Optimal. Leaf size=35 \[ \frac{\text{PolyLog}\left (2,e^{a+b x}\right )}{2 b}-\frac{\text{PolyLog}\left (2,-e^{a+b x}\right )}{2 b} \]
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Rubi [A] time = 0.0139707, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 5912} \[ \frac{\text{PolyLog}\left (2,e^{a+b x}\right )}{2 b}-\frac{\text{PolyLog}\left (2,-e^{a+b x}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 5912
Rubi steps
\begin{align*} \int \tanh ^{-1}\left (e^{a+b x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{x} \, dx,x,e^{a+b x}\right )}{b}\\ &=-\frac{\text{Li}_2\left (-e^{a+b x}\right )}{2 b}+\frac{\text{Li}_2\left (e^{a+b x}\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0773172, size = 68, normalized size = 1.94 \[ \frac{-\text{PolyLog}\left (2,-e^{a+b x}\right )+\text{PolyLog}\left (2,e^{a+b x}\right )+b x \left (\log \left (1-e^{a+b x}\right )-\log \left (e^{a+b x}+1\right )+2 \tanh ^{-1}\left (e^{a+b x}\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 67, normalized size = 1.9 \begin{align*}{\frac{\ln \left ({{\rm e}^{bx+a}} \right ){\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{b}}-{\frac{{\it dilog} \left ({{\rm e}^{bx+a}} \right ) }{2\,b}}-{\frac{{\it dilog} \left ({{\rm e}^{bx+a}}+1 \right ) }{2\,b}}-{\frac{\ln \left ({{\rm e}^{bx+a}} \right ) \ln \left ({{\rm e}^{bx+a}}+1 \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.961612, size = 144, normalized size = 4.11 \begin{align*} \frac{{\left (b x + a\right )} \operatorname{artanh}\left (e^{\left (b x + a\right )}\right )}{b} - \frac{{\left (b x + a\right )}{\left (\log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left (e^{\left (b x + a\right )} - 1\right )\right )} - \log \left (-e^{\left (b x + a\right )}\right ) \log \left (e^{\left (b x + a\right )} + 1\right ) +{\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) -{\rm Li}_2\left (e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (b x + a\right )} + 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54787, size = 419, normalized size = 11.97 \begin{align*} \frac{b x \log \left (-\frac{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1}\right ) - b x \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) +{\left (b x + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) +{\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) -{\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{atanh}{\left (e^{a + b x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{artanh}\left (e^{\left (b x + a\right )}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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