Optimal. Leaf size=137 \[ -\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d i}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d i}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d i} \]
[Out]
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Rubi [A]
time = 0.08, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2551, 2354,
2421, 6724} \begin {gather*} -\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 i}+\frac {2 B^2 n^2 \text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d i}-\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 i} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2354
Rule 2421
Rule 2551
Rule 6724
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}{189 c+189 d x} \, dx &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d x)}{(a+b x) (c+d x)} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d x)}{(a+b x) (c+d x)} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d 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 (189 c+189 d x)}{(b c-a d) (a+b x)}\right ) \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\frac {1}{189} (2 B n) \int \frac {\left (-A-B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (189 c+189 d x)}{c+d x} \, dx-\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 (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\frac {1}{189} (2 B n) \int \left (\frac {A \log (189 c+189 d x)}{-c-d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (189 c+189 d x)}{-c-d x}\right ) \, dx-\frac {(2 b B n) \int \left (\frac {A \log (189 c+189 d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (189 c+189 d x)}{a+b x}\right ) \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\frac {1}{189} (2 A B n) \int \frac {\log (189 c+189 d x)}{-c-d x} \, dx-\frac {1}{189} \left (2 B^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (189 c+189 d x)}{-c-d x} \, dx-\frac {(2 A b B n) \int \frac {\log (189 c+189 d x)}{a+b x} \, dx}{189 d}-\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 (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+(2 A B n) \int \frac {\log \left (\frac {189 d (a+b x)}{-189 b c+189 a d}\right )}{189 c+189 d x} \, dx-\frac {(2 A B n) \text {Subst}\left (\int -\frac {189 \log (x)}{x} \, dx,x,189 c+189 d x\right )}{35721 d}-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (189 c+189 d x)}{a+b x} \, dx}{189 d}-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left ((c+d x)^{-n}\right ) \log (189 c+189 d x)}{a+b x} \, dx}{189 d}+\frac {1}{189} \left (B^2 n^2\right ) \int \frac {\log ^2(189 c+189 d x)}{c+d x} \, dx-\frac {\left (b B^2 n^2\right ) \int \frac {\log ^2(189 c+189 d x)}{a+b x} \, dx}{189 d}-\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 (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {(2 A B n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,189 c+189 d x\right )}{189 d}+\frac {(2 A B n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}-\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {189 b c-189 a d}{b}+\frac {189 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{189 d}-\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right ) \log \left (-\frac {-189 b c+189 a d}{b}+\frac {189 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{189 d}+\left (2 B^2 n^2\right ) \int \frac {\log \left (\frac {189 d (a+b x)}{-189 b c+189 a d}\right ) \log (189 c+189 d x)}{189 c+189 d x} \, dx+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {189 \log ^2(x)}{x} \, dx,x,189 c+189 d x\right )}{35721 d}+\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 {189 d (a+b x)}{-189 b c+189 a d}\right )}{189 c+189 d x} \, dx\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\frac {B^2 \text {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {189 b c-189 a d}{b}+\frac {189 d x}{b}} \, dx,x,a+b x\right )}{b}+\frac {\left (2 B^2 n\right ) \text {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 )}{189 b}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-189 b c+189 a d}{b}+\frac {189 d x}{b}\right )}{-\frac {-189 b c+189 a d}{b}+\frac {189 d x}{b}} \, dx,x,a+b x\right )}{b}+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,189 c+189 d x\right )}{189 d}+\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {189 d \left (\frac {-189 b c+189 a d}{189 d}+\frac {b x}{189 d}\right )}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}+\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 ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \log (189 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\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 )}{189 d}+\frac {\left (2 B^2 n\right ) \text {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 )}{189 d}-\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {189 d x}{189 b c-189 a d}\right )}{x} \, dx,x,a+b x\right )}{189 d}+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (189 (c+d x))\right )}{189 d}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-189 b c+189 a d}{189 d}+\frac {b x}{189 d}\right )}{x} \, dx,x,189 (c+d x)\right )}{189 d}+\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \log (189 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\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 )}{189 d}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\frac {\left (b B^2\right ) \text {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 )}{189 d^2}+\frac {\left (b B^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-189 b c+189 a d}{189 d}+\frac {b x}{189 d}} \, dx,x,189 (c+d x)\right )}{35721 d^2}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {189 d x}{189 b c-189 a d}\right )}{x} \, dx,x,a+b x\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \log (189 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\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 )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{189 d}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\frac {\left (2 B^2 n\right ) \text {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 )}{189 d}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 (c+d x)\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\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 )}{189 d}+\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 )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{189 d}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{189 d}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 (c+d x)\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\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 )}{189 d}+\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 )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(306\) vs. \(2(137)=274\).
time = 0.12, size = 306, normalized size = 2.23 \begin {gather*} \frac {\left (A+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 )\right )^2 \log (c+d x)-B n \left (A+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 )\right ) \left (\log ^2\left (\frac {c}{d}+x\right )+2 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {a+b x}{c+d x}\right )\right ) \log (c+d x)-2 \left (\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )-B^2 n^2 \left (\log ^2\left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+2 \log \left (\frac {a+b x}{c+d x}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )}{d i} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{d i x +c i}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {A^{2}}{c + d x}\, dx + \int \frac {B^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c + d x}\, dx + \int \frac {2 A B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx}{i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{c\,i+d\,i\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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