Optimal. Leaf size=113 \[ -\frac{1}{2} m \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} b m n \text{PolyLog}\left (3,-\frac{f x^2}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.125981, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2375, 2337, 2374, 6589} \[ -\frac{1}{2} m \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} b m n \text{PolyLog}\left (3,-\frac{f x^2}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2375
Rule 2337
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x} \, dx &=\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{(f m) \int \frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{b n}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x^2}{e}\right )}{2 b n}+m \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x^2}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x^2}{e}\right )}{2 b n}-\frac{1}{2} m \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x^2}{e}\right )+\frac{1}{2} (b m n) \int \frac{\text{Li}_2\left (-\frac{f x^2}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x^2}{e}\right )}{2 b n}-\frac{1}{2} m \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x^2}{e}\right )+\frac{1}{4} b m n \text{Li}_3\left (-\frac{f x^2}{e}\right )\\ \end{align*}
Mathematica [C] time = 0.0868114, size = 297, normalized size = 2.63 \[ \frac{1}{2} \left (a m \text{PolyLog}\left (2,\frac{f x^2}{e}+1\right )-2 b m \log \left (c x^n\right ) \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 b m \log \left (c x^n\right ) \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 b m n \text{PolyLog}\left (3,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 b m n \text{PolyLog}\left (3,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+a \log \left (-\frac{f x^2}{e}\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 b m \log (x) \log \left (c x^n\right ) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )-b n \log ^2(x) \log \left (d \left (e+f x^2\right )^m\right )+b m n \log ^2(x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+b m n \log ^2(x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.792, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) }{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (b n \log \left (x\right )^{2} - 2 \, b \log \left (x\right ) \log \left (x^{n}\right ) - 2 \,{\left (b \log \left (c\right ) + a\right )} \log \left (x\right )\right )} \log \left ({\left (f x^{2} + e\right )}^{m}\right ) - \int -\frac{b f m n x^{2} \log \left (x\right )^{2} + b e \log \left (c\right ) \log \left (d\right ) - 2 \,{\left (b f m \log \left (c\right ) + a f m\right )} x^{2} \log \left (x\right ) +{\left (b f \log \left (c\right ) \log \left (d\right ) + a f \log \left (d\right )\right )} x^{2} + a e \log \left (d\right ) -{\left (2 \, b f m x^{2} \log \left (x\right ) - b f x^{2} \log \left (d\right ) - b e \log \left (d\right )\right )} \log \left (x^{n}\right )}{f x^{3} + e x}\,{d x} \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 \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x}, 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 )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]