Optimal. Leaf size=45 \[ \frac{e^{a c+b c x} \coth ^{-1}(\tanh (c (a+b x)))}{b c}-\frac{e^{a c+b c x}}{b c} \]
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Rubi [A] time = 0.0596168, 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} \[ \frac{e^{a c+b c x} \coth ^{-1}(\tanh (c (a+b x)))}{b c}-\frac{e^{a c+b c x}}{b c} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 6276
Rubi steps
\begin{align*} \int e^{c (a+b x)} \coth ^{-1}(\tanh (a c+b c x)) \, dx &=\frac{\operatorname{Subst}\left (\int e^x \coth ^{-1}(\tanh (x)) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac{e^{a c+b c x} \coth ^{-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} \coth ^{-1}(\tanh (c (a+b x)))}{b c}\\ \end{align*}
Mathematica [A] time = 0.0752806, 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 [C] time = 0.25, size = 351, normalized size = 7.8 \begin{align*}{\frac{{{\rm e}^{c \left ( bx+a \right ) }}\ln \left ({{\rm e}^{c \left ( bx+a \right ) }} \right ) }{bc}}+{\frac{{\frac{i}{4}}{{\rm e}^{c \left ( bx+a \right ) }}}{bc} \left ( -2\,\pi \, \left ({\it csgn} \left ({\frac{i}{{{\rm e}^{2\,c \left ( bx+a \right ) }}+1}} \right ) \right ) ^{3}+2\,\pi \, \left ({\it csgn} \left ({\frac{i}{{{\rm e}^{2\,c \left ( bx+a \right ) }}+1}} \right ) \right ) ^{2}-\pi \, \left ({\it csgn} \left ({\frac{i{{\rm e}^{2\,c \left ( bx+a \right ) }}}{{{\rm e}^{2\,c \left ( bx+a \right ) }}+1}} \right ) \right ) ^{3}+\pi \, \left ({\it csgn} \left ({\frac{i{{\rm e}^{2\,c \left ( bx+a \right ) }}}{{{\rm e}^{2\,c \left ( bx+a \right ) }}+1}} \right ) \right ) ^{2}{\it csgn} \left ( i{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) +\pi \, \left ({\it csgn} \left ({\frac{i{{\rm e}^{2\,c \left ( bx+a \right ) }}}{{{\rm e}^{2\,c \left ( bx+a \right ) }}+1}} \right ) \right ) ^{2}{\it csgn} \left ({\frac{i}{{{\rm e}^{2\,c \left ( bx+a \right ) }}+1}} \right ) -\pi \,{\it csgn} \left ({\frac{i{{\rm e}^{2\,c \left ( bx+a \right ) }}}{{{\rm e}^{2\,c \left ( bx+a \right ) }}+1}} \right ){\it csgn} \left ( i{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ){\it csgn} \left ({\frac{i}{{{\rm e}^{2\,c \left ( bx+a \right ) }}+1}} \right ) -\pi \, \left ({\it csgn} \left ( i{{\rm e}^{c \left ( bx+a \right ) }} \right ) \right ) ^{2}{\it csgn} \left ( i{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) +2\,\pi \,{\it csgn} \left ( i{{\rm e}^{c \left ( bx+a \right ) }} \right ) \left ({\it csgn} \left ( i{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) \right ) ^{2}-\pi \, \left ({\it csgn} \left ( i{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) \right ) ^{3}+4\,i-2\,\pi \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05829, size = 58, normalized size = 1.29 \begin{align*} \frac{\operatorname{arcoth}\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 = 1.7068, 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 [A] time = 25.7929, size = 63, normalized size = 1.4 \begin{align*} \begin{cases} \frac{i \pi x}{2} & \text{for}\: c = 0 \wedge \left (b = 0 \vee c = 0\right ) \\x e^{a c} \operatorname{acoth}{\left (\tanh{\left (a c \right )} \right )} & \text{for}\: b = 0 \\\frac{e^{a c} e^{b c x} \operatorname{acoth}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arcoth}\left (\tanh \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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