Optimal. Leaf size=141 \[ -\frac{1}{4} b d f n \text{PolyLog}\left (2,-d f x^2\right )+d f \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} d f \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{1}{4} b d f n \log \left (d f x^2+1\right )-\frac{b n \log \left (d f x^2+1\right )}{4 x^2}-\frac{1}{2} b d f n \log ^2(x)+\frac{1}{2} b d f n \log (x) \]
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Rubi [A] time = 0.127723, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2454, 2395, 36, 29, 31, 2376, 2301, 2391} \[ -\frac{1}{4} b d f n \text{PolyLog}\left (2,-d f x^2\right )+d f \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} d f \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{1}{4} b d f n \log \left (d f x^2+1\right )-\frac{b n \log \left (d f x^2+1\right )}{4 x^2}-\frac{1}{2} b d f n \log ^2(x)+\frac{1}{2} b d f n \log (x) \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2395
Rule 36
Rule 29
Rule 31
Rule 2376
Rule 2301
Rule 2391
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^3} \, dx &=d f \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} d f \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-(b n) \int \left (\frac{d f \log (x)}{x}-\frac{\log \left (1+d f x^2\right )}{2 x^3}-\frac{d f \log \left (1+d f x^2\right )}{2 x}\right ) \, dx\\ &=d f \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} d f \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}+\frac{1}{2} (b n) \int \frac{\log \left (1+d f x^2\right )}{x^3} \, dx+\frac{1}{2} (b d f n) \int \frac{\log \left (1+d f x^2\right )}{x} \, dx-(b d f n) \int \frac{\log (x)}{x} \, dx\\ &=-\frac{1}{2} b d f n \log ^2(x)+d f \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} d f \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-\frac{1}{4} b d f n \text{Li}_2\left (-d f x^2\right )+\frac{1}{4} (b n) \operatorname{Subst}\left (\int \frac{\log (1+d f x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac{1}{2} b d f n \log ^2(x)+d f \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f x^2\right )}{4 x^2}-\frac{1}{2} d f \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-\frac{1}{4} b d f n \text{Li}_2\left (-d f x^2\right )+\frac{1}{4} (b d f n) \operatorname{Subst}\left (\int \frac{1}{x (1+d f x)} \, dx,x,x^2\right )\\ &=-\frac{1}{2} b d f n \log ^2(x)+d f \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f x^2\right )}{4 x^2}-\frac{1}{2} d f \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-\frac{1}{4} b d f n \text{Li}_2\left (-d f x^2\right )+\frac{1}{4} (b d f n) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{4} \left (b d^2 f^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{1+d f x} \, dx,x,x^2\right )\\ &=\frac{1}{2} b d f n \log (x)-\frac{1}{2} b d f n \log ^2(x)+d f \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b d f n \log \left (1+d f x^2\right )-\frac{b n \log \left (1+d f x^2\right )}{4 x^2}-\frac{1}{2} d f \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-\frac{1}{4} b d f n \text{Li}_2\left (-d f x^2\right )\\ \end{align*}
Mathematica [C] time = 0.0956682, size = 241, normalized size = 1.71 \[ b d f n \left (\frac{1}{2} \left (-\text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )-\log (x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )+\frac{1}{2} \left (-\text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )-\log (x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )\right )+\frac{\log ^2(x)}{2}\right )-\frac{1}{2} a d f \log \left (d f x^2+1\right )-\frac{a \log \left (d f x^2+1\right )}{2 x^2}+a d f \log (x)+\frac{1}{2} b d f \log (x) \left (2 \left (\log \left (c x^n\right )-n \log (x)\right )+n\right )-\frac{1}{4} b d f \left (2 \left (\log \left (c x^n\right )-n \log (x)\right )+n\right ) \log \left (d f x^2+1\right )-\frac{b \left (2 \left (\log \left (c x^n\right )-n \log (x)\right )+2 n \log (x)+n\right ) \log \left (d f x^2+1\right )}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.09, size = 619, normalized size = 4.4 \begin{align*} \text{result too large to display} \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{{\left (b{\left (n + 2 \, \log \left (c\right )\right )} + 2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} \log \left (d f x^{2} + 1\right )}{4 \, x^{2}} + \int \frac{2 \, b d f \log \left (x^{n}\right ) + 2 \, a d f +{\left (d f n + 2 \, d f \log \left (c\right )\right )} b}{2 \,{\left (d f x^{3} + x\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 \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a \log \left (d f x^{2} + 1\right )}{x^{3}}, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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