Optimal. Leaf size=46 \[ \frac {\log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {b \text {Li}_2\left (\frac {2}{c x+1}-1\right )}{2 d} \]
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Rubi [A] time = 0.07, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5932, 2447} \[ \frac {\log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {b \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2447
Rule 5932
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{x (d+c d x)} \, dx &=\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {(b c) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {b \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 55, normalized size = 1.20 \[ \frac {-2 a \log (c x+1)+2 a \log (x)-b \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )+2 b \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x\right ) + a}{c d x^{2} + d x}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 156, normalized size = 3.39 \[ \frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (c x +1\right )}{d}-\frac {b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d}+\frac {b \arctanh \left (c x \right ) \ln \left (c x \right )}{d}+\frac {b \ln \left (c x +1\right )^{2}}{4 d}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d}+\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 d}+\frac {b \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 d}-\frac {b \dilog \left (c x \right )}{2 d}-\frac {b \dilog \left (c x +1\right )}{2 d}-\frac {b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d} \]
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 \[ -a {\left (\frac {\log \left (c x + 1\right )}{d} - \frac {\log \relax (x)}{d}\right )} + \frac {1}{2} \, b \int \frac {\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{c d x^{2} + d x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x\,\left (d+c\,d\,x\right )} \,d x \]
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 \[ \frac {\int \frac {a}{c x^{2} + x}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c x^{2} + x}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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