Optimal. Leaf size=19 \[ \frac{\log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A] time = 0.0189866, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2282, 12, 260} \[ \frac{\log \left (1-e^{2 a+2 b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 260
Rubi steps
\begin{align*} \int e^{a+b x} \text{csch}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{2 x}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0108075, size = 19, normalized size = 1. \[ \frac{\log \left (1-e^{2 a+2 b x}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 19, normalized size = 1. \begin{align*} x+{\frac{\ln \left ( \sinh \left ( bx+a \right ) \right ) }{b}}+{\frac{a}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11127, size = 36, normalized size = 1.89 \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0032, size = 76, normalized size = 4. \begin{align*} \frac{\log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \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*} e^{a} \int e^{b x} \operatorname{csch}{\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 = 1.13005, size = 23, normalized size = 1.21 \begin{align*} \frac{\log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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