Optimal. Leaf size=411 \[ -\frac {3 b^2 n^2 \text {Li}_2\left (-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {3 b^2 n^2 \text {Li}_3\left (-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}+\frac {3 b^2 n^2 \left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3 b n \text {Li}_2\left (-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {3 b n \left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}+\frac {\left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 d f}+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{8 d f}+\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{8 d f}+\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{8 d f}-\frac {3 b^3 n^3 \left (d f x^2+1\right ) \log \left (d f x^2+1\right )}{8 d f}+\frac {3}{2} b^3 n^3 x^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.04, antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 21, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.808, Rules used = {2454, 2389, 2295, 2377, 2305, 2304, 2353, 2302, 30, 6742, 2374, 2383, 6589, 14, 2351, 2301, 2376, 2475, 2411, 43, 2315} \[ -\frac {3 b^2 n^2 \text {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {3 b^2 n^2 \text {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}+\frac {3 b n \text {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}+\frac {3 b^3 n^3 \text {PolyLog}\left (2,-d f x^2\right )}{8 d f}+\frac {3 b^3 n^3 \text {PolyLog}\left (3,-d f x^2\right )}{8 d f}+\frac {3 b^3 n^3 \text {PolyLog}\left (4,-d f x^2\right )}{8 d f}+\frac {3 b^2 n^2 \left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {3 b n \left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}+\frac {\left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 d f}+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \left (d f x^2+1\right ) \log \left (d f x^2+1\right )}{8 d f}+\frac {3}{2} b^3 n^3 x^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 30
Rule 43
Rule 2295
Rule 2301
Rule 2302
Rule 2304
Rule 2305
Rule 2315
Rule 2351
Rule 2353
Rule 2374
Rule 2376
Rule 2377
Rule 2383
Rule 2389
Rule 2411
Rule 2454
Rule 2475
Rule 6589
Rule 6742
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx &=-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}-(3 b n) \int \left (-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f x}\right ) \, dx\\ &=-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}+\frac {1}{2} (3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {(3 b n) \int \frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x} \, dx}{2 d f}\\ &=\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}-\frac {(3 b n) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}+d f x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )\right ) \, dx}{2 d f}-\frac {1}{2} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac {3}{8} b^3 n^3 x^2-\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}-\frac {1}{2} (3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right ) \, dx-\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x} \, dx}{2 d f}\\ &=\frac {3}{8} b^3 n^3 x^2-\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}+\left (3 b^2 n^2\right ) \int \left (-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f x}\right ) \, dx-\frac {\left (3 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{x} \, dx}{2 d f}\\ &=\frac {3}{8} b^3 n^3 x^2-\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{4 d f}-\frac {1}{2} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {\left (3 b^2 n^2\right ) \int \frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x} \, dx}{2 d f}+\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-d f x^2\right )}{x} \, dx}{4 d f}\\ &=\frac {3}{4} b^3 n^3 x^2-\frac {3}{2} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{8 d f}+\frac {\left (3 b^2 n^2\right ) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}+d f x \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )\right ) \, dx}{2 d f}\\ &=\frac {3}{4} b^3 n^3 x^2-\frac {3}{2} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{8 d f}+\frac {1}{2} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right ) \, dx+\frac {\left (3 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x} \, dx}{2 d f}\\ &=\frac {3}{4} b^3 n^3 x^2-\frac {9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^2 n^2 \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d f}-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{8 d f}-\frac {1}{2} \left (3 b^3 n^3\right ) \int \left (-\frac {x}{2}+\frac {\left (1+d f x^2\right ) \log \left (1+d f x^2\right )}{2 d f x}\right ) \, dx+\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-d f x^2\right )}{x} \, dx}{4 d f}\\ &=\frac {9}{8} b^3 n^3 x^2-\frac {9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^2 n^2 \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d f}-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{8 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{8 d f}-\frac {\left (3 b^3 n^3\right ) \int \frac {\left (1+d f x^2\right ) \log \left (1+d f x^2\right )}{x} \, dx}{4 d f}\\ &=\frac {9}{8} b^3 n^3 x^2-\frac {9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^2 n^2 \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d f}-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{8 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{8 d f}-\frac {\left (3 b^3 n^3\right ) \operatorname {Subst}\left (\int \frac {(1+d f x) \log (1+d f x)}{x} \, dx,x,x^2\right )}{8 d f}\\ &=\frac {9}{8} b^3 n^3 x^2-\frac {9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^2 n^2 \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d f}-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{8 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{8 d f}-\frac {\left (3 b^3 n^3\right ) \operatorname {Subst}\left (\int \frac {x \log (x)}{-\frac {1}{d f}+\frac {x}{d f}} \, dx,x,1+d f x^2\right )}{8 d^2 f^2}\\ &=\frac {9}{8} b^3 n^3 x^2-\frac {9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^2 n^2 \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d f}-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{8 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{8 d f}-\frac {\left (3 b^3 n^3\right ) \operatorname {Subst}\left (\int \left (d f \log (x)+\frac {d f \log (x)}{-1+x}\right ) \, dx,x,1+d f x^2\right )}{8 d^2 f^2}\\ &=\frac {9}{8} b^3 n^3 x^2-\frac {9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^2 n^2 \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d f}-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{8 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{8 d f}-\frac {\left (3 b^3 n^3\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1+d f x^2\right )}{8 d f}-\frac {\left (3 b^3 n^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{-1+x} \, dx,x,1+d f x^2\right )}{8 d f}\\ &=\frac {3}{2} b^3 n^3 x^2-\frac {9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \left (1+d f x^2\right ) \log \left (1+d f x^2\right )}{8 d f}+\frac {3 b^2 n^2 \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d f}-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}+\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{8 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{8 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{8 d f}\\ \end {align*}
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Mathematica [C] time = 0.60, size = 1004, normalized size = 2.44 \[ \frac {-b^3 \left (4 d f x^2 \log ^3(x)-4 \log \left (1-i \sqrt {d} \sqrt {f} x\right ) \log ^3(x)-4 \log \left (i \sqrt {d} \sqrt {f} x+1\right ) \log ^3(x)-6 d f x^2 \log ^2(x)-12 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right ) \log ^2(x)-12 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right ) \log ^2(x)+6 d f x^2 \log (x)+24 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right ) \log (x)+24 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right ) \log (x)-3 d f x^2-24 \text {Li}_4\left (-i \sqrt {d} \sqrt {f} x\right )-24 \text {Li}_4\left (i \sqrt {d} \sqrt {f} x\right )\right ) n^3+3 b^2 \left (-2 a+b n+2 b n \log (x)-2 b \log \left (c x^n\right )\right ) \left (2 d f \log ^2(x) x^2+d f x^2-2 d f \log (x) x^2-2 \log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-2 \log ^2(x) \log \left (i \sqrt {d} \sqrt {f} x+1\right )-4 \log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-4 \log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+4 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+4 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )\right ) n^2+6 b \left (2 a^2-2 b n a+4 b \left (\log \left (c x^n\right )-n \log (x)\right ) a+b^2 n^2+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 (\frac {1}{2} d f x^2-d f \log (x) x^2+\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (i \sqrt {d} \sqrt {f} x+1\right )+\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right ) n-d f x^2 \left (4 a^3-6 b n a^2+12 b \left (\log \left (c x^n\right )-n \log (x)\right ) a^2+6 b^2 n^2 a+12 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 a+12 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right ) a-3 b^3 n^3+4 b^3 \left (\log \left (c x^n\right )-n \log (x)\right )^3-6 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 )+d f x^2 \left (4 a^3-6 b n a^2+6 b^2 n^2 a-3 b^3 n^3+4 b^3 \log ^3\left (c x^n\right )-6 b^2 (b n-2 a) \log ^2\left (c x^n\right )+6 b \left (2 a^2-2 b n a+b^2 n^2\right ) \log \left (c x^n\right )\right ) \log \left (d f x^2+1\right )+\left (4 a^3-6 b n a^2+12 b \left (\log \left (c x^n\right )-n \log (x)\right ) a^2+6 b^2 n^2 a+12 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 a+12 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right ) a-3 b^3 n^3+4 b^3 \left (\log \left (c x^n\right )-n \log (x)\right )^3-6 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 ) \log \left (d f x^2+1\right )}{8 d f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} x \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b x \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{3} x \log \left (d f x^{2} + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{3} x \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )\, dx \]
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 \[ \frac {1}{8} \, {\left (4 \, b^{3} x^{2} \log \left (x^{n}\right )^{3} - 6 \, {\left (b^{3} {\left (n - 2 \, \log \relax (c)\right )} - 2 \, a b^{2}\right )} x^{2} \log \left (x^{n}\right )^{2} + 6 \, {\left ({\left (n^{2} - 2 \, n \log \relax (c) + 2 \, \log \relax (c)^{2}\right )} b^{3} - 2 \, a b^{2} {\left (n - 2 \, \log \relax (c)\right )} + 2 \, a^{2} b\right )} x^{2} \log \left (x^{n}\right ) + {\left (6 \, {\left (n^{2} - 2 \, n \log \relax (c) + 2 \, \log \relax (c)^{2}\right )} a b^{2} - {\left (3 \, n^{3} - 6 \, n^{2} \log \relax (c) + 6 \, n \log \relax (c)^{2} - 4 \, \log \relax (c)^{3}\right )} b^{3} - 6 \, a^{2} b {\left (n - 2 \, \log \relax (c)\right )} + 4 \, a^{3}\right )} x^{2}\right )} \log \left (d f x^{2} + 1\right ) - \int \frac {4 \, b^{3} d f x^{3} \log \left (x^{n}\right )^{3} + 6 \, {\left (2 \, a b^{2} d f - {\left (d f n - 2 \, d f \log \relax (c)\right )} b^{3}\right )} x^{3} \log \left (x^{n}\right )^{2} + 6 \, {\left (2 \, a^{2} b d f - 2 \, {\left (d f n - 2 \, d f \log \relax (c)\right )} a b^{2} + {\left (d f n^{2} - 2 \, d f n \log \relax (c) + 2 \, d f \log \relax (c)^{2}\right )} b^{3}\right )} x^{3} \log \left (x^{n}\right ) + {\left (4 \, a^{3} d f - 6 \, {\left (d f n - 2 \, d f \log \relax (c)\right )} a^{2} b + 6 \, {\left (d f n^{2} - 2 \, d f n \log \relax (c) + 2 \, d f \log \relax (c)^{2}\right )} a b^{2} - {\left (3 \, d f n^{3} - 6 \, d f n^{2} \log \relax (c) + 6 \, d f n \log \relax (c)^{2} - 4 \, d f \log \relax (c)^{3}\right )} b^{3}\right )} x^{3}}{4 \, {\left (d f x^{2} + 1\right )}}\,{d x} \]
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 \[ \int x\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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