Integrand size = 32, antiderivative size = 518 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f-g x^2\right )} \, dx=\frac {n \log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f}-\frac {n \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f}-\frac {\log (x) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 f}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 f}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f}+\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 f}-\frac {n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f}-\frac {n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )}{2 f}+\frac {n \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )}{f}+\frac {n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f}+\frac {n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )}{2 f}-\frac {n \operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )}{f} \]
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Time = 0.44 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2593, 272, 36, 29, 31, 2463, 2441, 2352, 266, 2440, 2438} \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f-g x^2\right )} \, dx=\frac {\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 f}-\frac {\log (x) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{f}-\frac {n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f}-\frac {n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{\sqrt {g} a+b \sqrt {f}}\right )}{2 f}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{a \sqrt {g}+b \sqrt {f}}\right )}{2 f}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f}+\frac {n \operatorname {PolyLog}\left (2,\frac {b x}{a}+1\right )}{f}+\frac {n \log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f}+\frac {n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f}+\frac {n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{\sqrt {g} c+d \sqrt {f}}\right )}{2 f}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{c \sqrt {g}+d \sqrt {f}}\right )}{2 f}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f}-\frac {n \operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{f}-\frac {n \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f} \]
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 272
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2593
Rubi steps \begin{align*} \text {integral}& = 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 ) \text {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 ) \text {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 ) \text {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 \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}+\frac {n \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}-\frac {n \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}-\frac {n \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} \\ & = \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*}
Time = 0.15 (sec) , antiderivative size = 487, normalized size of antiderivative = 0.94 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f-g x^2\right )} \, dx=-\frac {2 n \log (x) \log \left (1+\frac {b x}{a}\right )-2 \log (x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 n \log (x) \log \left (1+\frac {d x}{c}\right )-n \log \left (\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )+n \log \left (\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )-n \log \left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+n \log \left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+2 n \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-2 n \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )-n \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )+n \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )-n \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )+n \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f} \]
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Time = 1.08 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.09
method | result | size |
parts | \(\frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) \ln \left (x \right )}{f}-\frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) \ln \left (-g \,x^{2}+f \right )}{2 f}-\frac {n \left (\frac {\left (a d -c b \right ) \left (\frac {\left (\frac {\ln \left (d x +c \right ) \ln \left (-g \,x^{2}+f \right )}{d}+\frac {2 g \left (-\frac {\ln \left (d x +c \right ) \left (\ln \left (\frac {d \sqrt {f g}-\left (d x +c \right ) g +c g}{d \sqrt {f g}+c g}\right )+\ln \left (\frac {d \sqrt {f g}+\left (d x +c \right ) g -c g}{d \sqrt {f g}-c g}\right )\right )}{2 g}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {f g}-\left (d x +c \right ) g +c g}{d \sqrt {f g}+c g}\right )+\operatorname {dilog}\left (\frac {d \sqrt {f g}+\left (d x +c \right ) g -c g}{d \sqrt {f g}-c g}\right )}{2 g}\right )}{d}\right ) d}{a d -c b}-\frac {\left (\frac {\ln \left (b x +a \right ) \ln \left (-g \,x^{2}+f \right )}{b}+\frac {2 g \left (-\frac {\ln \left (b x +a \right ) \left (\ln \left (\frac {b \sqrt {f g}-g \left (b x +a \right )+a g}{b \sqrt {f g}+a g}\right )+\ln \left (\frac {b \sqrt {f g}+g \left (b x +a \right )-a g}{b \sqrt {f g}-a g}\right )\right )}{2 g}-\frac {\operatorname {dilog}\left (\frac {b \sqrt {f g}-g \left (b x +a \right )+a g}{b \sqrt {f g}+a g}\right )+\operatorname {dilog}\left (\frac {b \sqrt {f g}+g \left (b x +a \right )-a g}{b \sqrt {f g}-a g}\right )}{2 g}\right )}{b}\right ) b}{a d -c b}\right )}{f}-\frac {2 \left (a d -c b \right ) \left (\frac {d \left (\frac {\operatorname {dilog}\left (\frac {d x +c}{c}\right )}{d}+\frac {\ln \left (x \right ) \ln \left (\frac {d x +c}{c}\right )}{d}\right )}{a d -c b}-\frac {b \left (\frac {\operatorname {dilog}\left (\frac {b x +a}{a}\right )}{b}+\frac {\ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{b}\right )}{a d -c b}\right )}{f}\right )}{2}\) | \(567\) |
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\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f-g x^2\right )} \, dx=\int { -\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (g x^{2} - f\right )} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f-g x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f-g x^2\right )} \, dx=\int { -\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (g x^{2} - f\right )} x} \,d x } \]
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\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f-g x^2\right )} \, dx=\int { -\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (g x^{2} - f\right )} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f-g x^2\right )} \, dx=\int \frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{x\,\left (f-g\,x^2\right )} \,d x \]
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