Optimal. Leaf size=518 \[ -\frac{n \text{PolyLog}\left (2,-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f}-\frac{n \text{PolyLog}\left (2,\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )}{2 f}+\frac{n \text{PolyLog}\left (2,\frac{b x}{a}+1\right )}{f}+\frac{n \text{PolyLog}\left (2,-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f}+\frac{n \text{PolyLog}\left (2,\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )}{2 f}-\frac{n \text{PolyLog}\left (2,\frac{d x}{c}+1\right )}{f}+\frac{\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 f}-\frac{\log (x) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{f}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )}{2 f}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f}+\frac{n \log \left (-\frac{b x}{a}\right ) \log (a+b x)}{f}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )}{2 f}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f}-\frac{n \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{f} \]
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Rubi [A] time = 0.602265, antiderivative size = 518, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2513, 266, 36, 29, 31, 2416, 2394, 2315, 260, 2393, 2391} \[ -\frac{n \text{PolyLog}\left (2,-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f}-\frac{n \text{PolyLog}\left (2,\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )}{2 f}+\frac{n \text{PolyLog}\left (2,\frac{b x}{a}+1\right )}{f}+\frac{n \text{PolyLog}\left (2,-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f}+\frac{n \text{PolyLog}\left (2,\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )}{2 f}-\frac{n \text{PolyLog}\left (2,\frac{d x}{c}+1\right )}{f}+\frac{\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 f}-\frac{\log (x) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{f}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )}{2 f}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f}+\frac{n \log \left (-\frac{b x}{a}\right ) \log (a+b x)}{f}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )}{2 f}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f}-\frac{n \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{f} \]
Antiderivative was successfully verified.
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Rule 2513
Rule 266
Rule 36
Rule 29
Rule 31
Rule 2416
Rule 2394
Rule 2315
Rule 260
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{x \left (f-g x^2\right )} \, dx &=n \int \frac{\log (a+b x)}{x \left (f-g x^2\right )} \, dx-n \int \frac{\log (c+d x)}{x \left (f-g x^2\right )} \, dx-\left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac{1}{x \left (f-g x^2\right )} \, dx\\ &=n \int \left (\frac{\log (a+b x)}{f x}-\frac{g x \log (a+b x)}{f \left (-f+g x^2\right )}\right ) \, dx-n \int \left (\frac{\log (c+d x)}{f x}-\frac{g x \log (c+d x)}{f \left (-f+g x^2\right )}\right ) \, dx-\frac{1}{2} \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{x (f-g x)} \, dx,x,x^2\right )\\ &=\frac{n \int \frac{\log (a+b x)}{x} \, dx}{f}-\frac{n \int \frac{\log (c+d x)}{x} \, dx}{f}-\frac{(g n) \int \frac{x \log (a+b x)}{-f+g x^2} \, dx}{f}+\frac{(g n) \int \frac{x \log (c+d x)}{-f+g x^2} \, dx}{f}-\frac{\left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 f}-\frac{\left (g \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{f-g x} \, dx,x,x^2\right )}{2 f}\\ &=\frac{n \log \left (-\frac{b x}{a}\right ) \log (a+b x)}{f}-\frac{n \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{f}-\frac{\log (x) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f}+\frac{\left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 f}-\frac{(b n) \int \frac{\log \left (-\frac{b x}{a}\right )}{a+b x} \, dx}{f}+\frac{(d n) \int \frac{\log \left (-\frac{d x}{c}\right )}{c+d x} \, dx}{f}-\frac{(g n) \int \left (-\frac{\log (a+b x)}{2 \sqrt{g} \left (\sqrt{f}-\sqrt{g} x\right )}+\frac{\log (a+b x)}{2 \sqrt{g} \left (\sqrt{f}+\sqrt{g} x\right )}\right ) \, dx}{f}+\frac{(g n) \int \left (-\frac{\log (c+d x)}{2 \sqrt{g} \left (\sqrt{f}-\sqrt{g} x\right )}+\frac{\log (c+d x)}{2 \sqrt{g} \left (\sqrt{f}+\sqrt{g} x\right )}\right ) \, dx}{f}\\ &=\frac{n \log \left (-\frac{b x}{a}\right ) \log (a+b x)}{f}-\frac{n \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{f}-\frac{\log (x) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f}+\frac{\left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 f}+\frac{n \text{Li}_2\left (1+\frac{b x}{a}\right )}{f}-\frac{n \text{Li}_2\left (1+\frac{d x}{c}\right )}{f}+\frac{\left (\sqrt{g} n\right ) \int \frac{\log (a+b x)}{\sqrt{f}-\sqrt{g} x} \, dx}{2 f}-\frac{\left (\sqrt{g} n\right ) \int \frac{\log (a+b x)}{\sqrt{f}+\sqrt{g} x} \, dx}{2 f}-\frac{\left (\sqrt{g} n\right ) \int \frac{\log (c+d x)}{\sqrt{f}-\sqrt{g} x} \, dx}{2 f}+\frac{\left (\sqrt{g} n\right ) \int \frac{\log (c+d x)}{\sqrt{f}+\sqrt{g} x} \, dx}{2 f}\\ &=\frac{n \log \left (-\frac{b x}{a}\right ) \log (a+b x)}{f}-\frac{n \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{f}-\frac{\log (x) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{2 f}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{2 f}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f}+\frac{\left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 f}+\frac{n \text{Li}_2\left (1+\frac{b x}{a}\right )}{f}-\frac{n \text{Li}_2\left (1+\frac{d x}{c}\right )}{f}+\frac{(b n) \int \frac{\log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{a+b x} \, dx}{2 f}+\frac{(b n) \int \frac{\log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{a+b x} \, dx}{2 f}-\frac{(d n) \int \frac{\log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{c+d x} \, dx}{2 f}-\frac{(d n) \int \frac{\log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{c+d x} \, dx}{2 f}\\ &=\frac{n \log \left (-\frac{b x}{a}\right ) \log (a+b x)}{f}-\frac{n \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{f}-\frac{\log (x) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{2 f}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{2 f}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f}+\frac{\left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 f}+\frac{n \text{Li}_2\left (1+\frac{b x}{a}\right )}{f}-\frac{n \text{Li}_2\left (1+\frac{d x}{c}\right )}{f}+\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{b \sqrt{f}-a \sqrt{g}}\right )}{x} \, dx,x,a+b x\right )}{2 f}+\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{b \sqrt{f}+a \sqrt{g}}\right )}{x} \, dx,x,a+b x\right )}{2 f}-\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{d \sqrt{f}-c \sqrt{g}}\right )}{x} \, dx,x,c+d x\right )}{2 f}-\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{d \sqrt{f}+c \sqrt{g}}\right )}{x} \, dx,x,c+d x\right )}{2 f}\\ &=\frac{n \log \left (-\frac{b x}{a}\right ) \log (a+b x)}{f}-\frac{n \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{f}-\frac{\log (x) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{2 f}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{2 f}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f}+\frac{\left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 f}-\frac{n \text{Li}_2\left (-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f}-\frac{n \text{Li}_2\left (\frac{\sqrt{g} (a+b x)}{b \sqrt{f}+a \sqrt{g}}\right )}{2 f}+\frac{n \text{Li}_2\left (1+\frac{b x}{a}\right )}{f}+\frac{n \text{Li}_2\left (-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f}+\frac{n \text{Li}_2\left (\frac{\sqrt{g} (c+d x)}{d \sqrt{f}+c \sqrt{g}}\right )}{2 f}-\frac{n \text{Li}_2\left (1+\frac{d x}{c}\right )}{f}\\ \end{align*}
Mathematica [A] time = 0.186835, size = 487, normalized size = 0.94 \[ -\frac{-n \text{PolyLog}\left (2,\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )-n \text{PolyLog}\left (2,\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )+2 n \text{PolyLog}\left (2,-\frac{b x}{a}\right )+n \text{PolyLog}\left (2,\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )+n \text{PolyLog}\left (2,\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )-2 n \text{PolyLog}\left (2,-\frac{d x}{c}\right )+\log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 \log (x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )-n \log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )+2 n \log (x) \log \left (\frac{b x}{a}+1\right )+n \log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )+n \log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )-2 n \log (x) \log \left (\frac{d x}{c}+1\right )}{2 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.457, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( -g{x}^{2}+f \right ) }\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) }\, dx \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*} -\int \frac{\log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{{\left (g x^{2} - f\right )} 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{\log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{g x^{3} - f x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{{\left (g x^{2} - f\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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