Optimal. Leaf size=411 \[ -\frac{3 b^2 n^2 \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac{3 b^2 n^2 \text{PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}+\frac{3 b n \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}+\frac{3 b^3 n^3 \text{PolyLog}\left (2,-d f x^2\right )}{8 d f}+\frac{3 b^3 n^3 \text{PolyLog}\left (3,-d f x^2\right )}{8 d f}+\frac{3 b^3 n^3 \text{PolyLog}\left (4,-d f x^2\right )}{8 d f}+\frac{3 b^2 n^2 \left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac{9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{3 b n \left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}+\frac{\left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 d f}+\frac{3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b^3 n^3 \left (d f x^2+1\right ) \log \left (d f x^2+1\right )}{8 d f}+\frac{3}{2} b^3 n^3 x^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.04221, antiderivative size = 411, normalized size of antiderivative = 1., 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 2454
Rule 2389
Rule 2295
Rule 2377
Rule 2305
Rule 2304
Rule 2353
Rule 2302
Rule 30
Rule 6742
Rule 2374
Rule 2383
Rule 6589
Rule 14
Rule 2351
Rule 2301
Rule 2376
Rule 2475
Rule 2411
Rule 43
Rule 2315
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*}
Mathematica [C] time = 0.537744, 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{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right ) \log ^2(x)-12 \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right ) \log ^2(x)+6 d f x^2 \log (x)+24 \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right ) \log (x)+24 \text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right ) \log (x)-3 d f x^2-24 \text{PolyLog}\left (4,-i \sqrt{d} \sqrt{f} x\right )-24 \text{PolyLog}\left (4,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{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )-4 \log (x) \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+4 \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right )+4 \text{PolyLog}\left (3,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{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+\text{PolyLog}\left (2,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.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.138, size = 0, normalized size = 0. \begin{align*} \int x \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{8} \,{\left (4 \, b^{3} x^{2} \log \left (x^{n}\right )^{3} - 6 \,{\left (b^{3}{\left (n - 2 \, \log \left (c\right )\right )} - 2 \, a b^{2}\right )} x^{2} \log \left (x^{n}\right )^{2} + 6 \,{\left ({\left (n^{2} - 2 \, n \log \left (c\right ) + 2 \, \log \left (c\right )^{2}\right )} b^{3} - 2 \, a b^{2}{\left (n - 2 \, \log \left (c\right )\right )} + 2 \, a^{2} b\right )} x^{2} \log \left (x^{n}\right ) +{\left (6 \,{\left (n^{2} - 2 \, n \log \left (c\right ) + 2 \, \log \left (c\right )^{2}\right )} a b^{2} -{\left (3 \, n^{3} - 6 \, n^{2} \log \left (c\right ) + 6 \, n \log \left (c\right )^{2} - 4 \, \log \left (c\right )^{3}\right )} b^{3} - 6 \, a^{2} b{\left (n - 2 \, \log \left (c\right )\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 \left (c\right )\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 \left (c\right )\right )} a b^{2} +{\left (d f n^{2} - 2 \, d f n \log \left (c\right ) + 2 \, d f \log \left (c\right )^{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 \left (c\right )\right )} a^{2} b + 6 \,{\left (d f n^{2} - 2 \, d f n \log \left (c\right ) + 2 \, d f \log \left (c\right )^{2}\right )} a b^{2} -{\left (3 \, d f n^{3} - 6 \, d f n^{2} \log \left (c\right ) + 6 \, d f n \log \left (c\right )^{2} - 4 \, d f \log \left (c\right )^{3}\right )} b^{3}\right )} x^{3}}{4 \,{\left (d f x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]