Optimal. Leaf size=39 \[ \frac{1}{4} b n \text{PolyLog}\left (3,-d f x^2\right )-\frac{1}{2} \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \]
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Rubi [A] time = 0.032042, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2374, 6589} \[ \frac{1}{4} b n \text{PolyLog}\left (3,-d f x^2\right )-\frac{1}{2} \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right ) \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 ) \text{Li}_2\left (-d f x^2\right )+\frac{1}{2} (b n) \int \frac{\text{Li}_2\left (-d f x^2\right )}{x} \, dx\\ &=-\frac{1}{2} \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )+\frac{1}{4} b n \text{Li}_3\left (-d f x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0103969, size = 50, normalized size = 1.28 \[ -\frac{1}{2} a \text{PolyLog}\left (2,-d f x^2\right )-\frac{1}{2} b \log \left (c x^n\right ) \text{PolyLog}\left (2,-d f x^2\right )+\frac{1}{4} b n \text{PolyLog}\left (3,-d f x^2\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.156, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \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}{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 (d f x^{2} + 1\right ) - \int -\frac{b d f n x \log \left (x\right )^{2} - 2 \, b d f x \log \left (x\right ) \log \left (x^{n}\right ) - 2 \,{\left (b d f \log \left (c\right ) + a d f\right )} x \log \left (x\right )}{d f x^{2} + 1}\,{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 \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a \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 )} \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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