Optimal. Leaf size=45 \[ \frac{e^{a c+b c x} \tanh ^{-1}(\tanh (c (a+b x)))}{b c}-\frac{e^{a c+b c x}}{b c} \]
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Rubi [A] time = 0.0543779, 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, 6275} \[ \frac{e^{a c+b c x} \tanh ^{-1}(\tanh (c (a+b x)))}{b c}-\frac{e^{a c+b c x}}{b c} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 6275
Rubi steps
\begin{align*} \int e^{c (a+b x)} \tanh ^{-1}(\tanh (a c+b c x)) \, dx &=\frac{\operatorname{Subst}\left (\int e^x \tanh ^{-1}(\tanh (x)) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac{e^{a c+b c x} \tanh ^{-1}(\tanh (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} \tanh ^{-1}(\tanh (c (a+b x)))}{b c}\\ \end{align*}
Mathematica [A] time = 0.0816431, size = 46, normalized size = 1.02 \[ \frac{e^{c (a+b x)} \left (\tanh ^{-1}\left (\frac{e^{2 c (a+b x)}-1}{e^{2 c (a+b x)}+1}\right )-1\right )}{b c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.038, 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 ({\it Artanh} \left ( \tanh \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 = 0.994307, size = 58, normalized size = 1.29 \begin{align*} \frac{\operatorname{artanh}\left (\tanh \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} - \frac{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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Fricas [A] time = 1.73324, size = 112, normalized size = 2.49 \begin{align*} \frac{{\left (b c x + a c - 1\right )} \cosh \left (b c x + a c\right ) +{\left (b c x + a c - 1\right )} \sinh \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 [A] time = 11.3783, size = 58, normalized size = 1.29 \begin{align*} \begin{cases} 0 & \text{for}\: c = 0 \wedge \left (b = 0 \vee c = 0\right ) \\x e^{a c} \operatorname{atanh}{\left (\tanh{\left (a c \right )} \right )} & \text{for}\: b = 0 \\\frac{e^{a c} e^{b c x} \operatorname{atanh}{\left (\tanh{\left (a c + b c x \right )} \right )}}{b c} - \frac{e^{a c} e^{b c x}}{b c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10506, 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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