Optimal. Leaf size=546 \[ x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 b \sqrt {-e} m n \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-\frac {2 b \sqrt {-e} m n \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-\frac {\sqrt {-e} m \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {f}}+\frac {\sqrt {-e} m \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )-\frac {4 b \sqrt {e} m n (a-b n) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+4 a b m n x+4 b m n x (a-b n)-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {4 b^2 \sqrt {e} m n \log \left (c x^n\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+8 b^2 m n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 i b^2 \sqrt {e} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 i b^2 \sqrt {e} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 b^2 \sqrt {-e} m n^2 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {2 b^2 \sqrt {-e} m n^2 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-8 b^2 m n^2 x \]
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Rubi [A] time = 0.81, antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 16, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {2296, 2295, 2371, 6, 321, 205, 2351, 2324, 12, 4848, 2391, 2353, 2330, 2317, 2374, 6589} \[ \frac {2 b \sqrt {-e} m n \text {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-\frac {2 b \sqrt {-e} m n \text {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}+\frac {2 i b^2 \sqrt {e} m n^2 \text {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 i b^2 \sqrt {e} m n^2 \text {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 b^2 \sqrt {-e} m n^2 \text {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {2 b^2 \sqrt {-e} m n^2 \text {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {\sqrt {-e} m \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {f}}+\frac {\sqrt {-e} m \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )-\frac {4 b \sqrt {e} m n (a-b n) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+4 a b m n x+4 b m n x (a-b n)-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {4 b^2 \sqrt {e} m n \log \left (c x^n\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+8 b^2 m n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-8 b^2 m n^2 x \]
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 205
Rule 321
Rule 2295
Rule 2296
Rule 2317
Rule 2324
Rule 2330
Rule 2351
Rule 2353
Rule 2371
Rule 2374
Rule 2391
Rule 4848
Rule 6589
Rubi steps
\begin {align*} \int \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-(2 f m) \int \left (-\frac {2 a b n x^2}{e+f x^2}+\frac {2 b^2 n^2 x^2}{e+f x^2}-\frac {2 b^2 n x^2 \log \left (c x^n\right )}{e+f x^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2}\right ) \, dx\\ &=-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-(2 f m) \int \left (\frac {\left (-2 a b n+2 b^2 n^2\right ) x^2}{e+f x^2}-\frac {2 b^2 n x^2 \log \left (c x^n\right )}{e+f x^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2}\right ) \, dx\\ &=-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-(2 f m) \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx+\left (4 b^2 f m n\right ) \int \frac {x^2 \log \left (c x^n\right )}{e+f x^2} \, dx+(4 b f m n (a-b n)) \int \frac {x^2}{e+f x^2} \, dx\\ &=4 b m n (a-b n) x-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-(2 f m) \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 \left (e+f x^2\right )}\right ) \, dx+\left (4 b^2 f m n\right ) \int \left (\frac {\log \left (c x^n\right )}{f}-\frac {e \log \left (c x^n\right )}{f \left (e+f x^2\right )}\right ) \, dx-(4 b e m n (a-b n)) \int \frac {1}{e+f x^2} \, dx\\ &=4 b m n (a-b n) x-\frac {4 b \sqrt {e} m n (a-b n) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-(2 m) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx+(2 e m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx+\left (4 b^2 m n\right ) \int \log \left (c x^n\right ) \, dx-\left (4 b^2 e m n\right ) \int \frac {\log \left (c x^n\right )}{e+f x^2} \, dx\\ &=-4 b^2 m n^2 x+4 b m n (a-b n) x-\frac {4 b \sqrt {e} m n (a-b n) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+4 b^2 m n x \log \left (c x^n\right )-\frac {4 b^2 \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+(2 e m) \int \left (\frac {\sqrt {-e} \left (a+b \log \left (c x^n\right )\right )^2}{2 e \left (\sqrt {-e}-\sqrt {f} x\right )}+\frac {\sqrt {-e} \left (a+b \log \left (c x^n\right )\right )^2}{2 e \left (\sqrt {-e}+\sqrt {f} x\right )}\right ) \, dx+(4 b m n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (4 b^2 e m n^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f} x} \, dx\\ &=4 a b m n x-4 b^2 m n^2 x+4 b m n (a-b n) x-\frac {4 b \sqrt {e} m n (a-b n) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+4 b^2 m n x \log \left (c x^n\right )-\frac {4 b^2 \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\left (\sqrt {-e} m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}-\sqrt {f} x} \, dx+\left (\sqrt {-e} m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}+\sqrt {f} x} \, dx+\left (4 b^2 m n\right ) \int \log \left (c x^n\right ) \, dx+\frac {\left (4 b^2 \sqrt {e} m n^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {f}}\\ &=4 a b m n x-8 b^2 m n^2 x+4 b m n (a-b n) x-\frac {4 b \sqrt {e} m n (a-b n) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+8 b^2 m n x \log \left (c x^n\right )-\frac {4 b^2 \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^2-\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {\left (2 b \sqrt {-e} m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{\sqrt {f}}-\frac {\left (2 b \sqrt {-e} m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{\sqrt {f}}+\frac {\left (2 i b^2 \sqrt {e} m n^2\right ) \int \frac {\log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {f}}-\frac {\left (2 i b^2 \sqrt {e} m n^2\right ) \int \frac {\log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {f}}\\ &=4 a b m n x-8 b^2 m n^2 x+4 b m n (a-b n) x-\frac {4 b \sqrt {e} m n (a-b n) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+8 b^2 m n x \log \left (c x^n\right )-\frac {4 b^2 \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^2-\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {2 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {2 i b^2 \sqrt {e} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 i b^2 \sqrt {e} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {\left (2 b^2 \sqrt {-e} m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{\sqrt {f}}+\frac {\left (2 b^2 \sqrt {-e} m n^2\right ) \int \frac {\text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{\sqrt {f}}\\ &=4 a b m n x-8 b^2 m n^2 x+4 b m n (a-b n) x-\frac {4 b \sqrt {e} m n (a-b n) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+8 b^2 m n x \log \left (c x^n\right )-\frac {4 b^2 \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^2-\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {2 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {2 i b^2 \sqrt {e} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 i b^2 \sqrt {e} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 b^2 \sqrt {-e} m n^2 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {2 b^2 \sqrt {-e} m n^2 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 993, normalized size = 1.82 \[ \frac {-2 \sqrt {f} m x a^2+2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) a^2+\sqrt {f} x \log \left (d \left (f x^2+e\right )^m\right ) a^2+8 b \sqrt {f} m n x a-4 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) a-4 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x) a-4 b \sqrt {f} m x \log \left (c x^n\right ) a+4 b \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right ) a+2 i b \sqrt {e} m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) a-2 i b \sqrt {e} m n \log (x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right ) a-2 b \sqrt {f} n x \log \left (d \left (f x^2+e\right )^m\right ) a+2 b \sqrt {f} x \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) a+2 b^2 \sqrt {e} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2(x)-2 b^2 \sqrt {f} m x \log ^2\left (c x^n\right )+2 b^2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2\left (c x^n\right )-12 b^2 \sqrt {f} m n^2 x+4 b^2 \sqrt {e} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+4 b^2 \sqrt {e} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)+8 b^2 \sqrt {f} m n x \log \left (c x^n\right )-4 b^2 \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )-4 b^2 \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x) \log \left (c x^n\right )-i b^2 \sqrt {e} m n^2 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 i b^2 \sqrt {e} m n^2 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 i b^2 \sqrt {e} m n \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+i b^2 \sqrt {e} m n^2 \log ^2(x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right )+2 i b^2 \sqrt {e} m n^2 \log (x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right )-2 i b^2 \sqrt {e} m n \log (x) \log \left (c x^n\right ) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right )+b^2 \sqrt {f} x \log ^2\left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )+2 b^2 \sqrt {f} n^2 x \log \left (d \left (f x^2+e\right )^m\right )-2 b^2 \sqrt {f} n x \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )-2 i b \sqrt {e} m n \left (a-b n+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 i b \sqrt {e} m n \left (a-b n+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 i b^2 \sqrt {e} m n^2 \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 i b^2 \sqrt {e} m n^2 \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 130.68, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )\, dx \]
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 \[ {\left (b^{2} m x \log \left (x^{n}\right )^{2} - 2 \, {\left ({\left (m n - m \log \relax (c)\right )} b^{2} - a b m\right )} x \log \left (x^{n}\right ) - {\left (2 \, {\left (m n - m \log \relax (c)\right )} a b - {\left (2 \, m n^{2} - 2 \, m n \log \relax (c) + m \log \relax (c)^{2}\right )} b^{2} - a^{2} m\right )} x\right )} \log \left (f x^{2} + e\right ) + \int \frac {b^{2} e \log \relax (c)^{2} \log \relax (d) + 2 \, a b e \log \relax (c) \log \relax (d) + a^{2} e \log \relax (d) - {\left ({\left (2 \, f m - f \log \relax (d)\right )} a^{2} - 2 \, {\left (2 \, f m n - {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)\right )} a b + {\left (4 \, f m n^{2} - 4 \, f m n \log \relax (c) + {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{2}\right )} x^{2} - {\left ({\left (2 \, f m - f \log \relax (d)\right )} b^{2} x^{2} - b^{2} e \log \relax (d)\right )} \log \left (x^{n}\right )^{2} + 2 \, {\left (b^{2} e \log \relax (c) \log \relax (d) + a b e \log \relax (d) - {\left ({\left (2 \, f m - f \log \relax (d)\right )} a b - {\left (2 \, f m n - {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)\right )} b^{2}\right )} x^{2}\right )} \log \left (x^{n}\right )}{f x^{2} + e}\,{d x} \]
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 \[ \int \ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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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