Optimal. Leaf size=60 \[ -\frac{1}{2} b d \text{PolyLog}(2,-c x)+\frac{1}{2} b d \text{PolyLog}(2,c x)+a c d x+a d \log (x)+\frac{1}{2} b d \log \left (1-c^2 x^2\right )+b c d x \tanh ^{-1}(c x) \]
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Rubi [A] time = 0.0693332, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5940, 5910, 260, 5912} \[ -\frac{1}{2} b d \text{PolyLog}(2,-c x)+\frac{1}{2} b d \text{PolyLog}(2,c x)+a c d x+a d \log (x)+\frac{1}{2} b d \log \left (1-c^2 x^2\right )+b c d x \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5910
Rule 260
Rule 5912
Rubi steps
\begin{align*} \int \frac{(d+c d x) \left (a+b \tanh ^{-1}(c x)\right )}{x} \, dx &=\int \left (c d \left (a+b \tanh ^{-1}(c x)\right )+\frac{d \left (a+b \tanh ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx+(c d) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=a c d x+a d \log (x)-\frac{1}{2} b d \text{Li}_2(-c x)+\frac{1}{2} b d \text{Li}_2(c x)+(b c d) \int \tanh ^{-1}(c x) \, dx\\ &=a c d x+b c d x \tanh ^{-1}(c x)+a d \log (x)-\frac{1}{2} b d \text{Li}_2(-c x)+\frac{1}{2} b d \text{Li}_2(c x)-\left (b c^2 d\right ) \int \frac{x}{1-c^2 x^2} \, dx\\ &=a c d x+b c d x \tanh ^{-1}(c x)+a d \log (x)+\frac{1}{2} b d \log \left (1-c^2 x^2\right )-\frac{1}{2} b d \text{Li}_2(-c x)+\frac{1}{2} b d \text{Li}_2(c x)\\ \end{align*}
Mathematica [A] time = 0.0754164, size = 54, normalized size = 0.9 \[ \frac{1}{2} d \left (-b \text{PolyLog}(2,-c x)+b \text{PolyLog}(2,c x)+2 a c x+2 a \log (x)+b \log \left (1-c^2 x^2\right )+2 b c x \tanh ^{-1}(c x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 86, normalized size = 1.4 \begin{align*} da\ln \left ( cx \right ) +acdx+db{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) +bcdx{\it Artanh} \left ( cx \right ) +{\frac{db\ln \left ( cx-1 \right ) }{2}}+{\frac{db\ln \left ( cx+1 \right ) }{2}}-{\frac{db{\it dilog} \left ( cx \right ) }{2}}-{\frac{db{\it dilog} \left ( cx+1 \right ) }{2}}-{\frac{db\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{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 c d x + \frac{1}{2} \,{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d + \frac{1}{2} \, b d \int \frac{\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{x}\,{d x} + a d \log \left (x\right ) \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{a c d x + a d +{\left (b c d x + b d\right )} \operatorname{artanh}\left (c x\right )}{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*} d \left (\int a c\, dx + \int \frac{a}{x}\, dx + \int b c \operatorname{atanh}{\left (c x \right )}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{x}\, dx\right ) \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{{\left (c d x + d\right )}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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