Optimal. Leaf size=478 \[ -\frac{2 b \sqrt{f} m n \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{-e}}+\frac{2 b \sqrt{f} m n \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{-e}}-\frac{2 i b^2 \sqrt{f} m n^2 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{2 i b^2 \sqrt{f} m n^2 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{2 b^2 \sqrt{f} m n^2 \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{2 b^2 \sqrt{f} m n^2 \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}+\frac{\sqrt{f} m \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-e}}-\frac{\sqrt{f} m \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-e}}+\frac{4 b \sqrt{f} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}-\frac{2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}+\frac{4 b^2 \sqrt{f} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.515391, antiderivative size = 478, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 2305
Rule 2304
Rule 2378
Rule 205
Rule 2324
Rule 12
Rule 4848
Rule 2391
Rule 2330
Rule 2317
Rule 2374
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*}
Mathematica [A] time = 0.317987, 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{PolyLog}\left (2,-\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{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 i b^2 \sqrt{f} m n^2 x \text{PolyLog}\left (3,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 i b^2 \sqrt{f} m n^2 x \text{PolyLog}\left (3,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 7.095, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) }{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]