Optimal. Leaf size=220 \[ -\frac{B^2 i n^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^2 d}+\frac{B i n (b c-a d)^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d}-\frac{B i n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2}+\frac{i (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d}+\frac{B^2 i n^2 (b c-a d)^2 \log (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.480723, antiderivative size = 307, normalized size of antiderivative = 1.4, number of steps used = 15, number of rules used = 12, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{B^2 i n^2 (b c-a d)^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 d}-\frac{B i n (b c-a d)^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d}+\frac{i (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d}-\frac{A B i n x (b c-a d)}{b}-\frac{B^2 i n (a+b x) (b c-a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2}+\frac{B^2 i n^2 (b c-a d)^2 \log ^2(a+b x)}{2 b^2 d}+\frac{B^2 i n^2 (b c-a d)^2 \log (c+d x)}{b^2 d}-\frac{B^2 i n^2 (b c-a d)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 2528
Rule 2486
Rule 31
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int (162 c+162 d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\frac{81 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac{(B n) \int \frac{26244 (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{162 d}\\ &=\frac{81 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac{(162 B (b c-a d) n) \int \frac{(c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{d}\\ &=\frac{81 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac{(162 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 )}{b}+\frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b (a+b x)}\right ) \, dx}{d}\\ &=\frac{81 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac{(162 B (b c-a d) n) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b}-\frac{\left (162 B (b c-a d)^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b d}\\ &=-\frac{162 A B (b c-a d) n x}{b}-\frac{162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac{81 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac{\left (162 B^2 (b c-a d) n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b}+\frac{\left (162 B^2 (b c-a d)^2 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 d}\\ &=-\frac{162 A B (b c-a d) n x}{b}-\frac{162 B^2 (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2}-\frac{162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac{81 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{\left (162 B^2 (b c-a d)^2 n^2\right ) \int \frac{1}{c+d x} \, dx}{b^2}+\frac{\left (162 B^2 (b c-a d)^2 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^2 d}\\ &=-\frac{162 A B (b c-a d) n x}{b}-\frac{162 B^2 (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2}-\frac{162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac{81 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{162 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^2 d}-\frac{\left (162 B^2 (b c-a d)^2 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2}+\frac{\left (162 B^2 (b c-a d)^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b d}\\ &=-\frac{162 A B (b c-a d) n x}{b}-\frac{162 B^2 (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2}-\frac{162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac{81 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{162 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^2 d}-\frac{162 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 d}+\frac{\left (162 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 d}+\frac{\left (162 B^2 (b c-a d)^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b d}\\ &=-\frac{162 A B (b c-a d) n x}{b}+\frac{81 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^2 d}-\frac{162 B^2 (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2}-\frac{162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac{81 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{162 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^2 d}-\frac{162 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 d}+\frac{\left (162 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 d}\\ &=-\frac{162 A B (b c-a d) n x}{b}+\frac{81 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^2 d}-\frac{162 B^2 (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2}-\frac{162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac{81 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{162 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^2 d}-\frac{162 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 d}-\frac{162 B^2 (b c-a d)^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 d}\\ \end{align*}
Mathematica [A] time = 0.211707, size = 216, normalized size = 0.98 \[ \frac{i \left (\frac{B n (b c-a d) \left (2 B n (a d-b c) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )-2 (b c-a d) \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+B n \log \left (\frac{b (c+d x)}{b c-a d}\right )+A\right )-2 \left (B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+B n (a d-b c) \log (c+d x)+A b d x\right )+B n (b c-a d) \log ^2(a+b x)\right )}{b^2}+(c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.324, size = 0, normalized size = 0. \begin{align*} \int \left ( dix+ci \right ) \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 [B] time = 3.5242, size = 1114, normalized size = 5.06 \begin{align*} \text{result too large to display} \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 (A^{2} d i x + A^{2} c i +{\left (B^{2} d i x + B^{2} c i\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B d i x + A B c i\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ), 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{\left (d i x + c i\right )}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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