Optimal. Leaf size=41 \[ \frac {\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)}{c}-\frac {\text {Li}_2\left (\frac {2}{a x+1}-1\right )}{2 c} \]
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Rubi [A] time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1593, 5932, 2447} \[ \frac {\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)}{c}-\frac {\text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 2447
Rule 5932
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{c x+a c x^2} \, dx &=\int \frac {\tanh ^{-1}(a x)}{x (c+a c x)} \, dx\\ &=\frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {a \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 39, normalized size = 0.95 \[ \frac {\tanh ^{-1}(a x) \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )}{c}-\frac {\text {Li}_2\left (e^{-2 \tanh ^{-1}(a x)}\right )}{2 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (a x\right )}{a c x^{2} + c x}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (a x\right )}{a c x^{2} + c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 126, normalized size = 3.07 \[ \frac {\arctanh \left (a x \right ) \ln \left (a x \right )}{c}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{c}+\frac {\ln \left (a x +1\right )^{2}}{4 c}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{2 c}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{2 c}+\frac {\dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{2 c}-\frac {\dilog \left (a x \right )}{2 c}-\frac {\dilog \left (a x +1\right )}{2 c}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 120, normalized size = 2.93 \[ \frac {1}{4} \, a {\left (\frac {\log \left (a x + 1\right )^{2}}{a c} - \frac {2 \, {\left (\log \left (a x + 1\right ) \log \left (-\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a c} - \frac {2 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )}}{a c} + \frac {2 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )}}{a c}\right )} - {\left (\frac {\log \left (a x + 1\right )}{c} - \frac {\log \relax (x)}{c}\right )} \operatorname {artanh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )}{a\,c\,x^2+c\,x} \,d x \]
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 \[ \frac {\int \frac {\operatorname {atanh}{\left (a x \right )}}{a x^{2} + x}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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