Optimal. Leaf size=62 \[ \frac{4}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A] time = 0.0653165, antiderivative size = 62, normalized size of antiderivative = 1., 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, 444, 43} \[ \frac{4}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 444
Rule 43
Rubi steps
\begin{align*} \int e^{a+b x} \coth ^2(a+b x) \text{csch}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{2 x \left (1+x^2\right )^2}{\left (-1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x \left (1+x^2\right )^2}{\left (-1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1+x)^2}{(-1+x)^3} \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{4}{(-1+x)^3}+\frac{4}{(-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{4}{b \left (1-e^{2 a+2 b x}\right )}+\frac{\log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0814723, size = 46, normalized size = 0.74 \[ \frac{\frac{2-4 e^{2 (a+b x)}}{\left (e^{2 (a+b x)}-1\right )^2}+\log \left (1-e^{2 (a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 43, normalized size = 0.7 \begin{align*} x+{\frac{a}{b}}-{\frac{{\rm coth} \left (bx+a\right )}{b}}+{\frac{\ln \left ( \sinh \left ( bx+a \right ) \right ) }{b}}-{\frac{ \left ({\rm coth} \left (bx+a\right ) \right ) ^{2}}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01903, size = 93, normalized size = 1.5 \begin{align*} \frac{\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac{\log \left (e^{\left (b x + a\right )} - 1\right )}{b} - \frac{2 \,{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}}{b{\left (e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62851, size = 711, normalized size = 11.47 \begin{align*} -\frac{4 \, \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 ) + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 4 \, \sinh \left (b x + a\right )^{2} - 2}{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 = 1.14078, size = 81, normalized size = 1.31 \begin{align*} \frac{\log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} - \frac{3 \, e^{\left (4 \, b x + 4 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1}{2 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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