Integrand size = 32, antiderivative size = 596 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2 \left (f-g x^2\right )} \, dx=\frac {b n \log (x)}{a f}-\frac {d n \log (x)}{c f}-\frac {b n \log (a+b x)}{a f}-\frac {n \log (a+b x)}{f x}+\frac {d n \log (c+d x)}{c f}+\frac {n \log (c+d x)}{f x}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac {\sqrt {g} \text {arctanh}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^{3/2}}-\frac {\sqrt {g} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f^{3/2}}-\frac {\sqrt {g} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f^{3/2}}-\frac {\sqrt {g} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )}{2 f^{3/2}}-\frac {\sqrt {g} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )}{2 f^{3/2}} \]
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Time = 0.43 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2593, 331, 214, 2463, 2442, 36, 29, 31, 2456, 2441, 2440, 2438} \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2 \left (f-g x^2\right )} \, dx=-\frac {\sqrt {g} \text {arctanh}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{f^{3/2}}+\frac {-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)}{f x}+\frac {\sqrt {g} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f^{3/2}}-\frac {\sqrt {g} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{\sqrt {g} a+b \sqrt {f}}\right )}{2 f^{3/2}}-\frac {\sqrt {g} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{a \sqrt {g}+b \sqrt {f}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f^{3/2}}+\frac {b n \log (x)}{a f}-\frac {b n \log (a+b x)}{a f}-\frac {n \log (a+b x)}{f x}-\frac {\sqrt {g} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{\sqrt {g} c+d \sqrt {f}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{c \sqrt {g}+d \sqrt {f}}\right )}{2 f^{3/2}}-\frac {\sqrt {g} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f^{3/2}}-\frac {d n \log (x)}{c f}+\frac {d n \log (c+d x)}{c f}+\frac {n \log (c+d x)}{f x} \]
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Rule 29
Rule 31
Rule 36
Rule 214
Rule 331
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2456
Rule 2463
Rule 2593
Rubi steps \begin{align*} \text {integral}& = n \int \frac {\log (a+b x)}{x^2 \left (f-g x^2\right )} \, dx-n \int \frac {\log (c+d x)}{x^2 \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^2 \left (f-g x^2\right )} \, dx \\ & = \frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}+n \int \left (\frac {\log (a+b x)}{f x^2}+\frac {g \log (a+b x)}{f \left (f-g x^2\right )}\right ) \, dx-n \int \left (\frac {\log (c+d x)}{f x^2}+\frac {g \log (c+d x)}{f \left (f-g x^2\right )}\right ) \, dx-\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 ) \int \frac {1}{f-g x^2} \, dx}{f} \\ & = \frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac {\sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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^{3/2}}+\frac {n \int \frac {\log (a+b x)}{x^2} \, dx}{f}-\frac {n \int \frac {\log (c+d x)}{x^2} \, dx}{f}+\frac {(g n) \int \frac {\log (a+b x)}{f-g x^2} \, dx}{f}-\frac {(g n) \int \frac {\log (c+d x)}{f-g x^2} \, dx}{f} \\ & = -\frac {n \log (a+b x)}{f x}+\frac {n \log (c+d x)}{f x}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac {\sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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^{3/2}}+\frac {(b n) \int \frac {1}{x (a+b x)} \, dx}{f}-\frac {(d n) \int \frac {1}{x (c+d x)} \, dx}{f}+\frac {(g n) \int \left (\frac {\log (a+b x)}{2 \sqrt {f} \left (\sqrt {f}-\sqrt {g} x\right )}+\frac {\log (a+b x)}{2 \sqrt {f} \left (\sqrt {f}+\sqrt {g} x\right )}\right ) \, dx}{f}-\frac {(g n) \int \left (\frac {\log (c+d x)}{2 \sqrt {f} \left (\sqrt {f}-\sqrt {g} x\right )}+\frac {\log (c+d x)}{2 \sqrt {f} \left (\sqrt {f}+\sqrt {g} x\right )}\right ) \, dx}{f} \\ & = -\frac {n \log (a+b x)}{f x}+\frac {n \log (c+d x)}{f x}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac {\sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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^{3/2}}+\frac {(b n) \int \frac {1}{x} \, dx}{a f}-\frac {\left (b^2 n\right ) \int \frac {1}{a+b x} \, dx}{a f}-\frac {(d n) \int \frac {1}{x} \, dx}{c f}+\frac {\left (d^2 n\right ) \int \frac {1}{c+d x} \, dx}{c f}+\frac {(g n) \int \frac {\log (a+b x)}{\sqrt {f}-\sqrt {g} x} \, dx}{2 f^{3/2}}+\frac {(g n) \int \frac {\log (a+b x)}{\sqrt {f}+\sqrt {g} x} \, dx}{2 f^{3/2}}-\frac {(g n) \int \frac {\log (c+d x)}{\sqrt {f}-\sqrt {g} x} \, dx}{2 f^{3/2}}-\frac {(g n) \int \frac {\log (c+d x)}{\sqrt {f}+\sqrt {g} x} \, dx}{2 f^{3/2}} \\ & = \frac {b n \log (x)}{a f}-\frac {d n \log (x)}{c f}-\frac {b n \log (a+b x)}{a f}-\frac {n \log (a+b x)}{f x}+\frac {d n \log (c+d x)}{c f}+\frac {n \log (c+d x)}{f x}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac {\sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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^{3/2}}-\frac {\sqrt {g} 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^{3/2}}+\frac {\sqrt {g} 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^{3/2}}+\frac {\sqrt {g} 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^{3/2}}-\frac {\sqrt {g} 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^{3/2}}+\frac {\left (b \sqrt {g} n\right ) \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^{3/2}}-\frac {\left (b \sqrt {g} n\right ) \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^{3/2}}-\frac {\left (d \sqrt {g} n\right ) \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^{3/2}}+\frac {\left (d \sqrt {g} n\right ) \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^{3/2}} \\ & = \frac {b n \log (x)}{a f}-\frac {d n \log (x)}{c f}-\frac {b n \log (a+b x)}{a f}-\frac {n \log (a+b x)}{f x}+\frac {d n \log (c+d x)}{c f}+\frac {n \log (c+d x)}{f x}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac {\sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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^{3/2}}-\frac {\sqrt {g} 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^{3/2}}+\frac {\sqrt {g} 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^{3/2}}+\frac {\sqrt {g} 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^{3/2}}-\frac {\sqrt {g} 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^{3/2}}-\frac {\left (\sqrt {g} n\right ) \text {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^{3/2}}+\frac {\left (\sqrt {g} n\right ) \text {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^{3/2}}+\frac {\left (\sqrt {g} n\right ) \text {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^{3/2}}-\frac {\left (\sqrt {g} n\right ) \text {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^{3/2}} \\ & = \frac {b n \log (x)}{a f}-\frac {d n \log (x)}{c f}-\frac {b n \log (a+b x)}{a f}-\frac {n \log (a+b x)}{f x}+\frac {d n \log (c+d x)}{c f}+\frac {n \log (c+d x)}{f x}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac {\sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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^{3/2}}-\frac {\sqrt {g} 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^{3/2}}+\frac {\sqrt {g} 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^{3/2}}+\frac {\sqrt {g} 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^{3/2}}-\frac {\sqrt {g} 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^{3/2}}+\frac {\sqrt {g} n \text {Li}_2\left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f^{3/2}}-\frac {\sqrt {g} n \text {Li}_2\left (\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )}{2 f^{3/2}}-\frac {\sqrt {g} n \text {Li}_2\left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \text {Li}_2\left (\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )}{2 f^{3/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 479, normalized size of antiderivative = 0.80 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2 \left (f-g x^2\right )} \, dx=\frac {-\frac {2 \sqrt {f} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x}+\frac {2 \sqrt {f} n ((b c-a d) \log (x)-b c \log (a+b x)+a d \log (c+d x))}{a c}-\sqrt {g} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )+\sqrt {g} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+\sqrt {g} n \left (\left (\log \left (\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )-\log \left (\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )\right )-\sqrt {g} n \left (\left (\log \left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )-\log \left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )\right )}{2 f^{3/2}} \]
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\[\int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{x^{2} \left (-g \,x^{2}+f \right )}d x\]
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\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2 \left (f-g x^2\right )} \, dx=\int { -\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (g x^{2} - f\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2 \left (f-g x^2\right )} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (484) = 968\).
Time = 0.40 (sec) , antiderivative size = 969, normalized size of antiderivative = 1.63 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2 \left (f-g x^2\right )} \, dx=\frac {1}{2} \, {\left (2 \, a c d {\left (\frac {b^{2} \log \left (b x + a\right )}{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} f} + \frac {d}{{\left (b c^{2} d - a c d^{2}\right )} f x + {\left (b c^{3} - a c^{2} d\right )} f} - \frac {{\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} f} - \frac {\log \left (x\right )}{a c^{2} f}\right )} + 2 \, b d^{2} {\left (\frac {c}{{\left (b c d^{2} - a d^{3}\right )} f x + {\left (b c^{2} d - a c d^{2}\right )} f} + \frac {a \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f} - \frac {a \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f}\right )} - 2 \, b c d {\left (\frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f} + \frac {1}{{\left (b c d - a d^{2}\right )} f x + {\left (b c^{2} - a c d\right )} f}\right )} - 2 \, a d^{2} {\left (\frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f} + \frac {1}{{\left (b c d - a d^{2}\right )} f x + {\left (b c^{2} - a c d\right )} f}\right )} - 2 \, b c {\left (\frac {b \log \left (b x + a\right )}{{\left (a b c - a^{2} d\right )} f} - \frac {d \log \left (d x + c\right )}{{\left (b c^{2} - a c d\right )} f} - \frac {\log \left (x\right )}{a c f}\right )} + 2 \, b d {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} f} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} f}\right )} + \frac {{\left (\log \left (\sqrt {g} x - \sqrt {f}\right ) \log \left (\frac {b \sqrt {g} x - b \sqrt {f}}{b \sqrt {f} + a \sqrt {g}} + 1\right ) + {\rm Li}_2\left (-\frac {b \sqrt {g} x - b \sqrt {f}}{b \sqrt {f} + a \sqrt {g}}\right )\right )} \sqrt {g}}{f^{\frac {3}{2}}} - \frac {{\left (\log \left (\sqrt {g} x + \sqrt {f}\right ) \log \left (-\frac {b \sqrt {g} x + b \sqrt {f}}{b \sqrt {f} - a \sqrt {g}} + 1\right ) + {\rm Li}_2\left (\frac {b \sqrt {g} x + b \sqrt {f}}{b \sqrt {f} - a \sqrt {g}}\right )\right )} \sqrt {g}}{f^{\frac {3}{2}}} - \frac {{\left (\log \left (\sqrt {g} x - \sqrt {f}\right ) \log \left (\frac {d \sqrt {g} x - d \sqrt {f}}{d \sqrt {f} + c \sqrt {g}} + 1\right ) + {\rm Li}_2\left (-\frac {d \sqrt {g} x - d \sqrt {f}}{d \sqrt {f} + c \sqrt {g}}\right )\right )} \sqrt {g}}{f^{\frac {3}{2}}} + \frac {{\left (\log \left (\sqrt {g} x + \sqrt {f}\right ) \log \left (-\frac {d \sqrt {g} x + d \sqrt {f}}{d \sqrt {f} - c \sqrt {g}} + 1\right ) + {\rm Li}_2\left (\frac {d \sqrt {g} x + d \sqrt {f}}{d \sqrt {f} - c \sqrt {g}}\right )\right )} \sqrt {g}}{f^{\frac {3}{2}}}\right )} n - \frac {1}{2} \, {\left (\frac {g \log \left (\frac {g x - \sqrt {f g}}{g x + \sqrt {f g}}\right )}{\sqrt {f g} f} + \frac {2}{f x}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) \]
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\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2 \left (f-g x^2\right )} \, dx=\int { -\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (g x^{2} - f\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2 \left (f-g x^2\right )} \, dx=\int \frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{x^2\,\left (f-g\,x^2\right )} \,d x \]
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