Optimal. Leaf size=530 \[ \frac {3 b^2 n^2 \text {Li}_2(-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {3 b^2 n^2 \text {Li}_3(-e x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac {3}{4} b^2 n^2 x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac {9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {9 a b^2 n^2 x}{2 e}-\frac {3 b n \text {Li}_2(-e x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {3}{4} b n x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {9 b^3 n^2 x \log \left (c x^n\right )}{2 e}-\frac {3 b^3 n^3 \text {Li}_2(-e x)}{4 e^2}-\frac {3 b^3 n^3 \text {Li}_3(-e x)}{2 e^2}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{e^2}+\frac {3 b^3 n^3 \log (e x+1)}{8 e^2}-\frac {3}{8} b^3 n^3 x^2 \log (e x+1)-\frac {45 b^3 n^3 x}{8 e}+\frac {3}{4} b^3 n^3 x^2 \]
[Out]
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Rubi [A] time = 0.49, antiderivative size = 530, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2395, 43, 2377, 2296, 2295, 2305, 2304, 2374, 2383, 6589, 2376, 2391} \[ \frac {3 b^2 n^2 \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {3 b^2 n^2 \text {PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {3 b n \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac {3 b^3 n^3 \text {PolyLog}(2,-e x)}{4 e^2}-\frac {3 b^3 n^3 \text {PolyLog}(3,-e x)}{2 e^2}-\frac {3 b^3 n^3 \text {PolyLog}(4,-e x)}{e^2}-\frac {3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac {3}{4} b^2 n^2 x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac {9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {9 a b^2 n^2 x}{2 e}+\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac {3}{4} b n x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}+\frac {1}{2} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac {3 b^3 n^3 \log (e x+1)}{8 e^2}-\frac {3}{8} b^3 n^3 x^2 \log (e x+1)-\frac {45 b^3 n^3 x}{8 e}+\frac {3}{4} b^3 n^3 x^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2374
Rule 2376
Rule 2377
Rule 2383
Rule 2391
Rule 2395
Rule 6589
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx &=\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-(3 b n) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac {1}{4} x \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2 x}+\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)\right ) \, dx\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {1}{4} (3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {1}{2} (3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx+\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx}{2 e^2}-\frac {(3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{2 e}\\ &=-\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {1}{4} x^2 \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 (1+e x)}{4 e^2}-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{2 e^2}-\frac {1}{4} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (3 b^2 n^2\right ) \int \left (\frac {a+b \log \left (c x^n\right )}{2 e}-\frac {1}{4} x \left (a+b \log \left (c x^n\right )\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2 x}+\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right ) \log (1+e 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(-e x)}{x} \, dx}{e^2}+\frac {\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=\frac {3 a b^2 n^2 x}{e}+\frac {3}{16} b^3 n^3 x^2-\frac {3}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {1}{4} x^2 \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 (1+e x)}{4 e^2}-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{2 e^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^2}-\frac {1}{4} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{2} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx-\frac {\left (3 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{2 e^2}+\frac {\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 e}+\frac {\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{e}-\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_3(-e x)}{x} \, dx}{e^2}\\ &=\frac {9 a b^2 n^2 x}{2 e}-\frac {3 b^3 n^3 x}{e}+\frac {3}{8} b^3 n^3 x^2+\frac {3 b^3 n^2 x \log \left (c x^n\right )}{e}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^2}+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^2}-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{2 e^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^2}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{e^2}+\frac {\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{2 e}-\frac {1}{2} \left (3 b^3 n^3\right ) \int \left (\frac {1}{2 e}-\frac {x}{4}-\frac {\log (1+e x)}{2 e^2 x}+\frac {1}{2} x \log (1+e x)\right ) \, dx-\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_2(-e x)}{x} \, dx}{2 e^2}\\ &=\frac {9 a b^2 n^2 x}{2 e}-\frac {21 b^3 n^3 x}{4 e}+\frac {9}{16} b^3 n^3 x^2+\frac {9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^2}+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^2}-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{2 e^2}-\frac {3 b^3 n^3 \text {Li}_3(-e x)}{2 e^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^2}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{e^2}-\frac {1}{4} \left (3 b^3 n^3\right ) \int x \log (1+e x) \, dx+\frac {\left (3 b^3 n^3\right ) \int \frac {\log (1+e x)}{x} \, dx}{4 e^2}\\ &=\frac {9 a b^2 n^2 x}{2 e}-\frac {21 b^3 n^3 x}{4 e}+\frac {9}{16} b^3 n^3 x^2+\frac {9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{8} b^3 n^3 x^2 \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^2}+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^2}-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b^3 n^3 \text {Li}_2(-e x)}{4 e^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{2 e^2}-\frac {3 b^3 n^3 \text {Li}_3(-e x)}{2 e^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^2}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{e^2}+\frac {1}{8} \left (3 b^3 e n^3\right ) \int \frac {x^2}{1+e x} \, dx\\ &=\frac {9 a b^2 n^2 x}{2 e}-\frac {21 b^3 n^3 x}{4 e}+\frac {9}{16} b^3 n^3 x^2+\frac {9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{8} b^3 n^3 x^2 \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^2}+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^2}-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b^3 n^3 \text {Li}_2(-e x)}{4 e^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{2 e^2}-\frac {3 b^3 n^3 \text {Li}_3(-e x)}{2 e^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^2}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{e^2}+\frac {1}{8} \left (3 b^3 e n^3\right ) \int \left (-\frac {1}{e^2}+\frac {x}{e}+\frac {1}{e^2 (1+e x)}\right ) \, dx\\ &=\frac {9 a b^2 n^2 x}{2 e}-\frac {45 b^3 n^3 x}{8 e}+\frac {3}{4} b^3 n^3 x^2+\frac {9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 n^3 \log (1+e x)}{8 e^2}-\frac {3}{8} b^3 n^3 x^2 \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^2}+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^2}-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b^3 n^3 \text {Li}_2(-e x)}{4 e^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{2 e^2}-\frac {3 b^3 n^3 \text {Li}_3(-e x)}{2 e^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^2}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{e^2}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 806, normalized size = 1.52 \[ \frac {-2 e^2 x^2 a^3+4 e x a^3+4 e^2 x^2 \log (e x+1) a^3-4 \log (e x+1) a^3+6 b e^2 n x^2 a^2-18 b e n x a^2-6 b e^2 x^2 \log \left (c x^n\right ) a^2+12 b e x \log \left (c x^n\right ) a^2-6 b e^2 n x^2 \log (e x+1) a^2+6 b n \log (e x+1) a^2+12 b e^2 x^2 \log \left (c x^n\right ) \log (e x+1) a^2-12 b \log \left (c x^n\right ) \log (e x+1) a^2-9 b^2 e^2 n^2 x^2 a-6 b^2 e^2 x^2 \log ^2\left (c x^n\right ) a+12 b^2 e x \log ^2\left (c x^n\right ) a+42 b^2 e n^2 x a+12 b^2 e^2 n x^2 \log \left (c x^n\right ) a-36 b^2 e n x \log \left (c x^n\right ) a-6 b^2 n^2 \log (e x+1) a+6 b^2 e^2 n^2 x^2 \log (e x+1) a-12 b^2 \log ^2\left (c x^n\right ) \log (e x+1) a+12 b^2 e^2 x^2 \log ^2\left (c x^n\right ) \log (e x+1) a-12 b^2 e^2 n x^2 \log \left (c x^n\right ) \log (e x+1) a+12 b^2 n \log \left (c x^n\right ) \log (e x+1) a-2 b^3 e^2 x^2 \log ^3\left (c x^n\right )+4 b^3 e x \log ^3\left (c x^n\right )+6 b^3 e^2 n^3 x^2+6 b^3 e^2 n x^2 \log ^2\left (c x^n\right )-18 b^3 e n x \log ^2\left (c x^n\right )-45 b^3 e n^3 x-9 b^3 e^2 n^2 x^2 \log \left (c x^n\right )+42 b^3 e n^2 x \log \left (c x^n\right )+3 b^3 n^3 \log (e x+1)-4 b^3 \log ^3\left (c x^n\right ) \log (e x+1)+4 b^3 e^2 x^2 \log ^3\left (c x^n\right ) \log (e x+1)-3 b^3 e^2 n^3 x^2 \log (e x+1)-6 b^3 e^2 n x^2 \log ^2\left (c x^n\right ) \log (e x+1)+6 b^3 n \log ^2\left (c x^n\right ) \log (e x+1)-6 b^3 n^2 \log \left (c x^n\right ) \log (e x+1)+6 b^3 e^2 n^2 x^2 \log \left (c x^n\right ) \log (e x+1)-6 b n \left (2 a^2-2 b n a+b^2 n^2+2 b^2 \log ^2\left (c x^n\right )-2 b (b n-2 a) \log \left (c x^n\right )\right ) \text {Li}_2(-e x)+12 b^2 n^2 \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)-24 b^3 n^3 \text {Li}_4(-e x)}{8 e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x \log \left (c x^{n}\right )^{3} \log \left (e x + 1\right ) + 3 \, a b^{2} x \log \left (c x^{n}\right )^{2} \log \left (e x + 1\right ) + 3 \, a^{2} b x \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a^{3} x \log \left (e x + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left (e x + 1\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{3} x \ln \left (e x +1\right )\, dx \]
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 \[ -\frac {{\left (b^{3} e^{2} x^{2} - 2 \, b^{3} e x - 2 \, {\left (b^{3} e^{2} x^{2} - b^{3}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{3}}{4 \, e^{2}} + \frac {{\left (2 \, x^{2} \log \left (e x + 1\right ) - e {\left (\frac {e x^{2} - 2 \, x}{e^{2}} + \frac {2 \, \log \left (e x + 1\right )}{e^{3}}\right )}\right )} b^{3} e^{2} \log \relax (c)^{3} + 3 \, {\left (2 \, x^{2} \log \left (e x + 1\right ) - e {\left (\frac {e x^{2} - 2 \, x}{e^{2}} + \frac {2 \, \log \left (e x + 1\right )}{e^{3}}\right )}\right )} a b^{2} e^{2} \log \relax (c)^{2} + 3 \, {\left (2 \, x^{2} \log \left (e x + 1\right ) - e {\left (\frac {e x^{2} - 2 \, x}{e^{2}} + \frac {2 \, \log \left (e x + 1\right )}{e^{3}}\right )}\right )} a^{2} b e^{2} \log \relax (c) + {\left (2 \, x^{2} \log \left (e x + 1\right ) - e {\left (\frac {e x^{2} - 2 \, x}{e^{2}} + \frac {2 \, \log \left (e x + 1\right )}{e^{3}}\right )}\right )} a^{3} e^{2} + \int \frac {3 \, {\left (4 \, {\left (b^{3} e^{2} \log \relax (c)^{2} + 2 \, a b^{2} e^{2} \log \relax (c) + a^{2} b e^{2}\right )} x^{2} \log \left (e x + 1\right ) \log \left (x^{n}\right ) + {\left (b^{3} e^{2} n x^{2} - 2 \, b^{3} e n x + 2 \, {\left (b^{3} n + {\left (2 \, a b^{2} e^{2} - {\left (e^{2} n - 2 \, e^{2} \log \relax (c)\right )} b^{3}\right )} x^{2}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}\right )}}{x}\,{d x}}{4 \, e^{2}} \]
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 \[ \int x\,\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]
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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