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 = 0.06, 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, 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*}
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Mathematica [A] time = 0.09, 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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fricas [A] time = 0.53, 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 = 0.14, size = 35, normalized size = 0.78 \[ \frac {{\left (b^{2} c^{2} x + a b c^{2} - b c\right )} e^{\left (b c x + a c\right )}}{b^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 68, normalized size = 1.51 \[ \frac {\left (x b c +a c \right ) {\mathrm e}^{x b c +a c}-{\mathrm e}^{x b c +a c}+{\mathrm e}^{x b c +a c} \left (\mathrm {arccoth}\left (\coth \left (x b c +a c \right )\right )-x b c -a c \right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 42, normalized size = 0.93 \[ \frac {a e^{\left (b c x + a c\right )}}{b} + \frac {{\left (b c x e^{\left (a c\right )} - e^{\left (a c\right )}\right )} e^{\left (b c x\right )}}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 28, normalized size = 0.62 \[ \frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\left (\mathrm {acoth}\left (\mathrm {coth}\left (a\,c+b\,c\,x\right )\right )-1\right )}{b\,c} \]
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 \[ e^{a c} \int e^{b c x} \operatorname {acoth}{\left (\coth {\left (a c + b c x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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