Optimal. Leaf size=46 \[ \frac{\log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac{b \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )}{2 d} \]
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Rubi [A] time = 0.0735335, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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 5932
Rule 2447
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*}
Mathematica [A] time = 0.110306, size = 55, normalized size = 1.2 \[ \frac{-b \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )-2 a \log (c x+1)+2 a \log (x)+2 b \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.046, size = 156, normalized size = 3.4 \begin{align*}{\frac{a\ln \left ( cx \right ) }{d}}-{\frac{a\ln \left ( cx+1 \right ) }{d}}-{\frac{b{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{d}}+{\frac{b{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) }{d}}+{\frac{b}{2\,d}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{b\ln \left ( cx+1 \right ) }{2\,d}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }+{\frac{b}{2\,d}{\it dilog} \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }+{\frac{b \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{4\,d}}-{\frac{b{\it dilog} \left ( cx \right ) }{2\,d}}-{\frac{b{\it dilog} \left ( cx+1 \right ) }{2\,d}}-{\frac{b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -a{\left (\frac{\log \left (c x + 1\right )}{d} - \frac{\log \left (x\right )}{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{artanh}\left (c x\right ) + a}{c d x^{2} + d x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c x^{2} + x}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{c x^{2} + x}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{artanh}\left (c x\right ) + a}{{\left (c d x + d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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