Optimal. Leaf size=80 \[ \frac{e^{2 a+2 b x}}{2 b}+\frac{6}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac{3 \log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A] time = 0.06267, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2282, 444, 43} \[ \frac{e^{2 a+2 b x}}{2 b}+\frac{6}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac{3 \log \left (1-e^{2 a+2 b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 444
Rule 43
Rubi steps
\begin{align*} \int e^{2 (a+b x)} \coth ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x \left (1+x^2\right )^3}{\left (-1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1+x)^3}{(-1+x)^3} \, dx,x,e^{2 a+2 b x}\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{8}{(-1+x)^3}+\frac{12}{(-1+x)^2}+\frac{6}{-1+x}\right ) \, dx,x,e^{2 a+2 b x}\right )}{2 b}\\ &=\frac{e^{2 a+2 b x}}{2 b}-\frac{2}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac{6}{b \left (1-e^{2 a+2 b x}\right )}+\frac{3 \log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.109898, size = 60, normalized size = 0.75 \[ \frac{\frac{8-12 e^{2 (a+b x)}}{\left (e^{2 (a+b x)}-1\right )^2}+e^{2 (a+b x)}+6 \log \left (1-e^{2 (a+b x)}\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.127, size = 70, normalized size = 0.9 \begin{align*}{\frac{{{\rm e}^{2\,bx+2\,a}}}{2\,b}}-6\,{\frac{a}{b}}-2\,{\frac{3\,{{\rm e}^{2\,bx+2\,a}}-2}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) ^{2}}}+3\,{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0164, size = 143, normalized size = 1.79 \begin{align*} \frac{6 \,{\left (b x + a\right )}}{b} + \frac{3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac{3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac{10 \, e^{\left (-2 \, b x - 2 \, a\right )} - 5 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1}{2 \, b{\left (e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.867, size = 1099, normalized size = 13.74 \begin{align*} \frac{\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} +{\left (15 \, \cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )^{4} - 2 \, \cosh \left (b x + a\right )^{4} + 4 \,{\left (5 \, \cosh \left (b x + a\right )^{3} - 2 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} +{\left (15 \, \cosh \left (b x + a\right )^{4} - 12 \, \cosh \left (b x + a\right )^{2} - 11\right )} \sinh \left (b x + a\right )^{2} - 11 \, \cosh \left (b x + a\right )^{2} + 6 \,{\left (\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} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \,{\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{5} - 4 \, \cosh \left (b x + a\right )^{3} - 11 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 8}{2 \,{\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 2 \, b \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 4 \,{\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23137, size = 95, normalized size = 1.19 \begin{align*} -\frac{\frac{9 \, e^{\left (4 \, b x + 4 \, a\right )} - 6 \, e^{\left (2 \, b x + 2 \, a\right )} + 1}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} - e^{\left (2 \, b x + 2 \, a\right )} - 6 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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