Optimal. Leaf size=45 \[ \frac{3 e^{a+b x}}{2 b}+\frac{e^{3 a+3 b x}}{6 b}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.0341525, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2282, 12, 390, 207} \[ \frac{3 e^{a+b x}}{2 b}+\frac{e^{3 a+3 b x}}{6 b}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 390
Rule 207
Rubi steps
\begin{align*} \int e^{2 (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 \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}{-1+x^2} \, dx,x,e^{a+b x}\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (3+x^2+\frac{4}{-1+x^2}\right ) \, dx,x,e^{a+b x}\right )}{2 b}\\ &=\frac{3 e^{a+b x}}{2 b}+\frac{e^{3 a+3 b x}}{6 b}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{3 e^{a+b x}}{2 b}+\frac{e^{3 a+3 b x}}{6 b}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.262552, size = 58, normalized size = 1.29 \[ -\frac{e^{a+b x} \left (-\frac{1}{3} e^{2 (a+b x)}+\frac{4 \tanh ^{-1}\left (\sqrt{e^{2 (a+b x)}}\right )}{\sqrt{e^{2 (a+b x)}}}-3\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 54, normalized size = 1.2 \begin{align*}{\frac{{{\rm e}^{3\,bx+3\,a}}}{6\,b}}+{\frac{3\,{{\rm e}^{bx+a}}}{2\,b}}-{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{b}}+{\frac{\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00814, size = 82, normalized size = 1.82 \begin{align*} \frac{{\left (9 \, e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{6 \, 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 = 1.84004, size = 298, normalized size = 6.62 \begin{align*} \frac{\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + 3 \,{\left (\cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right ) + 9 \, \cosh \left (b x + a\right ) - 6 \, \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 6 \, \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right )}{6 \, 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 = 1.15038, size = 73, normalized size = 1.62 \begin{align*} \frac{{\left (e^{\left (3 \, b x + 9 \, a\right )} + 9 \, e^{\left (b x + 7 \, a\right )}\right )} e^{\left (-6 \, a\right )} - 6 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 6 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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