Optimal. Leaf size=45 \[ -\frac {e^{a c+b c x}}{b c}+\frac {e^{a c+b c x} \coth ^{-1}(\coth (c (a+b x)))}{b c} \]
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Rubi [A]
time = 0.04, 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 = {2225, 6411}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 6411
Rubi steps
\begin {align*} \int e^{c (a+b x)} \coth ^{-1}(\coth (a c+b c x)) \, dx &=\frac {\text {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 {\text {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 = 0.06, size = 46, normalized size = 1.02 \begin {gather*} \frac {e^{c (a+b x)} \left (-1+\coth ^{-1}\left (\frac {1+e^{2 c (a+b x)}}{-1+e^{2 c (a+b x)}}\right )\right )}{b c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 68, normalized size = 1.51
method | result | size |
default | \(\frac {{\mathrm e}^{x b c +a c} \left (x b c +a c \right )-{\mathrm e}^{x b c +a c}+{\mathrm e}^{x b c +a c} \left (\mathrm {arccoth}\left (\coth \left (x b c +a c \right )\right )-x b c -a c \right )}{b c}\) | \(68\) |
risch | \(\frac {{\mathrm e}^{c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}\right )}{c b}-\frac {i \left (\pi \mathrm {csgn}\left (i {\mathrm e}^{c \left (b x +a \right )}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right )-2 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{c \left (b x +a \right )}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right )^{3}+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )^{3}-4 i\right ) {\mathrm e}^{c \left (b x +a \right )}}{4 c b}\) | \(302\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 42, normalized size = 0.93 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 25, normalized size = 0.56 \begin {gather*} \frac {{\left (b c x + a c - 1\right )} e^{\left (b c x + a c\right )}}{b c} \end {gather*}
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 \begin {gather*} e^{a c} \int e^{b c x} \operatorname {acoth}{\left (\coth {\left (a c + b c x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 35, normalized size = 0.78 \begin {gather*} \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 {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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