Optimal. Leaf size=459 \[ -2 b \sqrt{-d} \sqrt{f} n \text{PolyLog}\left (2,-\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )+2 b \sqrt{-d} \sqrt{f} n \text{PolyLog}\left (2,\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-2 i b^2 \sqrt{d} \sqrt{f} n^2 \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+2 i b^2 \sqrt{d} \sqrt{f} n^2 \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+2 b^2 \sqrt{-d} \sqrt{f} n^2 \text{PolyLog}\left (3,-\sqrt{-d} \sqrt{f} x\right )-2 b^2 \sqrt{-d} \sqrt{f} n^2 \text{PolyLog}\left (3,\sqrt{-d} \sqrt{f} x\right )-\frac{2 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+\sqrt{-d} \sqrt{f} \log \left (1-\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2-\sqrt{-d} \sqrt{f} \log \left (\sqrt{-d} \sqrt{f} x+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b \sqrt{d} \sqrt{f} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b^2 n^2 \log \left (d f x^2+1\right )}{x}+4 b^2 \sqrt{d} \sqrt{f} n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.555646, antiderivative size = 459, 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, 203, 2324, 12, 4848, 2391, 2330, 2317, 2374, 6589} \[ -2 b \sqrt{-d} \sqrt{f} n \text{PolyLog}\left (2,-\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )+2 b \sqrt{-d} \sqrt{f} n \text{PolyLog}\left (2,\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-2 i b^2 \sqrt{d} \sqrt{f} n^2 \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+2 i b^2 \sqrt{d} \sqrt{f} n^2 \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+2 b^2 \sqrt{-d} \sqrt{f} n^2 \text{PolyLog}\left (3,-\sqrt{-d} \sqrt{f} x\right )-2 b^2 \sqrt{-d} \sqrt{f} n^2 \text{PolyLog}\left (3,\sqrt{-d} \sqrt{f} x\right )-\frac{2 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+\sqrt{-d} \sqrt{f} \log \left (1-\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2-\sqrt{-d} \sqrt{f} \log \left (\sqrt{-d} \sqrt{f} x+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b \sqrt{d} \sqrt{f} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b^2 n^2 \log \left (d f x^2+1\right )}{x}+4 b^2 \sqrt{d} \sqrt{f} n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2305
Rule 2304
Rule 2378
Rule 203
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 (\frac{1}{d}+f x^2\right )\right )}{x^2} \, dx &=-\frac{2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-(2 f) \int \left (-\frac{2 b^2 d n^2}{1+d f x^2}-\frac{2 b d n \left (a+b \log \left (c x^n\right )\right )}{1+d f x^2}-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2}\right ) \, dx\\ &=-\frac{2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}+(2 d f) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2} \, dx+(4 b d f n) \int \frac{a+b \log \left (c x^n\right )}{1+d f x^2} \, dx+\left (4 b^2 d f n^2\right ) \int \frac{1}{1+d f x^2} \, dx\\ &=4 b^2 \sqrt{d} \sqrt{f} n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )+4 b \sqrt{d} \sqrt{f} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}+(2 d f) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1-\sqrt{-d} \sqrt{f} x\right )}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1+\sqrt{-d} \sqrt{f} x\right )}\right ) \, dx-\left (4 b^2 d f n^2\right ) \int \frac{\tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f} x} \, dx\\ &=4 b^2 \sqrt{d} \sqrt{f} n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )+4 b \sqrt{d} \sqrt{f} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}+(d f) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{1-\sqrt{-d} \sqrt{f} x} \, dx+(d f) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{1+\sqrt{-d} \sqrt{f} x} \, dx-\left (4 b^2 \sqrt{d} \sqrt{f} n^2\right ) \int \frac{\tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{x} \, dx\\ &=4 b^2 \sqrt{d} \sqrt{f} n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )+4 b \sqrt{d} \sqrt{f} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )+\sqrt{-d} \sqrt{f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right )-\sqrt{-d} \sqrt{f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt{-d} \sqrt{f} x\right )-\frac{2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\left (2 b \sqrt{-d} \sqrt{f} n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{x} \, dx+\left (2 b \sqrt{-d} \sqrt{f} n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{x} \, dx-\left (2 i b^2 \sqrt{d} \sqrt{f} n^2\right ) \int \frac{\log \left (1-i \sqrt{d} \sqrt{f} x\right )}{x} \, dx+\left (2 i b^2 \sqrt{d} \sqrt{f} n^2\right ) \int \frac{\log \left (1+i \sqrt{d} \sqrt{f} x\right )}{x} \, dx\\ &=4 b^2 \sqrt{d} \sqrt{f} n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )+4 b \sqrt{d} \sqrt{f} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )+\sqrt{-d} \sqrt{f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right )-\sqrt{-d} \sqrt{f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt{-d} \sqrt{f} x\right )-\frac{2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-2 b \sqrt{-d} \sqrt{f} n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )+2 b \sqrt{-d} \sqrt{f} n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )-2 i b^2 \sqrt{d} \sqrt{f} n^2 \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )+2 i b^2 \sqrt{d} \sqrt{f} n^2 \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )+\left (2 b^2 \sqrt{-d} \sqrt{f} n^2\right ) \int \frac{\text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )}{x} \, dx-\left (2 b^2 \sqrt{-d} \sqrt{f} n^2\right ) \int \frac{\text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )}{x} \, dx\\ &=4 b^2 \sqrt{d} \sqrt{f} n^2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )+4 b \sqrt{d} \sqrt{f} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )+\sqrt{-d} \sqrt{f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right )-\sqrt{-d} \sqrt{f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt{-d} \sqrt{f} x\right )-\frac{2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-2 b \sqrt{-d} \sqrt{f} n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )+2 b \sqrt{-d} \sqrt{f} n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )-2 i b^2 \sqrt{d} \sqrt{f} n^2 \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )+2 i b^2 \sqrt{d} \sqrt{f} n^2 \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )+2 b^2 \sqrt{-d} \sqrt{f} n^2 \text{Li}_3\left (-\sqrt{-d} \sqrt{f} x\right )-2 b^2 \sqrt{-d} \sqrt{f} n^2 \text{Li}_3\left (\sqrt{-d} \sqrt{f} x\right )\\ \end{align*}
Mathematica [A] time = 0.304211, size = 414, normalized size = 0.9 \[ 2 i b \sqrt{d} \sqrt{f} n \left (-\text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+\text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\log (x) \left (\log \left (1-i \sqrt{d} \sqrt{f} x\right )-\log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)+b n\right )+i b^2 \sqrt{d} \sqrt{f} n^2 \left (2 \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right )-2 \text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right )-2 \log (x) \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+2 \log (x) \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\log ^2(x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )-\log ^2(x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )-\frac{\log \left (d f x^2+1\right ) \left (a^2+2 b (a+b n) \log \left (c x^n\right )+2 a b n+b^2 \log ^2\left (c x^n\right )+2 b^2 n^2\right )}{x}+2 \sqrt{d} \sqrt{f} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a^2+2 a b \left (\log \left (c x^n\right )-n \log (x)\right )+2 a b n+b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+2 b^2 n \left (\log \left (c x^n\right )-n \log (x)\right )+2 b^2 n^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ({d}^{-1}+f{x}^{2} \right ) \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{b^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{2} \log \left (d f x^{2} + 1\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} + \frac{1}{d}\right )} d\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]