Optimal. Leaf size=45 \[ \frac {3 e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{6 b}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2282, 12, 390, 207} \[ \frac {3 e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{6 b}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 390
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \cosh (a+b x) \coth (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{2 \left (-1+x^2\right )} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{-1+x^2} \, dx,x,e^{a+b x}\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (3+x^2+\frac {4}{-1+x^2}\right ) \, dx,x,e^{a+b x}\right )}{2 b}\\ &=\frac {3 e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{6 b}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {3 e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{6 b}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 58, normalized size = 1.29 \[ -\frac {e^{a+b x} \left (-\frac {1}{3} e^{2 (a+b x)}+\frac {4 \tanh ^{-1}\left (\sqrt {e^{2 (a+b x)}}\right )}{\sqrt {e^{2 (a+b x)}}}-3\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 98, normalized size = 2.18 \[ \frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + 3 \, {\left (\cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right ) + 9 \, \cosh \left (b x + a\right ) - 6 \, \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 6 \, \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right )}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 54, normalized size = 1.20 \[ \frac {{\left (e^{\left (3 \, b x + 9 \, a\right )} + 9 \, e^{\left (b x + 7 \, a\right )}\right )} e^{\left (-6 \, a\right )} - 6 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 6 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 54, normalized size = 1.20 \[ \frac {{\mathrm e}^{3 b x +3 a}}{6 b}+\frac {3 \,{\mathrm e}^{b x +a}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 61, normalized size = 1.36 \[ \frac {{\left (9 \, e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{6 \, b} - \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 53, normalized size = 1.18 \[ \frac {3\,{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}+\frac {{\mathrm {e}}^{3\,a+3\,b\,x}}{6\,b} \]
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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