Optimal. Leaf size=70 \[ \frac{3 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )^2}-\frac{\tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.0602406, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {2282, 12, 455, 385, 206} \[ \frac{3 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )^2}-\frac{\tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 455
Rule 385
Rule 206
Rubi steps
\begin{align*} \int e^{a+b x} \coth (a+b x) \text{csch}^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{4 x^2 \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^2 \left (-1-x^2\right )}{\left (1-x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=-\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-2-4 x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=-\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac{3 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=-\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac{3 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{\tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0802675, size = 59, normalized size = 0.84 \[ \frac{e^{a+b x}-3 e^{3 (a+b x)}+\left (e^{2 (a+b x)}-1\right )^2 \left (-\tanh ^{-1}\left (e^{a+b x}\right )\right )}{b \left (e^{2 (a+b x)}-1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 69, normalized size = 1. \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{\sinh \left ( bx+a \right ) }}+\sinh \left ( bx+a \right ) -{\frac{\cosh \left ( bx+a \right ) }{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}}+{\frac{{\rm csch} \left (bx+a\right ){\rm coth} \left (bx+a\right )}{2}}-{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01084, size = 105, normalized size = 1.5 \begin{align*} -\frac{\log \left (e^{\left (b x + a\right )} + 1\right )}{2 \, b} + \frac{\log \left (e^{\left (b x + a\right )} - 1\right )}{2 \, b} - \frac{3 \, e^{\left (3 \, b x + 3 \, a\right )} - e^{\left (b x + a\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 = 1.5487, size = 1089, normalized size = 15.56 \begin{align*} -\frac{6 \, \cosh \left (b x + a\right )^{3} + 18 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 6 \, \sinh \left (b x + a\right )^{3} +{\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 (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) -{\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 (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \,{\left (9 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - 2 \, \cosh \left (b x + a\right )}{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 = 1.37044, size = 92, normalized size = 1.31 \begin{align*} -\frac{\log \left (e^{\left (b x + a\right )} + 1\right )}{2 \, b} + \frac{\log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{2 \, b} - \frac{3 \, e^{\left (3 \, b x + 3 \, a\right )} - e^{\left (b x + a\right )}}{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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