Optimal. Leaf size=70 \[ \frac{1}{2} b n \text{PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{4} b^2 n^2 \text{PolyLog}\left (4,-d f x^2\right ) \]
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Rubi [A] time = 0.066647, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {2374, 2383, 6589} \[ \frac{1}{2} b n \text{PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{4} b^2 n^2 \text{PolyLog}\left (4,-d f x^2\right ) \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2383
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} \, dx &=-\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )+(b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{x} \, dx\\ &=-\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )+\frac{1}{2} b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^2\right )-\frac{1}{2} \left (b^2 n^2\right ) \int \frac{\text{Li}_3\left (-d f x^2\right )}{x} \, dx\\ &=-\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )+\frac{1}{2} b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^2\right )-\frac{1}{4} b^2 n^2 \text{Li}_4\left (-d f x^2\right )\\ \end{align*}
Mathematica [C] time = 0.205319, size = 484, normalized size = 6.91 \[ \frac{1}{3} \left (3 b n \left (-2 \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right )-2 \text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right )+2 \log (x) \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+2 \log (x) \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\log ^2(x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )+\log ^2(x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right ) \left (-a-b \log \left (c x^n\right )+b n \log (x)\right )-3 \left (\text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+\text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\log (x) \left (\log \left (1-i \sqrt{d} \sqrt{f} x\right )+\log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2-b^2 n^2 \left (6 \text{PolyLog}\left (4,-i \sqrt{d} \sqrt{f} x\right )+6 \text{PolyLog}\left (4,i \sqrt{d} \sqrt{f} x\right )+3 \log ^2(x) \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+3 \log ^2(x) \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )-6 \log (x) \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right )-6 \log (x) \text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right )+\log ^3(x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )+\log ^3(x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )+\log (x) \log \left (d f x^2+1\right ) \left (-3 b n \log (x) \left (a+b \log \left (c x^n\right )\right )+3 \left (a+b \log \left (c x^n\right )\right )^2+b^2 n^2 \log ^2(x)\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.217, 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}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \,{\left (b^{2} n^{2} \log \left (x\right )^{3} + 3 \, b^{2} \log \left (x\right ) \log \left (x^{n}\right )^{2} - 3 \,{\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )^{2} - 3 \,{\left (b^{2} n \log \left (x\right )^{2} - 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x\right )\right )} \log \left (x^{n}\right ) + 3 \,{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2}\right )} \log \left (x\right )\right )} \log \left (d f x^{2} + 1\right ) - \int \frac{2 \,{\left (b^{2} d f n^{2} x \log \left (x\right )^{3} + 3 \, b^{2} d f x \log \left (x\right ) \log \left (x^{n}\right )^{2} - 3 \,{\left (b^{2} d f n \log \left (c\right ) + a b d f n\right )} x \log \left (x\right )^{2} + 3 \,{\left (b^{2} d f \log \left (c\right )^{2} + 2 \, a b d f \log \left (c\right ) + a^{2} d f\right )} x \log \left (x\right ) - 3 \,{\left (b^{2} d f n x \log \left (x\right )^{2} - 2 \,{\left (b^{2} d f \log \left (c\right ) + a b d f\right )} x \log \left (x\right )\right )} \log \left (x^{n}\right )\right )}}{3 \,{\left (d f x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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