Optimal. Leaf size=473 \[ -\frac{6 b^2 e m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{6 b^2 e m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f}+\frac{3 b e m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac{6 b^3 e m n^3 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{f}+\frac{6 b^3 e m n^3 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{f}+\frac{6 b^3 e m n^3 \text{PolyLog}\left (4,-\frac{f x}{e}\right )}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )+\frac{6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}-12 a b^2 m n^2 x-6 b^2 m n^2 x (a-b n)-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac{3 b e m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac{e m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f}+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+\frac{6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )}{f}-18 b^3 m n^2 x \log \left (c x^n\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+18 b^3 m n^3 x \]
[Out]
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Rubi [A] time = 0.652272, antiderivative size = 473, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {2296, 2295, 2371, 6, 43, 2351, 2317, 2391, 2353, 2374, 6589, 2383} \[ -\frac{6 b^2 e m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{6 b^2 e m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f}+\frac{3 b e m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac{6 b^3 e m n^3 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{f}+\frac{6 b^3 e m n^3 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{f}+\frac{6 b^3 e m n^3 \text{PolyLog}\left (4,-\frac{f x}{e}\right )}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )+\frac{6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}-12 a b^2 m n^2 x-6 b^2 m n^2 x (a-b n)-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac{3 b e m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac{e m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f}+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+\frac{6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )}{f}-18 b^3 m n^2 x \log \left (c x^n\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+18 b^3 m n^3 x \]
Antiderivative was successfully verified.
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Rule 2296
Rule 2295
Rule 2371
Rule 6
Rule 43
Rule 2351
Rule 2317
Rule 2391
Rule 2353
Rule 2374
Rule 6589
Rule 2383
Rubi steps
\begin{align*} \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx &=6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac{6 a b^2 n^2 x}{e+f x}-\frac{6 b^3 n^3 x}{e+f x}+\frac{6 b^3 n^2 x \log \left (c x^n\right )}{e+f x}-\frac{3 b n x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{e+f x}\right ) \, dx\\ &=6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac{\left (6 a b^2 n^2-6 b^3 n^3\right ) x}{e+f x}+\frac{6 b^3 n^2 x \log \left (c x^n\right )}{e+f x}-\frac{3 b n x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{e+f x}\right ) \, dx\\ &=6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-(f m) \int \frac{x \left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx+(3 b f m n) \int \frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx-\left (6 b^3 f m n^2\right ) \int \frac{x \log \left (c x^n\right )}{e+f x} \, dx-\left (6 b^2 f m n^2 (a-b n)\right ) \int \frac{x}{e+f x} \, dx\\ &=6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{f (e+f x)}\right ) \, dx+(3 b f m n) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{f (e+f x)}\right ) \, dx-\left (6 b^3 f m n^2\right ) \int \left (\frac{\log \left (c x^n\right )}{f}-\frac{e \log \left (c x^n\right )}{f (e+f x)}\right ) \, dx-\left (6 b^2 f m n^2 (a-b n)\right ) \int \left (\frac{1}{f}-\frac{e}{f (e+f x)}\right ) \, dx\\ &=-6 b^2 m n^2 (a-b n) x+\frac{6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-m \int \left (a+b \log \left (c x^n\right )\right )^3 \, dx+(e m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx+(3 b m n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(3 b e m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx-\left (6 b^3 m n^2\right ) \int \log \left (c x^n\right ) \, dx+\left (6 b^3 e m n^2\right ) \int \frac{\log \left (c x^n\right )}{e+f x} \, dx\\ &=6 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x-6 b^3 m n^2 x \log \left (c x^n\right )+3 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+\frac{6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac{6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (1+\frac{f x}{e}\right )}{f}-\frac{3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{f}+\frac{e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{f}+(3 b m n) \int \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}{e}\right )}{x} \, dx}{f}-\left (6 b^2 m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{\left (6 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}{e}\right )}{x} \, dx}{f}-\frac{\left (6 b^3 e m n^3\right ) \int \frac{\log \left (1+\frac{f x}{e}\right )}{x} \, dx}{f}\\ &=-6 a b^2 m n^2 x+6 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x-6 b^3 m n^2 x \log \left (c x^n\right )+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+\frac{6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac{6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (1+\frac{f x}{e}\right )}{f}-\frac{3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{f}+\frac{e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{f}+\frac{6 b^3 e m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{f}-\frac{6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{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}{e}\right )}{f}-\left (6 b^2 m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-\left (6 b^3 m n^2\right ) \int \log \left (c x^n\right ) \, dx-\frac{\left (6 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}{e}\right )}{x} \, dx}{f}+\frac{\left (6 b^3 e m n^3\right ) \int \frac{\text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx}{f}\\ &=-12 a b^2 m n^2 x+12 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x-12 b^3 m n^2 x \log \left (c x^n\right )+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+\frac{6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac{6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (1+\frac{f x}{e}\right )}{f}-\frac{3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{f}+\frac{e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{f}+\frac{6 b^3 e m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{f}-\frac{6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{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}{e}\right )}{f}+\frac{6 b^3 e m n^3 \text{Li}_3\left (-\frac{f x}{e}\right )}{f}-\frac{6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x}{e}\right )}{f}-\left (6 b^3 m n^2\right ) \int \log \left (c x^n\right ) \, dx+\frac{\left (6 b^3 e m n^3\right ) \int \frac{\text{Li}_3\left (-\frac{f x}{e}\right )}{x} \, dx}{f}\\ &=-12 a b^2 m n^2 x+18 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x-18 b^3 m n^2 x \log \left (c x^n\right )+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+\frac{6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac{6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (1+\frac{f x}{e}\right )}{f}-\frac{3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{f}+\frac{e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{f}+\frac{6 b^3 e m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{f}-\frac{6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{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}{e}\right )}{f}+\frac{6 b^3 e m n^3 \text{Li}_3\left (-\frac{f x}{e}\right )}{f}-\frac{6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x}{e}\right )}{f}+\frac{6 b^3 e m n^3 \text{Li}_4\left (-\frac{f x}{e}\right )}{f}\\ \end{align*}
Mathematica [B] time = 0.40768, size = 1122, normalized size = 2.37 \[ \frac{-f m x a^3+e m \log (e+f x) a^3+f x \log \left (d (e+f x)^m\right ) a^3+6 b f m n x a^2-3 b f m x \log \left (c x^n\right ) a^2-3 b e m n \log (e+f x) a^2-3 b e m n \log (x) \log (e+f x) a^2+3 b e m \log \left (c x^n\right ) \log (e+f x) a^2-3 b f n x \log \left (d (e+f x)^m\right ) a^2+3 b f x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right ) a^2+3 b e m n \log (x) \log \left (\frac{f x}{e}+1\right ) a^2-3 b^2 f m x \log ^2\left (c x^n\right ) a-18 b^2 f m n^2 x a+12 b^2 f m n x \log \left (c x^n\right ) a+6 b^2 e m n^2 \log (e+f x) a+3 b^2 e m n^2 \log ^2(x) \log (e+f x) a+3 b^2 e m \log ^2\left (c x^n\right ) \log (e+f x) a+6 b^2 e m n^2 \log (x) \log (e+f x) a-6 b^2 e m n \log \left (c x^n\right ) \log (e+f x) a-6 b^2 e m n \log (x) \log \left (c x^n\right ) \log (e+f x) a+3 b^2 f x \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right ) a+6 b^2 f n^2 x \log \left (d (e+f x)^m\right ) a-6 b^2 f n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right ) a-3 b^2 e m n^2 \log ^2(x) \log \left (\frac{f x}{e}+1\right ) a-6 b^2 e m n^2 \log (x) \log \left (\frac{f x}{e}+1\right ) a+6 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right ) a-b^3 f m x \log ^3\left (c x^n\right )+6 b^3 f m n x \log ^2\left (c x^n\right )+24 b^3 f m n^3 x-18 b^3 f m n^2 x \log \left (c x^n\right )-6 b^3 e m n^3 \log (e+f x)-b^3 e m n^3 \log ^3(x) \log (e+f x)+b^3 e m \log ^3\left (c x^n\right ) \log (e+f x)-3 b^3 e m n^3 \log ^2(x) \log (e+f x)-3 b^3 e m n \log ^2\left (c x^n\right ) \log (e+f x)-3 b^3 e m n \log (x) \log ^2\left (c x^n\right ) \log (e+f x)-6 b^3 e m n^3 \log (x) \log (e+f x)+6 b^3 e m n^2 \log \left (c x^n\right ) \log (e+f x)+3 b^3 e m n^2 \log ^2(x) \log \left (c x^n\right ) \log (e+f x)+6 b^3 e m n^2 \log (x) \log \left (c x^n\right ) \log (e+f x)+b^3 f x \log ^3\left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b^3 f n x \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-6 b^3 f n^3 x \log \left (d (e+f x)^m\right )+6 b^3 f n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^3 e m n^3 \log ^3(x) \log \left (\frac{f x}{e}+1\right )+3 b^3 e m n^3 \log ^2(x) \log \left (\frac{f x}{e}+1\right )+3 b^3 e m n \log (x) \log ^2\left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )+6 b^3 e m n^3 \log (x) \log \left (\frac{f x}{e}+1\right )-3 b^3 e m n^2 \log ^2(x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )-6 b^3 e m n^2 \log (x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )+3 b e m n \left (a^2-2 b n a+2 b^2 n^2+b^2 \log ^2\left (c x^n\right )+2 b (a-b n) \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{f x}{e}\right )-6 b^2 e m n^2 \left (a-b n+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,-\frac{f x}{e}\right )+6 b^3 e m n^3 \text{PolyLog}\left (4,-\frac{f x}{e}\right )}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 5.89, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ( fx+e \right ) ^{m} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left ({\left (f x + e\right )}^{m} d\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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