Optimal. Leaf size=59 \[ \frac {e^{-a-b x}}{4 b}+\frac {e^{a+b x}}{b}+\frac {e^{3 a+3 b x}}{12 b}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.05, antiderivative size = 59, 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, 461, 207} \[ \frac {e^{-a-b x}}{4 b}+\frac {e^{a+b x}}{b}+\frac {e^{3 a+3 b x}}{12 b}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 461
Rule 2282
Rubi steps
\begin {align*} \int e^{a+b x} \cosh ^2(a+b x) \coth (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{4 x^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 )^3}{x^2 \left (-1+x^2\right )} \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (4-\frac {1}{x^2}+x^2+\frac {8}{-1+x^2}\right ) \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac {e^{-a-b x}}{4 b}+\frac {e^{a+b x}}{b}+\frac {e^{3 a+3 b x}}{12 b}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {e^{-a-b x}}{4 b}+\frac {e^{a+b x}}{b}+\frac {e^{3 a+3 b x}}{12 b}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 68, normalized size = 1.15 \[ \frac {e^{-a-b x} \left (12 e^{2 (a+b x)}+e^{4 (a+b x)}-24 \sqrt {e^{2 (a+b x)}} \tanh ^{-1}\left (\sqrt {e^{2 (a+b x)}}\right )+3\right )}{12 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 170, normalized size = 2.88 \[ \frac {\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 6 \, {\left (\cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )^{2} + 12 \, \cosh \left (b x + a\right )^{2} - 12 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 12 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 4 \, {\left (\cosh \left (b x + a\right )^{3} + 6 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3}{12 \, {\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 57, normalized size = 0.97 \[ \frac {e^{\left (3 \, b x + 3 \, a\right )} + 12 \, e^{\left (b x + a\right )} + 3 \, e^{\left (-b x - a\right )} - 12 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 12 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 50, normalized size = 0.85 \[ \frac {\left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )+\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{3}+\cosh \left (b x +a \right )-2 \arctanh \left ({\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 65, normalized size = 1.10 \[ \frac {e^{\left (3 \, b x + 3 \, a\right )} + 12 \, e^{\left (b x + a\right )}}{12 \, b} + \frac {e^{\left (-b x - a\right )}}{4 \, b} - \frac {\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (b x + a\right )} - 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 66, normalized size = 1.12 \[ \frac {{\mathrm {e}}^{a+b\,x}}{b}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}+\frac {{\mathrm {e}}^{-a-b\,x}}{4\,b}+\frac {{\mathrm {e}}^{3\,a+3\,b\,x}}{12\,b} \]
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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