Optimal. Leaf size=53 \[ \frac{e^{a+b x}}{b}+\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.0381733, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 390, 288, 206} \[ \frac{e^{a+b x}}{b}+\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 390
Rule 288
Rule 206
Rubi steps
\begin{align*} \int e^{a+b x} \coth ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{4 x^2}{\left (1-x^2\right )^2}\right ) \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{e^{a+b x}}{b}+\frac{4 \operatorname{Subst}\left (\int \frac{x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{e^{a+b x}}{b}+\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{e^{a+b x}}{b}+\frac{2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end{align*}
Mathematica [C] time = 2.54259, size = 179, normalized size = 3.38 \[ \frac{e^{a+b x} \left (\frac{4}{105} \left (e^{a+b x}+e^{3 (a+b x)}\right )^2 \, _4F_3\left (\frac{3}{2},2,2,2;1,1,\frac{9}{2};e^{2 (a+b x)}\right )+\frac{1}{48} e^{-4 (a+b x)} \left (-713 e^{2 (a+b x)}-181 e^{4 (a+b x)}+61 e^{6 (a+b x)}+\frac{3 \left (196 e^{2 (a+b x)}-14 e^{4 (a+b x)}-52 e^{6 (a+b x)}+e^{8 (a+b x)}+125\right ) \tanh ^{-1}\left (\sqrt{e^{2 (a+b x)}}\right )}{\sqrt{e^{2 (a+b x)}}}-375\right )\right )}{b} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.019, size = 47, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \cosh \left ( bx+a \right ) -2\,{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) -{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{\sinh \left ( bx+a \right ) }}+2\,\sinh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999177, size = 84, normalized size = 1.58 \begin{align*} \frac{e^{\left (b x + a\right )}}{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} - \frac{2 \, e^{\left (b x + a\right )}}{b{\left (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 = 1.59806, size = 585, normalized size = 11.04 \begin{align*} \frac{\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} -{\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 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 )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 3 \,{\left (\cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - 3 \, \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - 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 = 1.1461, size = 85, normalized size = 1.6 \begin{align*} \frac{e^{\left (b x + a\right )}}{b} - \frac{\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac{\log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} - \frac{2 \, e^{\left (b x + a\right )}}{b{\left (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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