Optimal. Leaf size=26 \[ a \log (x)-\frac {1}{2} b \text {Li}_2(-c x)+\frac {1}{2} b \text {Li}_2(c x) \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5912} \[ -\frac {1}{2} b \text {PolyLog}(2,-c x)+\frac {1}{2} b \text {PolyLog}(2,c x)+a \log (x) \]
Antiderivative was successfully verified.
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Rule 5912
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx &=a \log (x)-\frac {1}{2} b \text {Li}_2(-c x)+\frac {1}{2} b \text {Li}_2(c x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.92 \[ a \log (x)+\frac {1}{2} b (\text {Li}_2(c x)-\text {Li}_2(-c x)) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x\right ) + a}{x}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (c x\right ) + a}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 47, normalized size = 1.81 \[ a \ln \left (c x \right )+b \ln \left (c x \right ) \arctanh \left (c x \right )-\frac {b \dilog \left (c x \right )}{2}-\frac {b \dilog \left (c x +1\right )}{2}-\frac {b \ln \left (c x \right ) \ln \left (c x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, b \int \frac {\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{x}\,{d x} + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x} \,d x \]
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 \[ \int \frac {a + b \operatorname {atanh}{\left (c x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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