Optimal. Leaf size=30 \[ a \log (x)+\frac {1}{2} b \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{2} b \text {Li}_2\left (\frac {c}{x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6095, 5912} \[ \frac {1}{2} b \text {PolyLog}\left (2,-\frac {c}{x}\right )-\frac {1}{2} b \text {PolyLog}\left (2,\frac {c}{x}\right )+a \log (x) \]
Antiderivative was successfully verified.
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Rule 5912
Rule 6095
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (\frac {c}{x}\right )}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a \log (x)+\frac {1}{2} b \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{2} b \text {Li}_2\left (\frac {c}{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.93 \[ a \log (x)+\frac {1}{2} b \left (\text {Li}_2\left (-\frac {c}{x}\right )-\text {Li}_2\left (\frac {c}{x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (\frac {c}{x}\right ) + a}{x}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (\frac {c}{x}\right ) + a}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 63, normalized size = 2.10 \[ -a \ln \left (\frac {c}{x}\right )-b \ln \left (\frac {c}{x}\right ) \arctanh \left (\frac {c}{x}\right )+\frac {b \dilog \left (\frac {c}{x}\right )}{2}+\frac {b \dilog \left (1+\frac {c}{x}\right )}{2}+\frac {b \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2} \]
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 \[ \frac {1}{2} \, b \int \frac {\log \left (\frac {c}{x} + 1\right ) - \log \left (-\frac {c}{x} + 1\right )}{x}\,{d x} + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )}{x} \,d x \]
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 \[ \int \frac {a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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