Optimal. Leaf size=938 \[ -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}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.07, antiderivative size = 938, normalized size of antiderivative = 1.00, number of steps
used = 42, number of rules used = 17, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {2333, 2332,
2418, 6, 327, 209, 2393, 2361, 12, 4940, 2438, 2395, 2367, 2354, 2421, 6724, 2430}
\begin {gather*} 36 n^3 x b^3-36 n^2 x \log \left (c x^n\right ) b^3+\frac {12 n^2 \text {ArcTan}\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) \text {ArcTan}\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 ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6
Rule 12
Rule 209
Rule 327
Rule 2332
Rule 2333
Rule 2354
Rule 2361
Rule 2367
Rule 2393
Rule 2395
Rule 2418
Rule 2421
Rule 2430
Rule 2438
Rule 4940
Rule 6724
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*}
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Mathematica [A]
time = 0.45, size = 1027, normalized size = 1.09 \begin {gather*} \frac {-2 \sqrt {d} \sqrt {f} x \left (a^3-3 a^2 b n+6 a b^2 n^2-6 b^3 n^3+6 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a^3-3 a^2 b n+6 a b^2 n^2-6 b^3 n^3+6 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+\sqrt {d} \sqrt {f} x \left (a^3-3 a^2 b n+6 a b^2 n^2-6 b^3 n^3+3 b \left (a^2-2 a b n+2 b^2 n^2\right ) \log \left (c x^n\right )+3 b^2 (a-b n) \log ^2\left (c x^n\right )+b^3 \log ^3\left (c x^n\right )\right ) \log \left (1+d f x^2\right )+3 b n \left (a^2-2 a b n+2 b^2 n^2+2 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+2 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \left (-2 \sqrt {d} \sqrt {f} x (-1+\log (x))-i \left (\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )\right )+i \left (\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right )\right )-6 b^2 n^2 \left (a-b n-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\sqrt {d} \sqrt {f} x \left (2-2 \log (x)+\log ^2(x)\right )+\frac {1}{2} i \left (\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-2 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )\right )-\frac {1}{2} i \left (\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-2 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )\right )\right )+2 b^3 n^3 \left (-\sqrt {d} \sqrt {f} x \left (-6+6 \log (x)-3 \log ^2(x)+\log ^3(x)\right )-\frac {1}{2} i \left (\log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (-i \sqrt {d} \sqrt {f} x\right )\right )+\frac {1}{2} i \left (\log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (i \sqrt {d} \sqrt {f} x\right )\right )\right )}{\sqrt {d} \sqrt {f}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int \ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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