Optimal. Leaf size=169 \[ -i b \sqrt{d} \sqrt{f} n \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+i b \sqrt{d} \sqrt{f} n \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+2 \sqrt{d} \sqrt{f} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (d f x^2+1\right )}{x}+2 b \sqrt{d} \sqrt{f} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \]
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Rubi [A] time = 0.118955, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2455, 205, 2376, 4848, 2391, 203} \[ -i b \sqrt{d} \sqrt{f} n \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+i b \sqrt{d} \sqrt{f} n \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+2 \sqrt{d} \sqrt{f} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (d f x^2+1\right )}{x}+2 b \sqrt{d} \sqrt{f} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \]
Antiderivative was successfully verified.
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Rule 2455
Rule 205
Rule 2376
Rule 4848
Rule 2391
Rule 203
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^2} \, dx &=2 \sqrt{d} \sqrt{f} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-(b n) \int \left (\frac{2 \sqrt{d} \sqrt{f} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{x}-\frac{\log \left (1+d f x^2\right )}{x^2}\right ) \, dx\\ &=2 \sqrt{d} \sqrt{f} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}+(b n) \int \frac{\log \left (1+d f x^2\right )}{x^2} \, dx-\left (2 b \sqrt{d} \sqrt{f} n\right ) \int \frac{\tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{x} \, dx\\ &=2 \sqrt{d} \sqrt{f} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f x^2\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\left (i b \sqrt{d} \sqrt{f} n\right ) \int \frac{\log \left (1-i \sqrt{d} \sqrt{f} x\right )}{x} \, dx+\left (i b \sqrt{d} \sqrt{f} n\right ) \int \frac{\log \left (1+i \sqrt{d} \sqrt{f} x\right )}{x} \, dx+(2 b d f n) \int \frac{1}{1+d f x^2} \, dx\\ &=2 b \sqrt{d} \sqrt{f} n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )+2 \sqrt{d} \sqrt{f} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f x^2\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-i b \sqrt{d} \sqrt{f} n \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )+i b \sqrt{d} \sqrt{f} n \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )\\ \end{align*}
Mathematica [A] time = 0.0897354, size = 221, normalized size = 1.31 \[ 2 b d f n \left (\frac{i \left (\text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\log (x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )\right )}{2 \sqrt{d} \sqrt{f}}-\frac{i \left (\text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+\log (x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )}{2 \sqrt{d} \sqrt{f}}\right )-\frac{a \log \left (d f x^2+1\right )}{x}+2 a \sqrt{d} \sqrt{f} \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )-\frac{b \left (\log \left (c x^n\right )+n\right ) \log \left (d f x^2+1\right )}{x}+2 b \sqrt{d} \sqrt{f} \left (\log \left (c x^n\right )+n (-\log (x))+n\right ) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.174, 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}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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^{2}}, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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