Optimal. Leaf size=137 \[ -\frac{2 B n \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d g}+\frac{2 B^2 n^2 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )}{d g}-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{d g} \]
[Out]
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Rubi [B] time = 3.30232, antiderivative size = 782, normalized size of antiderivative = 5.71, number of steps used = 45, number of rules used = 23, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.657, Rules used = {2524, 2528, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 12, 6742, 2499, 2396, 2433, 2374, 6589, 2302, 30, 2500, 2375, 2317, 2440, 2434} \[ -\frac{2 A B n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{2 B^2 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((a+b x)^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{d g}-\frac{2 B^2 n^2 \text{PolyLog}\left (3,-\frac{d (a+b x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{2 B^2 n \log \left ((a+b x)^n\right ) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n \log \left ((c+d x)^{-n}\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{\log (c g+d g x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{d g}-\frac{2 A B n \log (c g+d g x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d g}+\frac{B^2 n \log ^2(c g+d g x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d g}+\frac{2 B^2 n \log (c g+d g x) \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((a+b x)^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{d g}-\frac{B^2 n^2 \log (a+b x) \log ^2(g (c+d x))}{d g}+\frac{B^2 n^2 \log ^2(g (c+d x)) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d g}-\frac{B^2 n^2 \log ^2(c g+d g x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d g}-\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}-\frac{B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d g}+\frac{B^2 \log ^2\left ((c+d x)^{-n}\right ) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d g}+\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}+\frac{A B n \log ^2(g (c+d x))}{d g}+\frac{B^2 n^2 \log ^3(g (c+d x))}{3 d g} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2524
Rule 2528
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 6688
Rule 12
Rule 6742
Rule 2499
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 2302
Rule 30
Rule 2500
Rule 2375
Rule 2317
Rule 2440
Rule 2434
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{c g+d g x} \, dx &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}-\frac{(2 B n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c g+d g x)}{a+b x} \, dx}{d g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}-\frac{(2 B n) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c g+d g x)}{(a+b x) (c+d x)} \, dx}{d g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}-\frac{(2 B (b c-a d) n) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c g+d g x)}{(a+b x) (c+d x)} \, dx}{d g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}-\frac{(2 B (b c-a d) n) \int \left (\frac{d \left (-A-B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c g+d g x)}{(b c-a d) (c+d x)}+\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c g+d g x)}{(b c-a d) (a+b x)}\right ) \, dx}{d g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}-\frac{(2 B n) \int \frac{\left (-A-B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c g+d g x)}{c+d x} \, dx}{g}-\frac{(2 b B n) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c g+d g x)}{a+b x} \, dx}{d g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}-\frac{(2 B n) \int \left (\frac{A \log (c g+d g x)}{-c-d x}+\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c g+d g x)}{-c-d x}\right ) \, dx}{g}-\frac{(2 b B n) \int \left (\frac{A \log (c g+d g x)}{a+b x}+\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c g+d g x)}{a+b x}\right ) \, dx}{d g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}-\frac{(2 A B n) \int \frac{\log (c g+d g x)}{-c-d x} \, dx}{g}-\frac{\left (2 B^2 n\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c g+d g x)}{-c-d x} \, dx}{g}-\frac{(2 A b B n) \int \frac{\log (c g+d g x)}{a+b x} \, dx}{d g}-\frac{\left (2 b B^2 n\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c g+d g x)}{a+b x} \, dx}{d g}\\ &=-\frac{2 A B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+(2 A B n) \int \frac{\log \left (\frac{d g (a+b x)}{-b c g+a d g}\right )}{c g+d g x} \, dx+\frac{(2 A B n) \operatorname{Subst}\left (\int \frac{g \log (x)}{x} \, dx,x,c g+d g x\right )}{d g^2}-\frac{\left (2 b B^2 n\right ) \int \frac{\log \left ((a+b x)^n\right ) \log (c g+d g x)}{a+b x} \, dx}{d g}-\frac{\left (2 b B^2 n\right ) \int \frac{\log \left ((c+d x)^{-n}\right ) \log (c g+d g x)}{a+b x} \, dx}{d g}+\frac{\left (B^2 n^2\right ) \int \frac{\log ^2(c g+d g x)}{c+d x} \, dx}{g}-\frac{\left (b B^2 n^2\right ) \int \frac{\log ^2(c g+d g x)}{a+b x} \, dx}{d g}-\frac{\left (2 b B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac{\log (c g+d g x)}{a+b x} \, dx}{d g}\\ &=-\frac{2 A B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac{2 B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (c g+d g x)}{d g}-\frac{B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+\frac{(2 A B n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c g+d g x\right )}{d g}+\frac{(2 A B n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}-\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^n\right ) \log \left (\frac{b c g-a d g}{b}+\frac{d g x}{b}\right )}{x} \, dx,x,a+b x\right )}{d g}-\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (\left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )^{-n}\right ) \log \left (-\frac{-b c g+a d g}{b}+\frac{d g x}{b}\right )}{x} \, dx,x,a+b x\right )}{d g}+\left (2 B^2 n^2\right ) \int \frac{\log \left (\frac{d g (a+b x)}{-b c g+a d g}\right ) \log (c g+d g x)}{c g+d g x} \, dx+\frac{\left (B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{g \log ^2(x)}{x} \, dx,x,c g+d g x\right )}{d g^2}+\left (2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac{\log \left (\frac{d g (a+b x)}{-b c g+a d g}\right )}{c g+d g x} \, dx\\ &=-\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac{A B n \log ^2(g (c+d x))}{d g}-\frac{2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac{2 A B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac{2 B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (c g+d g x)}{d g}-\frac{B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}-\frac{2 A B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{B^2 \operatorname{Subst}\left (\int \frac{\log ^2\left (x^n\right )}{\frac{b c g-a d g}{b}+\frac{d g x}{b}} \, dx,x,a+b x\right )}{b}+\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )^{-n}\right )}{-\frac{-b c+a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{b g}-\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{-b c g+a d g}{b}+\frac{d g x}{b}\right )}{-\frac{-b c g+a d g}{b}+\frac{d g x}{b}} \, dx,x,a+b x\right )}{b}+\frac{\left (B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,c g+d g x\right )}{d g}+\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{d g \left (\frac{-b c g+a d g}{d g}+\frac{b x}{d g}\right )}{-b c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}+\frac{\left (2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}\\ &=\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac{A B n \log ^2(g (c+d x))}{d g}-\frac{2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac{2 A B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac{2 B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (c g+d g x)}{d g}-\frac{B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}-\frac{2 A B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n^2 \log (g (c+d x)) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{x} \, dx,x,c+d x\right )}{d g}-\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^n\right ) \log \left (1+\frac{d g x}{b c g-a d g}\right )}{x} \, dx,x,a+b x\right )}{d g}+\frac{\left (B^2 n^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (g (c+d x))\right )}{d g}-\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{-b c g+a d g}{d g}+\frac{b x}{d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}+\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-b c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}\\ &=\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac{A B n \log ^2(g (c+d x))}{d g}-\frac{B^2 n^2 \log (a+b x) \log ^2(g (c+d x))}{d g}+\frac{B^2 n^2 \log ^3(g (c+d x))}{3 d g}-\frac{2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac{B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d g}-\frac{2 A B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac{2 B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (c g+d g x)}{d g}-\frac{B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+\frac{2 B^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{d g}-\frac{2 A B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n^2 \log (g (c+d x)) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{2 B^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{\left (b B^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (x^{-n}\right )}{\frac{-b c+a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{d^2 g}+\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{-b c g+a d g}{d g}+\frac{b x}{d g}} \, dx,x,c g+d g x\right )}{d^2 g^2}-\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{d g x}{b c g-a d g}\right )}{x} \, dx,x,a+b x\right )}{d g}\\ &=\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac{A B n \log ^2(g (c+d x))}{d g}-\frac{B^2 n^2 \log (a+b x) \log ^2(g (c+d x))}{d g}+\frac{B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(g (c+d x))}{d g}+\frac{B^2 n^2 \log ^3(g (c+d x))}{3 d g}-\frac{2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac{B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{d g}-\frac{2 A B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac{2 B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (c g+d g x)}{d g}-\frac{B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+\frac{2 B^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{d g}-\frac{2 A B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n^2 \log (g (c+d x)) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{d g}+\frac{2 B^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d g}-\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1+\frac{b x}{-b c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}\\ &=\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac{A B n \log ^2(g (c+d x))}{d g}-\frac{B^2 n^2 \log (a+b x) \log ^2(g (c+d x))}{d g}+\frac{B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(g (c+d x))}{d g}+\frac{B^2 n^2 \log ^3(g (c+d x))}{3 d g}-\frac{2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac{B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{d g}-\frac{2 A B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac{2 B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (c g+d g x)}{d g}-\frac{B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+\frac{2 B^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{d g}-\frac{2 A B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{d g}+\frac{2 B^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d g}-\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-b c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}\\ &=\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac{A B n \log ^2(g (c+d x))}{d g}-\frac{B^2 n^2 \log (a+b x) \log ^2(g (c+d x))}{d g}+\frac{B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(g (c+d x))}{d g}+\frac{B^2 n^2 \log ^3(g (c+d x))}{3 d g}-\frac{2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac{B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{d g}-\frac{2 A B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac{2 B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (c g+d g x)}{d g}-\frac{B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+\frac{2 B^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{d g}-\frac{2 A B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}+\frac{2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{d g}-\frac{2 B^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{d g}\\ \end{align*}
Mathematica [B] time = 0.399526, size = 537, normalized size = 3.92 \[ \frac{-3 B n \left (-2 \left (\text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+\log \left (\frac{a}{b}+x\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )+2 \log (c+d x) \left (-\log \left (\frac{a+b x}{c+d x}\right )+\log \left (\frac{a}{b}+x\right )-\log \left (\frac{c}{d}+x\right )\right )+\log ^2\left (\frac{c}{d}+x\right )\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )+A\right )+B^2 n^2 \left (-6 \text{PolyLog}\left (3,\frac{d (a+b x)}{a d-b c}\right )-6 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )+3 \left (-\log \left (\frac{a+b x}{c+d x}\right )+\log \left (\frac{a}{b}+x\right )-\log \left (\frac{c}{d}+x\right )\right ) \left (\log ^2\left (\frac{c}{d}+x\right )-2 \left (\text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+\log \left (\frac{a}{b}+x\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )\right )+6 \log \left (\frac{c}{d}+x\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+6 \log \left (\frac{a}{b}+x\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+3 \log ^2\left (\frac{c}{d}+x\right ) \left (\log \left (\frac{d (a+b x)}{a d-b c}\right )-\log \left (\frac{a}{b}+x\right )\right )+3 \log ^2\left (\frac{a}{b}+x\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )+3 \log (c+d x) \left (\log \left (\frac{a+b x}{c+d x}\right )-\log \left (\frac{a}{b}+x\right )+\log \left (\frac{c}{d}+x\right )\right )^2+\log ^3\left (\frac{c}{d}+x\right )\right )+3 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )+A\right )^2}{3 d g} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.444, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dgx+cg} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, 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*} \frac{B^{2} \log \left (d x + c\right ) \log \left ({\left (d x + c\right )}^{n}\right )^{2}}{d g} + \frac{A^{2} \log \left (d g x + c g\right )}{d g} - \int -\frac{B^{2} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + B^{2} \log \left (e\right )^{2} + 2 \, A B \log \left (e\right ) + 2 \,{\left (B^{2} \log \left (e\right ) + A B\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \,{\left (B^{2} n \log \left (d x + c\right ) + B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) + B^{2} \log \left (e\right ) + A B\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{d g x + c g}\,{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{B^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A^{2}}{d g x + c g}, 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{{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{d g x + c g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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