Optimal. Leaf size=93 \[ \frac{b c \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )}{2 d}-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{c \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac{b c \log \left (1-c^2 x^2\right )}{2 d}+\frac{b c \log (x)}{d} \]
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Rubi [A] time = 0.153175, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5934, 5916, 266, 36, 29, 31, 5932, 2447} \[ \frac{b c \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )}{2 d}-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{c \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac{b c \log \left (1-c^2 x^2\right )}{2 d}+\frac{b c \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 5934
Rule 5916
Rule 266
Rule 36
Rule 29
Rule 31
Rule 5932
Rule 2447
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}(c x)}{x^2 (d+c d x)} \, dx &=-\left (c \int \frac{a+b \tanh ^{-1}(c x)}{x (d+c d x)} \, dx\right )+\frac{\int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx}{d}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{(b c) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx}{d}+\frac{\left (b c^2\right ) \int \frac{\log \left (2-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{b c \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{2 d}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{b c \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{2 d}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 d}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{d x}+\frac{b c \log (x)}{d}-\frac{b c \log \left (1-c^2 x^2\right )}{2 d}-\frac{c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{b c \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.191839, size = 93, normalized size = 1. \[ \frac{b c x \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )-2 \left (a c x \log (x)-a c x \log (c x+1)+a-b c x \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )+b \tanh ^{-1}(c x) \left (c x \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+1\right )\right )}{2 d x} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.053, size = 225, normalized size = 2.4 \begin{align*} -{\frac{a}{dx}}-{\frac{ac\ln \left ( cx \right ) }{d}}+{\frac{ac\ln \left ( cx+1 \right ) }{d}}-{\frac{b{\it Artanh} \left ( cx \right ) }{dx}}-{\frac{bc{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) }{d}}+{\frac{bc{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{d}}-{\frac{bc\ln \left ( cx-1 \right ) }{2\,d}}+{\frac{bc\ln \left ( cx \right ) }{d}}-{\frac{bc\ln \left ( cx+1 \right ) }{2\,d}}+{\frac{bc{\it dilog} \left ( cx \right ) }{2\,d}}+{\frac{bc{\it dilog} \left ( cx+1 \right ) }{2\,d}}+{\frac{bc\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{2\,d}}-{\frac{bc}{2\,d}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }+{\frac{bc\ln \left ( cx+1 \right ) }{2\,d}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }-{\frac{bc}{2\,d}{\it dilog} \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{bc \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{4\,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{c \log \left (c x + 1\right )}{d} - \frac{c \log \left (x\right )}{d} - \frac{1}{d x}\right )} + \frac{1}{2} \, b \int \frac{\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{c d x^{3} + d x^{2}}\,{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^{3} + d x^{2}}, 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^{3} + x^{2}}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{c x^{3} + x^{2}}\, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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