Optimal. Leaf size=37 \[ \frac {e^{2 a+2 b x}}{2 b}+\frac {\log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2282, 444, 43} \[ \frac {e^{2 a+2 b x}}{2 b}+\frac {\log \left (1-e^{2 a+2 b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \coth (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x \left (-1-x^2\right )}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {-1-x}{1-x} \, dx,x,e^{2 a+2 b x}\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {2}{-1+x}\right ) \, dx,x,e^{2 a+2 b x}\right )}{2 b}\\ &=\frac {e^{2 a+2 b x}}{2 b}+\frac {\log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 35, normalized size = 0.95 \[ \frac {e^{2 a+2 b x}+2 \log \left (1-e^{2 a+2 b x}\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 64, normalized size = 1.73 \[ \frac {\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 2 \, \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 30, normalized size = 0.81 \[ \frac {e^{\left (2 \, b x + 2 \, a\right )} + 2 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 38, normalized size = 1.03 \[ \frac {{\mathrm e}^{2 b x +2 a}}{2 b}-\frac {2 a}{b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 57, normalized size = 1.54 \[ \frac {2 \, {\left (b x + a\right )}}{b} + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{2 \, b} + \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 30, normalized size = 0.81 \[ \frac {{\mathrm {e}}^{2\,a+2\,b\,x}+2\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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