Optimal. Leaf size=478 \[ -\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b \sqrt {f} m n \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-e}}+\frac {2 b \sqrt {f} m n \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-e}}+\frac {\sqrt {f} m \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}}-\frac {\sqrt {f} m \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}}+\frac {4 b \sqrt {f} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 i b^2 \sqrt {f} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {2 i b^2 \sqrt {f} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {2 b^2 \sqrt {f} m n^2 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 b^2 \sqrt {f} m n^2 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}+\frac {4 b^2 \sqrt {f} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}} \]
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Rubi [A] time = 0.52, antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2305, 2304, 2378, 205, 2324, 12, 4848, 2391, 2330, 2317, 2374, 6589} \[ -\frac {2 b \sqrt {f} m n \text {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-e}}+\frac {2 b \sqrt {f} m n \text {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-e}}-\frac {2 i b^2 \sqrt {f} m n^2 \text {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {2 i b^2 \sqrt {f} m n^2 \text {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {2 b^2 \sqrt {f} m n^2 \text {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 b^2 \sqrt {f} m n^2 \text {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}+\frac {\sqrt {f} m \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}}-\frac {\sqrt {f} m \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}}+\frac {4 b \sqrt {f} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}+\frac {4 b^2 \sqrt {f} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 2304
Rule 2305
Rule 2317
Rule 2324
Rule 2330
Rule 2374
Rule 2378
Rule 2391
Rule 4848
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx &=-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-(2 f m) \int \left (-\frac {2 b^2 n^2}{e+f x^2}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{e+f x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2}\right ) \, dx\\ &=-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}+(2 f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx+(4 b f m n) \int \frac {a+b \log \left (c x^n\right )}{e+f x^2} \, dx+\left (4 b^2 f m n^2\right ) \int \frac {1}{e+f x^2} \, dx\\ &=\frac {4 b^2 \sqrt {f} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {4 b \sqrt {f} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}+(2 f 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-\left (4 b^2 f m n^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f} x} \, dx\\ &=\frac {4 b^2 \sqrt {f} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {4 b \sqrt {f} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}-\sqrt {f} x} \, dx}{\sqrt {-e}}-\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}+\sqrt {f} x} \, dx}{\sqrt {-e}}-\frac {\left (4 b^2 \sqrt {f} m n^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {e}}\\ &=\frac {4 b^2 \sqrt {f} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {4 b \sqrt {f} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {\sqrt {f} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {\sqrt {f} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (2 b \sqrt {f} 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 {-e}}+\frac {\left (2 b \sqrt {f} 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 {-e}}-\frac {\left (2 i b^2 \sqrt {f} m n^2\right ) \int \frac {\log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {e}}+\frac {\left (2 i b^2 \sqrt {f} m n^2\right ) \int \frac {\log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {e}}\\ &=\frac {4 b^2 \sqrt {f} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {4 b \sqrt {f} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {\sqrt {f} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {\sqrt {f} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b \sqrt {f} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}+\frac {2 b \sqrt {f} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 i b^2 \sqrt {f} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {2 i b^2 \sqrt {f} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {\left (2 b^2 \sqrt {f} m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{\sqrt {-e}}-\frac {\left (2 b^2 \sqrt {f} m n^2\right ) \int \frac {\text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{\sqrt {-e}}\\ &=\frac {4 b^2 \sqrt {f} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {4 b \sqrt {f} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {\sqrt {f} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {\sqrt {f} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b \sqrt {f} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}+\frac {2 b \sqrt {f} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 i b^2 \sqrt {f} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {2 i b^2 \sqrt {f} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {2 b^2 \sqrt {f} m n^2 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 b^2 \sqrt {f} m n^2 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 917, normalized size = 1.92 \[ \frac {2 \sqrt {f} m x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) a^2-\sqrt {e} \log \left (d \left (f x^2+e\right )^m\right ) a^2+4 b \sqrt {f} m n x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) a-4 b \sqrt {f} m n x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x) a+4 b \sqrt {f} m x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right ) a+2 i b \sqrt {f} m n x \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) a-2 i b \sqrt {f} m n x \log (x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right ) a-2 b \sqrt {e} n \log \left (d \left (f x^2+e\right )^m\right ) a-2 b \sqrt {e} \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) a+2 b^2 \sqrt {f} m n^2 x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2(x)+2 b^2 \sqrt {f} m x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2\left (c x^n\right )+4 b^2 \sqrt {f} m n^2 x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-4 b^2 \sqrt {f} m n^2 x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)+4 b^2 \sqrt {f} m n x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )-4 b^2 \sqrt {f} m n x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x) \log \left (c x^n\right )-i b^2 \sqrt {f} m n^2 x \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 i b^2 \sqrt {f} m n^2 x \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 i b^2 \sqrt {f} m n x \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+i b^2 \sqrt {f} m n^2 x \log ^2(x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right )-2 i b^2 \sqrt {f} m n^2 x \log (x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right )-2 i b^2 \sqrt {f} m n x \log (x) \log \left (c x^n\right ) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right )-2 b^2 \sqrt {e} n^2 \log \left (d \left (f x^2+e\right )^m\right )-b^2 \sqrt {e} \log ^2\left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )-2 b^2 \sqrt {e} n \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )-2 i b \sqrt {f} m n x \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 {f} m n x \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 {f} m n^2 x \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 i b^2 \sqrt {f} m n^2 x \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 69.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{2}}\, dx \]
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 \[ -\frac {{\left (b^{2} m \log \left (x^{n}\right )^{2} + 2 \, {\left (m n + m \log \relax (c)\right )} a b + {\left (2 \, m n^{2} + 2 \, m n \log \relax (c) + m \log \relax (c)^{2}\right )} b^{2} + a^{2} m + 2 \, {\left ({\left (m n + m \log \relax (c)\right )} b^{2} + a b m\right )} \log \left (x^{n}\right )\right )} \log \left (f x^{2} + e\right )}{x} + \int \frac {b^{2} e \log \relax (c)^{2} \log \relax (d) + 2 \, a b e \log \relax (c) \log \relax (d) + a^{2} e \log \relax (d) + {\left ({\left (2 \, f m + f \log \relax (d)\right )} a^{2} + 2 \, {\left (2 \, f m n + {\left (2 \, f m + f \log \relax (d)\right )} \log \relax (c)\right )} a b + {\left (4 \, f m n^{2} + 4 \, f m n \log \relax (c) + {\left (2 \, f m + f \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{2}\right )} x^{2} + {\left ({\left (2 \, f m + f \log \relax (d)\right )} b^{2} x^{2} + b^{2} e \log \relax (d)\right )} \log \left (x^{n}\right )^{2} + 2 \, {\left (b^{2} e \log \relax (c) \log \relax (d) + a b e \log \relax (d) + {\left ({\left (2 \, f m + f \log \relax (d)\right )} a b + {\left (2 \, f m n + {\left (2 \, f m + f \log \relax (d)\right )} \log \relax (c)\right )} b^{2}\right )} x^{2}\right )} \log \left (x^{n}\right )}{f x^{4} + e x^{2}}\,{d x} \]
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 \[ \int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2} \,d x \]
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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