Optimal. Leaf size=514 \[ \frac {3}{2} b^3 m n^3 x^2-\frac {9}{4} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 e m n^3 \log \left (e+f x^2\right )}{8 f}-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{4 f}-\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^2}{e}\right )}{2 f}+\frac {3 b^3 e m n^3 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{8 f}-\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b^3 e m n^3 \text {Li}_3\left (-\frac {f x^2}{e}\right )}{8 f}-\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b^3 e m n^3 \text {Li}_4\left (-\frac {f x^2}{e}\right )}{8 f} \]
[Out]
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Rubi [A]
time = 0.62, antiderivative size = 514, normalized size of antiderivative = 1.00, number of steps
used = 26, number of rules used = 12, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2342, 2341,
2425, 272, 45, 2393, 2375, 2438, 2395, 2421, 6724, 2430} \begin {gather*} -\frac {3 b^2 e m n^2 \text {PolyLog}\left (2,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {3 b^2 e m n^2 \text {PolyLog}\left (3,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f}+\frac {3 b e m n \text {PolyLog}\left (2,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac {3 b^3 e m n^3 \text {PolyLog}\left (2,-\frac {f x^2}{e}\right )}{8 f}+\frac {3 b^3 e m n^3 \text {PolyLog}\left (3,-\frac {f x^2}{e}\right )}{8 f}+\frac {3 b^3 e m n^3 \text {PolyLog}\left (4,-\frac {f x^2}{e}\right )}{8 f}+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {3 b^2 e m n^2 \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {9}{4} b^2 m 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 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {3 b e m n \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac {e m \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f}+\frac {3}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {3 b^3 e m n^3 \log \left (e+f x^2\right )}{8 f}+\frac {3}{2} b^3 m n^3 x^2 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rule 2341
Rule 2342
Rule 2375
Rule 2393
Rule 2395
Rule 2421
Rule 2425
Rule 2430
Rule 2438
Rule 6724
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-(2 f m) \int \left (-\frac {3 b^3 n^3 x^3}{8 \left (e+f x^2\right )}+\frac {3 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{4 \left (e+f x^2\right )}-\frac {3 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{4 \left (e+f x^2\right )}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-(f m) \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2} \, dx+\frac {1}{2} (3 b f m n) \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx-\frac {1}{2} \left (3 b^2 f m n^2\right ) \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e+f x^2} \, dx+\frac {1}{4} \left (3 b^3 f m n^3\right ) \int \frac {x^3}{e+f x^2} \, dx\\ &=-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-(f m) \int \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{f \left (e+f x^2\right )}\right ) \, dx+\frac {1}{2} (3 b f m n) \int \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{f \left (e+f x^2\right )}\right ) \, dx-\frac {1}{2} \left (3 b^2 f m n^2\right ) \int \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{f}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{f \left (e+f x^2\right )}\right ) \, dx+\frac {1}{8} \left (3 b^3 f m n^3\right ) \text {Subst}\left (\int \frac {x}{e+f x} \, dx,x,x^2\right )\\ &=-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-m \int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx+(e m) \int \frac {x \left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2} \, dx+\frac {1}{2} (3 b m n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {1}{2} (3 b e m n) \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx-\frac {1}{2} \left (3 b^2 m n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{2} \left (3 b^2 e m n^2\right ) \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{e+f x^2} \, dx+\frac {1}{8} \left (3 b^3 f m n^3\right ) \text {Subst}\left (\int \left (\frac {1}{f}-\frac {e}{f (e+f x)}\right ) \, dx,x,x^2\right )\\ &=\frac {3}{4} b^3 m n^3 x^2-\frac {3}{4} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 e m n^3 \log \left (e+f x^2\right )}{8 f}-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{4 f}-\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^2}{e}\right )}{2 f}+\frac {1}{2} (3 b m n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {(3 b e m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{x} \, dx}{2 f}-\frac {1}{2} \left (3 b^2 m n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {\left (3 b^2 e m n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{x} \, dx}{2 f}-\frac {\left (3 b^3 e m n^3\right ) \int \frac {\log \left (1+\frac {f x^2}{e}\right )}{x} \, dx}{4 f}\\ &=\frac {9}{8} b^3 m n^3 x^2-\frac {3}{2} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 e m n^3 \log \left (e+f x^2\right )}{8 f}-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{4 f}-\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^2}{e}\right )}{2 f}+\frac {3 b^3 e m n^3 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{8 f}-\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 f}-\frac {1}{2} \left (3 b^2 m n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {\left (3 b^2 e m n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )}{x} \, dx}{2 f}+\frac {\left (3 b^3 e m n^3\right ) \int \frac {\text {Li}_2\left (-\frac {f x^2}{e}\right )}{x} \, dx}{4 f}\\ &=\frac {3}{2} b^3 m n^3 x^2-\frac {9}{4} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 e m n^3 \log \left (e+f x^2\right )}{8 f}-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{4 f}-\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^2}{e}\right )}{2 f}+\frac {3 b^3 e m n^3 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{8 f}-\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b^3 e m n^3 \text {Li}_3\left (-\frac {f x^2}{e}\right )}{8 f}-\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x^2}{e}\right )}{4 f}+\frac {\left (3 b^3 e m n^3\right ) \int \frac {\text {Li}_3\left (-\frac {f x^2}{e}\right )}{x} \, dx}{4 f}\\ &=\frac {3}{2} b^3 m n^3 x^2-\frac {9}{4} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 e m n^3 \log \left (e+f x^2\right )}{8 f}-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{4 f}-\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^2}{e}\right )}{2 f}+\frac {3 b^3 e m n^3 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{8 f}-\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b^3 e m n^3 \text {Li}_3\left (-\frac {f x^2}{e}\right )}{8 f}-\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b^3 e m n^3 \text {Li}_4\left (-\frac {f x^2}{e}\right )}{8 f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.32, size = 1911, normalized size = 3.72 \begin {gather*} \frac {-4 a^3 f m x^2+12 a^2 b f m n x^2-18 a b^2 f m n^2 x^2+12 b^3 f m n^3 x^2-12 a^2 b f m x^2 \log \left (c x^n\right )+24 a b^2 f m n x^2 \log \left (c x^n\right )-18 b^3 f m n^2 x^2 \log \left (c x^n\right )-12 a b^2 f m x^2 \log ^2\left (c x^n\right )+12 b^3 f m n x^2 \log ^2\left (c x^n\right )-4 b^3 f m x^2 \log ^3\left (c x^n\right )+12 a^2 b e m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 a b^2 e m n^2 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 b^3 e m n^3 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 a b^2 e m n^2 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 b^3 e m n^3 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 b^3 e m n^3 \log ^3(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+24 a b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b^3 e m n^2 \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b^3 e m n^2 \log ^2(x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+12 b^3 e m n \log (x) \log ^2\left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+12 a^2 b e m n \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 a b^2 e m n^2 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 b^3 e m n^3 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 a b^2 e m n^2 \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 b^3 e m n^3 \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 b^3 e m n^3 \log ^3(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+24 a b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b^3 e m n^2 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b^3 e m n^2 \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+12 b^3 e m n \log (x) \log ^2\left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 a^3 e m \log \left (e+f x^2\right )-6 a^2 b e m n \log \left (e+f x^2\right )+6 a b^2 e m n^2 \log \left (e+f x^2\right )-3 b^3 e m n^3 \log \left (e+f x^2\right )-12 a^2 b e m n \log (x) \log \left (e+f x^2\right )+12 a b^2 e m n^2 \log (x) \log \left (e+f x^2\right )-6 b^3 e m n^3 \log (x) \log \left (e+f x^2\right )+12 a b^2 e m n^2 \log ^2(x) \log \left (e+f x^2\right )-6 b^3 e m n^3 \log ^2(x) \log \left (e+f x^2\right )-4 b^3 e m n^3 \log ^3(x) \log \left (e+f x^2\right )+12 a^2 b e m \log \left (c x^n\right ) \log \left (e+f x^2\right )-12 a b^2 e m n \log \left (c x^n\right ) \log \left (e+f x^2\right )+6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (e+f x^2\right )-24 a b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (e+f x^2\right )+12 b^3 e m n^2 \log (x) \log \left (c x^n\right ) \log \left (e+f x^2\right )+12 b^3 e m n^2 \log ^2(x) \log \left (c x^n\right ) \log \left (e+f x^2\right )+12 a b^2 e m \log ^2\left (c x^n\right ) \log \left (e+f x^2\right )-6 b^3 e m n \log ^2\left (c x^n\right ) \log \left (e+f x^2\right )-12 b^3 e m n \log (x) \log ^2\left (c x^n\right ) \log \left (e+f x^2\right )+4 b^3 e m \log ^3\left (c x^n\right ) \log \left (e+f x^2\right )+4 a^3 f x^2 \log \left (d \left (e+f x^2\right )^m\right )-6 a^2 b f n x^2 \log \left (d \left (e+f x^2\right )^m\right )+6 a b^2 f n^2 x^2 \log \left (d \left (e+f x^2\right )^m\right )-3 b^3 f n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )+12 a^2 b f x^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-12 a b^2 f n x^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 f n^2 x^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+12 a b^2 f x^2 \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 f n x^2 \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+4 b^3 f x^2 \log ^3\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+6 b e m n \left (2 a^2-2 a b n+b^2 n^2-2 b (-2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 b e m n \left (2 a^2-2 a b n+b^2 n^2-2 b (-2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )-24 a b^2 e m n^2 \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+12 b^3 e m n^3 \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-24 b^3 e m n^2 \log \left (c x^n\right ) \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-24 a b^2 e m n^2 \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )+12 b^3 e m n^3 \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )-24 b^3 e m n^2 \log \left (c x^n\right ) \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )+24 b^3 e m n^3 \text {Li}_4\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+24 b^3 e m n^3 \text {Li}_4\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{8 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.23, size = 47964, normalized size = 93.32
method | result | size |
risch | \(\text {Expression too large to display}\) | \(47964\) |
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 x\,\ln \left (d\,{\left (f\,x^2+e\right )}^m\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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