Optimal. Leaf size=114 \[ \frac {\left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d f}-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \text {Li}_2\left (-d f x^2\right )}{4 d f}-\frac {b n \left (d f x^2+1\right ) \log \left (d f x^2+1\right )}{4 d f}+\frac {1}{2} b n x^2 \]
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Rubi [A] time = 0.18, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2454, 2389, 2295, 2376, 2475, 2411, 43, 2351, 2315} \[ \frac {b n \text {PolyLog}\left (2,-d f x^2\right )}{4 d f}+\frac {\left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d f}-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \left (d f x^2+1\right ) \log \left (d f x^2+1\right )}{4 d f}+\frac {1}{2} b n x^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2315
Rule 2351
Rule 2376
Rule 2389
Rule 2411
Rule 2454
Rule 2475
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx &=-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}-(b n) \int \left (-\frac {x}{2}+\frac {\left (1+d f x^2\right ) \log \left (1+d f x^2\right )}{2 d f x}\right ) \, dx\\ &=\frac {1}{4} b n x^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}-\frac {(b n) \int \frac {\left (1+d f x^2\right ) \log \left (1+d f x^2\right )}{x} \, dx}{2 d f}\\ &=\frac {1}{4} b n x^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}-\frac {(b n) \operatorname {Subst}\left (\int \frac {(1+d f x) \log (1+d f x)}{x} \, dx,x,x^2\right )}{4 d f}\\ &=\frac {1}{4} b n x^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}-\frac {(b n) \operatorname {Subst}\left (\int \frac {x \log (x)}{-\frac {1}{d f}+\frac {x}{d f}} \, dx,x,1+d f x^2\right )}{4 d^2 f^2}\\ &=\frac {1}{4} b n x^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}-\frac {(b n) \operatorname {Subst}\left (\int \left (d f \log (x)+\frac {d f \log (x)}{-1+x}\right ) \, dx,x,1+d f x^2\right )}{4 d^2 f^2}\\ &=\frac {1}{4} b n x^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}-\frac {(b n) \operatorname {Subst}\left (\int \log (x) \, dx,x,1+d f x^2\right )}{4 d f}-\frac {(b n) \operatorname {Subst}\left (\int \frac {\log (x)}{-1+x} \, dx,x,1+d f x^2\right )}{4 d f}\\ &=\frac {1}{2} b n x^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \left (1+d f x^2\right ) \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}+\frac {b n \text {Li}_2\left (-d f x^2\right )}{4 d f}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 267, normalized size = 2.34 \[ \frac {1}{2} a \left (\frac {\left (d f x^2+1\right ) \log \left (d f x^2+1\right )}{d f}-x^2\right )+\frac {1}{4} b x^2 \left (2 \left (\log \left (c x^n\right )-n \log (x)\right )+2 n \log (x)-n\right ) \log \left (d f x^2+1\right )+\frac {b \left (2 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right ) \log \left (d f x^2+1\right )}{4 d f}+\frac {1}{4} b x^2 \left (n-2 \left (\log \left (c x^n\right )-n \log (x)\right )\right )-b d f n \left (-\frac {\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )}{2 d^2 f^2}-\frac {\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )}{2 d^2 f^2}+\frac {\frac {1}{2} x^2 \log (x)-\frac {x^2}{4}}{d f}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b x \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a x \log \left (d f x^{2} + 1\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 820, normalized size = 7.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, {\left (2 \, b x^{2} \log \left (x^{n}\right ) - {\left (b {\left (n - 2 \, \log \relax (c)\right )} - 2 \, a\right )} x^{2}\right )} \log \left (d f x^{2} + 1\right ) - \int \frac {2 \, b d f x^{3} \log \left (x^{n}\right ) + {\left (2 \, a d f - {\left (d f n - 2 \, d f \log \relax (c)\right )} b\right )} x^{3}}{2 \, {\left (d f x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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