Optimal. Leaf size=25 \[ \frac{e^{a+b x}}{b}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.0177557, antiderivative size = 25, 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, 388, 206} \[ \frac{e^{a+b x}}{b}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 388
Rule 206
Rubi steps
\begin{align*} \int e^{a+b x} \coth (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1-x^2}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{e^{a+b x}}{b}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{e^{a+b x}}{b}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0162444, size = 22, normalized size = 0.88 \[ \frac{e^{a+b x}-2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 27, normalized size = 1.1 \begin{align*}{\frac{\sinh \left ( bx+a \right ) +\cosh \left ( bx+a \right ) -2\,{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08431, size = 51, normalized size = 2.04 \begin{align*} \frac{e^{\left (b x + a\right )}}{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 = 1.56986, size = 158, normalized size = 6.32 \begin{align*} \frac{\cosh \left (b x + a\right ) - \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + \sinh \left (b x + a\right )}{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 = 1.14439, size = 53, normalized size = 2.12 \begin{align*} \frac{e^{\left (b x + a\right )}}{b} - \frac{\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac{\log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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