Optimal. Leaf size=130 \[ -\frac{b \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac{b \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right )}{2 f}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac{\log \left (\frac{2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{f} \]
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Rubi [A] time = 0.136252, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6111, 5920, 2402, 2315, 2447} \[ -\frac{b \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac{b \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right )}{2 f}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac{\log \left (\frac{2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{f} \]
Antiderivative was successfully verified.
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Rule 6111
Rule 5920
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}(c+d x)}{e+f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(x)}{\frac{d e-c f}{d}+\frac{f x}{d}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac{2}{1+c+d x}\right )}{f}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+x}\right )}{1-x^2} \, dx,x,c+d x\right )}{f}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (\frac{2 \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )}{\left (\frac{f}{d}+\frac{d e-c f}{d}\right ) (1+x)}\right )}{1-x^2} \, dx,x,c+d x\right )}{f}\\ &=-\frac{\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac{2}{1+c+d x}\right )}{f}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}-\frac{b \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}+\frac{b \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c+d x}\right )}{f}\\ &=-\frac{\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac{2}{1+c+d x}\right )}{f}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac{b \text{Li}_2\left (1-\frac{2}{1+c+d x}\right )}{2 f}-\frac{b \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}\\ \end{align*}
Mathematica [A] time = 0.0730369, size = 126, normalized size = 0.97 \[ \frac{-b \text{PolyLog}\left (2,\frac{f (c+d x-1)}{(c-1) f-d e}\right )+b \text{PolyLog}\left (2,\frac{f (c+d x+1)}{c f-d e+f}\right )+2 a \log (e+f x)-b \log (-c-d x+1) \log \left (\frac{d (e+f x)}{-c f+d e+f}\right )+b \log (c+d x+1) \log \left (\frac{d (e+f x)}{d e-(c+1) f}\right )}{2 f} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.211, size = 202, normalized size = 1.6 \begin{align*}{\frac{a\ln \left ( \left ( dx+c \right ) f-cf+de \right ) }{f}}+{\frac{b\ln \left ( \left ( dx+c \right ) f-cf+de \right ){\it Artanh} \left ( dx+c \right ) }{f}}-{\frac{b\ln \left ( \left ( dx+c \right ) f-cf+de \right ) }{2\,f}\ln \left ({\frac{ \left ( dx+c \right ) f+f}{cf-de+f}} \right ) }-{\frac{b}{2\,f}{\it dilog} \left ({\frac{ \left ( dx+c \right ) f+f}{cf-de+f}} \right ) }+{\frac{b\ln \left ( \left ( dx+c \right ) f-cf+de \right ) }{2\,f}\ln \left ({\frac{ \left ( dx+c \right ) f-f}{cf-de-f}} \right ) }+{\frac{b}{2\,f}{\it dilog} \left ({\frac{ \left ( dx+c \right ) f-f}{cf-de-f}} \right ) } \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} \, b \int \frac{\log \left (d x + c + 1\right ) - \log \left (-d x - c + 1\right )}{f x + e}\,{d x} + \frac{a \log \left (f x + e\right )}{f} \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 (d x + c\right ) + a}{f x + e}, 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*} \int \frac{a + b \operatorname{atanh}{\left (c + d x \right )}}{e + f x}\, dx \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 (d x + c\right ) + a}{f x + e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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