Optimal. Leaf size=938 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.54641, antiderivative size = 938, normalized size of antiderivative = 1., number of steps used = 42, number of rules used = 17, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.68, Rules used = {2296, 2295, 2371, 6, 321, 203, 2351, 2324, 12, 4848, 2391, 2353, 2330, 2317, 2374, 6589, 2383} \[ 36 n^3 x b^3-36 n^2 x \log \left (c x^n\right ) b^3+\frac{12 n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right ) b^3}{\sqrt{d} \sqrt{f}}-6 n^3 x \log \left (d f x^2+1\right ) b^3+6 n^2 x \log \left (c x^n\right ) \log \left (d f x^2+1\right ) b^3-\frac{6 i n^3 \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right ) b^3}{\sqrt{d} \sqrt{f}}+\frac{6 i n^3 \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right ) b^3}{\sqrt{d} \sqrt{f}}+\frac{6 n^3 \text{PolyLog}\left (3,-\sqrt{-d} \sqrt{f} x\right ) b^3}{\sqrt{-d} \sqrt{f}}-\frac{6 n^3 \text{PolyLog}\left (3,\sqrt{-d} \sqrt{f} x\right ) b^3}{\sqrt{-d} \sqrt{f}}+\frac{6 n^3 \text{PolyLog}\left (4,-\sqrt{-d} \sqrt{f} x\right ) b^3}{\sqrt{-d} \sqrt{f}}-\frac{6 n^3 \text{PolyLog}\left (4,\sqrt{-d} \sqrt{f} x\right ) b^3}{\sqrt{-d} \sqrt{f}}-24 a n^2 x b^2-12 n^2 (a-b n) x b^2+\frac{12 n^2 (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) b^2}{\sqrt{d} \sqrt{f}}+6 a n^2 x \log \left (d f x^2+1\right ) b^2-\frac{6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\sqrt{-d} \sqrt{f} x\right ) b^2}{\sqrt{-d} \sqrt{f}}+\frac{6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,\sqrt{-d} \sqrt{f} x\right ) b^2}{\sqrt{-d} \sqrt{f}}-\frac{6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,-\sqrt{-d} \sqrt{f} x\right ) b^2}{\sqrt{-d} \sqrt{f}}+\frac{6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,\sqrt{-d} \sqrt{f} x\right ) b^2}{\sqrt{-d} \sqrt{f}}+12 n x \left (a+b \log \left (c x^n\right )\right )^2 b+\frac{3 n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right ) b}{\sqrt{-d} \sqrt{f}}-\frac{3 n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\sqrt{-d} \sqrt{f} x+1\right ) b}{\sqrt{-d} \sqrt{f}}-3 n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d f x^2+1\right ) b+\frac{3 n \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,-\sqrt{-d} \sqrt{f} x\right ) b}{\sqrt{-d} \sqrt{f}}-\frac{3 n \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,\sqrt{-d} \sqrt{f} x\right ) b}{\sqrt{-d} \sqrt{f}}-2 x \left (a+b \log \left (c x^n\right )\right )^3-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (\sqrt{-d} \sqrt{f} x+1\right )}{\sqrt{-d} \sqrt{f}}+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d f x^2+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2296
Rule 2295
Rule 2371
Rule 6
Rule 321
Rule 203
Rule 2351
Rule 2324
Rule 12
Rule 4848
Rule 2391
Rule 2353
Rule 2330
Rule 2317
Rule 2374
Rule 6589
Rule 2383
Rubi steps
\begin{align*} \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac{1}{d}+f x^2\right )\right ) \, dx &=6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(2 f) \int \left (\frac{6 a b^2 d n^2 x^2}{1+d f x^2}-\frac{6 b^3 d n^3 x^2}{1+d f x^2}+\frac{6 b^3 d n^2 x^2 \log \left (c x^n\right )}{1+d f x^2}-\frac{3 b d n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2}+\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )^3}{1+d f x^2}\right ) \, dx\\ &=6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(2 f) \int \left (\frac{d \left (6 a b^2 n^2-6 b^3 n^3\right ) x^2}{1+d f x^2}+\frac{6 b^3 d n^2 x^2 \log \left (c x^n\right )}{1+d f x^2}-\frac{3 b d n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2}+\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )^3}{1+d f x^2}\right ) \, dx\\ &=6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(2 d f) \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{1+d f x^2} \, dx+(6 b d f n) \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2} \, dx-\left (12 b^3 d f n^2\right ) \int \frac{x^2 \log \left (c x^n\right )}{1+d f x^2} \, dx-\left (12 b^2 d f n^2 (a-b n)\right ) \int \frac{x^2}{1+d f x^2} \, dx\\ &=-12 b^2 n^2 (a-b n) x+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(2 d f) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{d f \left (1+d f x^2\right )}\right ) \, dx+(6 b d f n) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d f \left (1+d f x^2\right )}\right ) \, dx-\left (12 b^3 d f n^2\right ) \int \left (\frac{\log \left (c x^n\right )}{d f}-\frac{\log \left (c x^n\right )}{d f \left (1+d f x^2\right )}\right ) \, dx+\left (12 b^2 n^2 (a-b n)\right ) \int \frac{1}{1+d f x^2} \, dx\\ &=-12 b^2 n^2 (a-b n) x+\frac{12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-2 \int \left (a+b \log \left (c x^n\right )\right )^3 \, dx+2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{1+d f x^2} \, dx+(6 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(6 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2} \, dx-\left (12 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx+\left (12 b^3 n^2\right ) \int \frac{\log \left (c x^n\right )}{1+d f x^2} \, dx\\ &=12 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac{12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-12 b^3 n^2 x \log \left (c x^n\right )+\frac{12 b^3 n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right )}{\sqrt{d} \sqrt{f}}+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \left (a+b \log \left (c x^n\right )\right )^3+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+2 \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 \left (1-\sqrt{-d} \sqrt{f} x\right )}+\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 \left (1+\sqrt{-d} \sqrt{f} x\right )}\right ) \, dx+(6 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(6 b n) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1-\sqrt{-d} \sqrt{f} x\right )}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1+\sqrt{-d} \sqrt{f} x\right )}\right ) \, dx-\left (12 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-\left (12 b^3 n^3\right ) \int \frac{\tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f} x} \, dx\\ &=-12 a b^2 n^2 x+12 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac{12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-12 b^3 n^2 x \log \left (c x^n\right )+\frac{12 b^3 n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right )}{\sqrt{d} \sqrt{f}}+12 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \left (a+b \log \left (c x^n\right )\right )^3+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{1-\sqrt{-d} \sqrt{f} x} \, dx-(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{1+\sqrt{-d} \sqrt{f} x} \, dx-\left (12 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-\left (12 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx-\frac{\left (12 b^3 n^3\right ) \int \frac{\tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{d} \sqrt{f}}+\int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{1-\sqrt{-d} \sqrt{f} x} \, dx+\int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{1+\sqrt{-d} \sqrt{f} x} \, dx\\ &=-24 a b^2 n^2 x+24 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac{12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-24 b^3 n^2 x \log \left (c x^n\right )+\frac{12 b^3 n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right )}{\sqrt{d} \sqrt{f}}+12 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}-\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}-\left (12 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx-\frac{\left (6 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}+\frac{\left (6 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}-\frac{\left (6 i b^3 n^3\right ) \int \frac{\log \left (1-i \sqrt{d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{d} \sqrt{f}}+\frac{\left (6 i b^3 n^3\right ) \int \frac{\log \left (1+i \sqrt{d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{d} \sqrt{f}}\\ &=-24 a b^2 n^2 x+36 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac{12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-36 b^3 n^2 x \log \left (c x^n\right )+\frac{12 b^3 n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right )}{\sqrt{d} \sqrt{f}}+12 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{6 i b^3 n^3 \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+\frac{6 i b^3 n^3 \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-\frac{\left (6 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}+\frac{\left (6 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}+\frac{\left (6 b^3 n^3\right ) \int \frac{\text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}-\frac{\left (6 b^3 n^3\right ) \int \frac{\text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}\\ &=-24 a b^2 n^2 x+36 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac{12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-36 b^3 n^2 x \log \left (c x^n\right )+\frac{12 b^3 n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right )}{\sqrt{d} \sqrt{f}}+12 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{6 i b^3 n^3 \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+\frac{6 i b^3 n^3 \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+\frac{6 b^3 n^3 \text{Li}_3\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{6 b^3 n^3 \text{Li}_3\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{\left (6 b^3 n^3\right ) \int \frac{\text{Li}_3\left (-\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}-\frac{\left (6 b^3 n^3\right ) \int \frac{\text{Li}_3\left (\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}\\ &=-24 a b^2 n^2 x+36 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac{12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-36 b^3 n^2 x \log \left (c x^n\right )+\frac{12 b^3 n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right )}{\sqrt{d} \sqrt{f}}+12 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{6 i b^3 n^3 \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+\frac{6 i b^3 n^3 \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+\frac{6 b^3 n^3 \text{Li}_3\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{6 b^3 n^3 \text{Li}_3\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{6 b^3 n^3 \text{Li}_4\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{6 b^3 n^3 \text{Li}_4\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}\\ \end{align*}
Mathematica [A] time = 0.697263, size = 1027, normalized size = 1.09 \[ \frac{2 b^3 \left (-\sqrt{d} \sqrt{f} x \left (\log ^3(x)-3 \log ^2(x)+6 \log (x)-6\right )-\frac{1}{2} i \left (\log \left (i \sqrt{d} \sqrt{f} x+1\right ) \log ^3(x)+3 \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right ) \log ^2(x)-6 \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right ) \log (x)+6 \text{PolyLog}\left (4,-i \sqrt{d} \sqrt{f} x\right )\right )+\frac{1}{2} i \left (\log \left (1-i \sqrt{d} \sqrt{f} x\right ) \log ^3(x)+3 \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right ) \log ^2(x)-6 \text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right ) \log (x)+6 \text{PolyLog}\left (4,i \sqrt{d} \sqrt{f} x\right )\right )\right ) n^3-6 b^2 \left (a-b n-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\sqrt{d} \sqrt{f} x \left (\log ^2(x)-2 \log (x)+2\right )+\frac{1}{2} i \left (\log \left (i \sqrt{d} \sqrt{f} x+1\right ) \log ^2(x)+2 \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right ) \log (x)-2 \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right )\right )-\frac{1}{2} i \left (\log \left (1-i \sqrt{d} \sqrt{f} x\right ) \log ^2(x)+2 \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right ) \log (x)-2 \text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right )\right )\right ) n^2+3 b \left (a^2-2 b n a+2 b \left (\log \left (c x^n\right )-n \log (x)\right ) a+2 b^2 n^2+b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+2 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )\right ) \left (-2 \sqrt{d} \sqrt{f} x (\log (x)-1)-i \left (\log (x) \log \left (i \sqrt{d} \sqrt{f} x+1\right )+\text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )\right )+i \left (\log (x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )+\text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )\right )\right ) n-2 \sqrt{d} \sqrt{f} x \left (a^3-3 b n a^2+3 b \left (\log \left (c x^n\right )-n \log (x)\right ) a^2+6 b^2 n^2 a+3 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 a+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right ) a-6 b^3 n^3+b^3 \left (\log \left (c x^n\right )-n \log (x)\right )^3-3 b^3 n \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^3 n^2 \left (\log \left (c x^n\right )-n \log (x)\right )\right )+2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a^3-3 b n a^2+3 b \left (\log \left (c x^n\right )-n \log (x)\right ) a^2+6 b^2 n^2 a+3 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 a+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right ) a-6 b^3 n^3+b^3 \left (\log \left (c x^n\right )-n \log (x)\right )^3-3 b^3 n \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^3 n^2 \left (\log \left (c x^n\right )-n \log (x)\right )\right )+\sqrt{d} \sqrt{f} x \left (a^3-3 b n a^2+6 b^2 n^2 a-6 b^3 n^3+b^3 \log ^3\left (c x^n\right )+3 b^2 (a-b n) \log ^2\left (c x^n\right )+3 b \left (a^2-2 b n a+2 b^2 n^2\right ) \log \left (c x^n\right )\right ) \log \left (d f x^2+1\right )}{\sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.25, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ({d}^{-1}+f{x}^{2} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (b^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{3} \log \left (d f x^{2} + 1\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 (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac{1}{d}\right )} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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