Optimal. Leaf size=124 \[ -\frac{b \text{PolyLog}(2,-c x)}{2 d^2}+\frac{b \text{PolyLog}(2,c x)}{2 d^2}-\frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{d^2 (c x+1)}+\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac{a \log (x)}{d^2}+\frac{b}{2 d^2 (c x+1)}-\frac{b \tanh ^{-1}(c x)}{2 d^2} \]
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Rubi [A] time = 0.174434, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {5940, 5912, 5926, 627, 44, 207, 5918, 2402, 2315} \[ -\frac{b \text{PolyLog}(2,-c x)}{2 d^2}+\frac{b \text{PolyLog}(2,c x)}{2 d^2}-\frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{d^2 (c x+1)}+\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac{a \log (x)}{d^2}+\frac{b}{2 d^2 (c x+1)}-\frac{b \tanh ^{-1}(c x)}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5912
Rule 5926
Rule 627
Rule 44
Rule 207
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}(c x)}{x (d+c d x)^2} \, dx &=\int \left (\frac{a+b \tanh ^{-1}(c x)}{d^2 x}-\frac{c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)^2}-\frac{c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx}{d^2}-\frac{c \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{d^2}-\frac{c \int \frac{a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{d^2}\\ &=\frac{a+b \tanh ^{-1}(c x)}{d^2 (1+c x)}+\frac{a \log (x)}{d^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{d^2}-\frac{b \text{Li}_2(-c x)}{2 d^2}+\frac{b \text{Li}_2(c x)}{2 d^2}-\frac{(b c) \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{d^2}-\frac{(b c) \int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d^2}\\ &=\frac{a+b \tanh ^{-1}(c x)}{d^2 (1+c x)}+\frac{a \log (x)}{d^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{d^2}-\frac{b \text{Li}_2(-c x)}{2 d^2}+\frac{b \text{Li}_2(c x)}{2 d^2}-\frac{b \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c x}\right )}{d^2}-\frac{(b c) \int \frac{1}{(1-c x) (1+c x)^2} \, dx}{d^2}\\ &=\frac{a+b \tanh ^{-1}(c x)}{d^2 (1+c x)}+\frac{a \log (x)}{d^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{d^2}-\frac{b \text{Li}_2(-c x)}{2 d^2}+\frac{b \text{Li}_2(c x)}{2 d^2}-\frac{b \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 d^2}-\frac{(b c) \int \left (\frac{1}{2 (1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{d^2}\\ &=\frac{b}{2 d^2 (1+c x)}+\frac{a+b \tanh ^{-1}(c x)}{d^2 (1+c x)}+\frac{a \log (x)}{d^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{d^2}-\frac{b \text{Li}_2(-c x)}{2 d^2}+\frac{b \text{Li}_2(c x)}{2 d^2}-\frac{b \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 d^2}+\frac{(b c) \int \frac{1}{-1+c^2 x^2} \, dx}{2 d^2}\\ &=\frac{b}{2 d^2 (1+c x)}-\frac{b \tanh ^{-1}(c x)}{2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{d^2 (1+c x)}+\frac{a \log (x)}{d^2}+\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{d^2}-\frac{b \text{Li}_2(-c x)}{2 d^2}+\frac{b \text{Li}_2(c x)}{2 d^2}-\frac{b \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 d^2}\\ \end{align*}
Mathematica [A] time = 0.407352, size = 101, normalized size = 0.81 \[ \frac{b \left (-2 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \left (2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )\right )\right )+\frac{4 a}{c x+1}-4 a \log (c x+1)+4 a \log (x)}{4 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.053, size = 221, normalized size = 1.8 \begin{align*}{\frac{a\ln \left ( cx \right ) }{{d}^{2}}}+{\frac{a}{{d}^{2} \left ( cx+1 \right ) }}-{\frac{a\ln \left ( cx+1 \right ) }{{d}^{2}}}+{\frac{b{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) }{{d}^{2}}}+{\frac{b{\it Artanh} \left ( cx \right ) }{{d}^{2} \left ( cx+1 \right ) }}-{\frac{b{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{{d}^{2}}}-{\frac{b\ln \left ( cx+1 \right ) }{2\,{d}^{2}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }+{\frac{b}{2\,{d}^{2}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }+{\frac{b}{2\,{d}^{2}}{\it dilog} \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }+{\frac{b \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{4\,{d}^{2}}}+{\frac{b\ln \left ( cx-1 \right ) }{4\,{d}^{2}}}+{\frac{b}{2\,{d}^{2} \left ( cx+1 \right ) }}-{\frac{b\ln \left ( cx+1 \right ) }{4\,{d}^{2}}}-{\frac{b{\it dilog} \left ( cx \right ) }{2\,{d}^{2}}}-{\frac{b{\it dilog} \left ( cx+1 \right ) }{2\,{d}^{2}}}-{\frac{b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{2\,{d}^{2}}} \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{1}{c d^{2} x + d^{2}} - \frac{\log \left (c x + 1\right )}{d^{2}} + \frac{\log \left (x\right )}{d^{2}}\right )} + \frac{1}{2} \, b \int \frac{\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{c^{2} d^{2} x^{3} + 2 \, c d^{2} x^{2} + d^{2} 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^{2} d^{2} x^{3} + 2 \, c d^{2} x^{2} + d^{2} 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^{2} x^{3} + 2 c x^{2} + x}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{c^{2} x^{3} + 2 c x^{2} + x}\, dx}{d^{2}} \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 )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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