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 = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}
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Mathematica [A] time = 0.09, 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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fricas [A] time = 0.60, size = 25, normalized size = 0.56 \[ \frac {{\left (b c x + a c - 1\right )} e^{\left (b c x + a c\right )}}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 40, normalized size = 0.89 \[ \frac {{\left (e^{\left (b c x\right )} \log \left (-e^{\left (2 \, b c x + 2 \, a c\right )}\right ) - 2 \, e^{\left (b c x\right )}\right )} e^{\left (a c\right )}}{2 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 351, normalized size = 7.80 \[ \frac {{\mathrm e}^{c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}\right )}{b c}+\frac {i \left (2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )^{2}-2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )^{3}-\pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )^{2}-\pi \mathrm {csgn}\left (i {\mathrm e}^{c \left (b x +a \right )}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right )+2 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{c \left (b x +a \right )}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}-\pi \mathrm {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right )^{3}+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )^{3}+4 i-2 \pi \right ) {\mathrm e}^{c \left (b x +a \right )}}{4 b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 43, normalized size = 0.96 \[ \frac {\operatorname {artanh}\left (\coth \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} - \frac {e^{\left (b c x + a c\right )}}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 28, normalized size = 0.62 \[ \frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\left (\mathrm {atanh}\left (\mathrm {coth}\left (a\,c+b\,c\,x\right )\right )-1\right )}{b\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{a c} \int e^{b c x} \operatorname {atanh}{\left (\coth {\left (a c + b c x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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