Optimal. Leaf size=45 \[ \frac{e^{a c+b c x} \coth ^{-1}(\coth (c (a+b x)))}{b c}-\frac{e^{a c+b c x}}{b c} \]
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Rubi [A] time = 0.0586381, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2194, 6276} \[ \frac{e^{a c+b c x} \coth ^{-1}(\coth (c (a+b x)))}{b c}-\frac{e^{a c+b c x}}{b c} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 6276
Rubi steps
\begin{align*} \int e^{c (a+b x)} \coth ^{-1}(\coth (a c+b c x)) \, dx &=\frac{\operatorname{Subst}\left (\int e^x \coth ^{-1}(\coth (x)) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac{e^{a c+b c x} \coth ^{-1}(\coth (c (a+b x)))}{b c}-\frac{\operatorname{Subst}\left (\int e^x \, dx,x,a c+b c x\right )}{b c}\\ &=-\frac{e^{a c+b c x}}{b c}+\frac{e^{a c+b c x} \coth ^{-1}(\coth (c (a+b x)))}{b c}\\ \end{align*}
Mathematica [A] time = 0.0797835, size = 46, normalized size = 1.02 \[ \frac{e^{c (a+b x)} \left (\coth ^{-1}\left (\frac{e^{2 c (a+b x)}+1}{e^{2 c (a+b x)}-1}\right )-1\right )}{b c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.069, size = 68, normalized size = 1.5 \begin{align*}{\frac{ \left ( xbc+ac \right ){{\rm e}^{xbc+ac}}-{{\rm e}^{xbc+ac}}+{{\rm e}^{xbc+ac}} \left ({\rm arccoth} \left ({\rm coth} \left (xbc+ac\right )\right )-xbc-ac \right ) }{bc}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1811, size = 57, normalized size = 1.27 \begin{align*} \frac{a e^{\left (b c x + a c\right )}}{b} + \frac{{\left (b c x e^{\left (a c\right )} - e^{\left (a c\right )}\right )} e^{\left (b c x\right )}}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51155, size = 55, normalized size = 1.22 \begin{align*} \frac{{\left (b c x + a c - 1\right )} e^{\left (b c x + a c\right )}}{b c} \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 = 1.18341, size = 47, normalized size = 1.04 \begin{align*} \frac{{\left (b^{2} c^{2} x + a b c^{2} - b c\right )} e^{\left (b c x + a c\right )}}{b^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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