Optimal. Leaf size=51 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 c d}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c d} \]
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Rubi [A] time = 0.0477698, antiderivative size = 51, normalized size of antiderivative = 1., 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 = {5918, 2402, 2315} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 c d}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c d} \]
Antiderivative was successfully verified.
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Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}(c x)}{d+c d x} \, dx &=-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c d}+\frac{b \int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c d}+\frac{b \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c x}\right )}{c d}\\ &=-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c d}+\frac{b \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 c d}\\ \end{align*}
Mathematica [A] time = 0.0973929, size = 52, normalized size = 1.02 \[ \frac{b \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+2 a \log (c x+1)-2 b \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )}{2 c d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.039, size = 112, normalized size = 2.2 \begin{align*}{\frac{a\ln \left ( cx+1 \right ) }{cd}}+{\frac{b{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{cd}}+{\frac{b\ln \left ( cx+1 \right ) }{2\,cd}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }-{\frac{b}{2\,cd}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{b}{2\,cd}{\it dilog} \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{b \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{4\,cd}} \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*} \frac{1}{2} \,{\left (2 \, c \int \frac{x \log \left (c x + 1\right )}{c^{2} d x^{2} - d}\,{d x} - \frac{\log \left (c x + 1\right ) \log \left (-c x + 1\right )}{c d}\right )} b + \frac{a \log \left (c d x + d\right )}{c d} \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 + d}, 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 + 1}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{c x + 1}\, 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}{c d x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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