Optimal. Leaf size=411 \[ \frac {3}{2} b^3 n^3 x^2-\frac {9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \left (1+d f x^2\right ) \log \left (1+d f x^2\right )}{8 d f}+\frac {3 b^2 n^2 \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d f}-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}+\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{8 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{8 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{8 d f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.72, 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 = {2504, 2436,
2332, 2424, 2342, 2341, 2395, 2339, 30, 6874, 2421, 2430, 6724, 14, 2393, 2338, 2423, 2525, 2458,
45, 2352} \begin {gather*} -\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 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 30
Rule 45
Rule 2332
Rule 2338
Rule 2339
Rule 2341
Rule 2342
Rule 2352
Rule 2393
Rule 2395
Rule 2421
Rule 2423
Rule 2424
Rule 2430
Rule 2436
Rule 2458
Rule 2504
Rule 2525
Rule 6724
Rule 6874
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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \log (x) \, dx,x,1+d f x^2\right )}{8 d f}-\frac {\left (3 b^3 n^3\right ) \text {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] Result contains complex when optimal does not.
time = 0.40, size = 1004, normalized size = 2.44 \begin {gather*} \frac {-d f x^2 \left (4 a^3-6 a^2 b n+6 a b^2 n^2-3 b^3 n^3+12 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+12 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 )+12 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-6 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+4 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+d f x^2 \left (4 a^3-6 a^2 b n+6 a b^2 n^2-3 b^3 n^3+6 b \left (2 a^2-2 a b n+b^2 n^2\right ) \log \left (c x^n\right )-6 b^2 (-2 a+b n) \log ^2\left (c x^n\right )+4 b^3 \log ^3\left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\left (4 a^3-6 a^2 b n+6 a b^2 n^2-3 b^3 n^3+12 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+12 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 )+12 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-6 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+4 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right ) \log \left (1+d f x^2\right )+6 b n \left (2 a^2-2 a b n+b^2 n^2+2 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+4 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \left (\frac {1}{2} d f x^2-d f x^2 \log (x)+\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right )+3 b^2 n^2 \left (-2 a+b n+2 b n \log (x)-2 b \log \left (c x^n\right )\right ) \left (d f x^2-2 d f x^2 \log (x)+2 d f x^2 \log ^2(x)-2 \log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-2 \log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\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 )-b^3 n^3 \left (-3 d f x^2+6 d f x^2 \log (x)-6 d f x^2 \log ^2(x)+4 d f x^2 \log ^3(x)-4 \log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-4 \log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-12 \log ^2(x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-12 \log ^2(x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+24 \log (x) \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+24 \log (x) \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )-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 )}{8 d f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________