Optimal. Leaf size=35 \[ \frac {\text {Li}_2\left (e^{a+b x}\right )}{2 b}-\frac {\text {Li}_2\left (-e^{a+b x}\right )}{2 b} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}
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Mathematica [A] time = 0.08, size = 68, normalized size = 1.94 \[ \frac {-\text {Li}_2\left (-e^{a+b x}\right )+\text {Li}_2\left (e^{a+b x}\right )+b x \left (\log \left (1-e^{a+b x}\right )-\log \left (e^{a+b x}+1\right )+2 \tanh ^{-1}\left (e^{a+b x}\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 138, normalized size = 3.94 \[ \frac {b x \log \left (-\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1}\right ) - b x \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b x + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {artanh}\left (e^{\left (b x + a\right )}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 67, normalized size = 1.91 \[ \frac {\ln \left ({\mathrm e}^{b x +a}\right ) \arctanh \left ({\mathrm e}^{b x +a}\right )}{b}-\frac {\dilog \left ({\mathrm e}^{b x +a}\right )}{2 b}-\frac {\dilog \left ({\mathrm e}^{b x +a}+1\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 107, normalized size = 3.06 \[ \frac {{\left (b x + a\right )} \operatorname {artanh}\left (e^{\left (b x + a\right )}\right )}{b} - \frac {{\left (b x + a\right )} {\left (\log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left (e^{\left (b x + a\right )} - 1\right )\right )} - \log \left (-e^{\left (b x + a\right )}\right ) \log \left (e^{\left (b x + a\right )} + 1\right ) + {\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - {\rm Li}_2\left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )} + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {atanh}\left ({\mathrm {e}}^{a+b\,x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {atanh}{\left (e^{a + b x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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