Optimal. Leaf size=63 \[ \frac{6}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac{2 \log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A] time = 0.0721251, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2282, 12, 446, 77} \[ \frac{6}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac{2 \log \left (1-e^{2 a+2 b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 446
Rule 77
Rubi steps
\begin{align*} \int e^{2 (a+b x)} \coth (a+b x) \text{csch}^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{4 x^3 \left (-1-x^2\right )}{\left (1-x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{4 \operatorname{Subst}\left (\int \frac{x^3 \left (-1-x^2\right )}{\left (1-x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{(-1-x) x}{(1-x)^3} \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{2}{(-1+x)^3}+\frac{3}{(-1+x)^2}+\frac{1}{-1+x}\right ) \, dx,x,e^{2 a+2 b x}\right )}{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{2 \log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.079961, size = 47, normalized size = 0.75 \[ \frac{2 \left (\frac{2-3 e^{2 (a+b x)}}{\left (e^{2 (a+b x)}-1\right )^2}+\log \left (1-e^{2 (a+b x)}\right )\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 56, normalized size = 0.9 \begin{align*} -4\,{\frac{a}{b}}-2\,{\frac{3\,{{\rm e}^{2\,bx+2\,a}}-2}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) ^{2}}}+2\,{\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 = 1.03539, size = 116, normalized size = 1.84 \begin{align*} 4 \, x + \frac{4 \, a}{b} + \frac{2 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac{2 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac{2 \,{\left (e^{\left (-2 \, b x - 2 \, a\right )} - 2\right )}}{b{\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85458, size = 714, normalized size = 11.33 \begin{align*} -\frac{2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} -{\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 ) + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 3 \, \sinh \left (b x + a\right )^{2} - 2\right )}}{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} \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 = 1.18718, size = 65, normalized size = 1.03 \begin{align*} -\frac{\frac{3 \, e^{\left (4 \, b x + 4 \, a\right )} - 1}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} - 2 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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