Optimal. Leaf size=42 \[ \frac{e^{2 a+2 b x}}{4 b}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b}-\frac{x}{2} \]
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Rubi [A] time = 0.0433302, antiderivative size = 42, 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, 446, 72} \[ \frac{e^{2 a+2 b x}}{4 b}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b}-\frac{x}{2} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 446
Rule 72
Rubi steps
\begin{align*} \int e^{a+b x} \cosh (a+b x) \coth (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{2 x \left (-1+x^2\right )} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x \left (-1+x^2\right )} \, dx,x,e^{a+b x}\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1+x)^2}{(-1+x) x} \, dx,x,e^{2 a+2 b x}\right )}{4 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{4}{-1+x}-\frac{1}{x}\right ) \, dx,x,e^{2 a+2 b x}\right )}{4 b}\\ &=\frac{e^{2 a+2 b x}}{4 b}-\frac{x}{2}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0293295, size = 39, normalized size = 0.93 \[ \frac{e^{2 a+2 b x}+4 \log \left (1-e^{2 a+2 b x}\right )-2 b x}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 52, normalized size = 1.2 \begin{align*}{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{2\,b}}+{\frac{x}{2}}+{\frac{a}{2\,b}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2\,b}}+{\frac{\ln \left ( \sinh \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02406, size = 68, normalized size = 1.62 \begin{align*} -\frac{1}{2} \, x - \frac{a}{2 \, b} + \frac{e^{\left (2 \, b x + 2 \, a\right )}}{4 \, 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 = 1.65233, size = 190, normalized size = 4.52 \begin{align*} -\frac{2 \, b x - \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} - 4 \, \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{4 \, 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 = 1.17022, size = 57, normalized size = 1.36 \begin{align*} -\frac{b x + a}{2 \, b} + \frac{e^{\left (2 \, b x + 2 \, a\right )}}{4 \, b} + \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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