Optimal. Leaf size=45 \[ \frac{e^{a c+b c x} \tanh ^{-1}(\coth (c (a+b x)))}{b c}-\frac{e^{a c+b c x}}{b c} \]
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Rubi [A] time = 0.0543121, 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} \[ \frac{e^{a c+b c x} \tanh ^{-1}(\coth (c (a+b x)))}{b c}-\frac{e^{a c+b c x}}{b c} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 6275
Rubi steps
\begin{align*} \int e^{c (a+b x)} \tanh ^{-1}(\coth (a c+b c x)) \, dx &=\frac{\operatorname{Subst}\left (\int e^x \tanh ^{-1}(\coth (x)) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac{e^{a c+b c x} \tanh ^{-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} \tanh ^{-1}(\coth (c (a+b x)))}{b c}\\ \end{align*}
Mathematica [A] time = 0.0832905, 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 [C] time = 0.257, 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 ) ^{2}-2\,\pi \, \left ({\it csgn} \left ({\frac{i}{{{\rm e}^{2\,c \left ( bx+a \right ) }}-1}} \right ) \right ) ^{3}-\pi \,{\it csgn} \left ({\frac{i}{{{\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 ) }}}{{{\rm e}^{2\,c \left ( bx+a \right ) }}-1}} \right ) +\pi \,{\it csgn} \left ({\frac{i}{{{\rm e}^{2\,c \left ( bx+a \right ) }}-1}} \right ) \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}-\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}+\pi \,{\it csgn} \left ( i{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) \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}-\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}+4\,i-2\,\pi \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986189, size = 58, normalized size = 1.29 \begin{align*} \frac{\operatorname{artanh}\left (\coth \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 = 1.93226, 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 = 1.19302, size = 54, normalized size = 1.2 \begin{align*} \frac{{\left (e^{\left (b c x\right )} \log \left (-e^{\left (2 \, b c x + 2 \, a c\right )}\right ) - 2 \, e^{\left (b c x\right )}\right )} e^{\left (a c\right )}}{2 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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