - \frac{e^{\left (2 \, b x + 2 \, a\right )} + 1}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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