Optimal. Leaf size=70 \[ \frac {1}{2} b n \text {Li}_3\left (-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} \text {Li}_2\left (-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{4} b^2 n^2 \text {Li}_4\left (-d f x^2\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 70, normalized size of antiderivative = 1.00, 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*}
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Mathematica [C] time = 0.21, size = 484, normalized size = 6.91 \[ \frac {1}{3} \left (\log (x) \log \left (d f x^2+1\right ) \left (-3 b n \log (x) \left (a+b \log \left (c x^n\right )\right )+3 \left (a+b \log \left (c x^n\right )\right )^2+b^2 n^2 \log ^2(x)\right )+3 b n \left (-2 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )-2 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )\right ) \left (-a-b \log \left (c x^n\right )+b n \log (x)\right )-3 \left (\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log (x) \left (\log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2-b^2 n^2 \left (6 \text {Li}_4\left (-i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )+\log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{2} \log \left (d f x^{2} + 1\right )}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.50, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )}{x}\, dx \]
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 \[ \frac {1}{3} \, {\left (b^{2} n^{2} \log \relax (x)^{3} + 3 \, b^{2} \log \relax (x) \log \left (x^{n}\right )^{2} - 3 \, {\left (b^{2} n \log \relax (c) + a b n\right )} \log \relax (x)^{2} - 3 \, {\left (b^{2} n \log \relax (x)^{2} - 2 \, {\left (b^{2} \log \relax (c) + a b\right )} \log \relax (x)\right )} \log \left (x^{n}\right ) + 3 \, {\left (b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c) + a^{2}\right )} \log \relax (x)\right )} \log \left (d f x^{2} + 1\right ) - \int \frac {2 \, {\left (b^{2} d f n^{2} x \log \relax (x)^{3} + 3 \, b^{2} d f x \log \relax (x) \log \left (x^{n}\right )^{2} - 3 \, {\left (b^{2} d f n \log \relax (c) + a b d f n\right )} x \log \relax (x)^{2} + 3 \, {\left (b^{2} d f \log \relax (c)^{2} + 2 \, a b d f \log \relax (c) + a^{2} d f\right )} x \log \relax (x) - 3 \, {\left (b^{2} d f n x \log \relax (x)^{2} - 2 \, {\left (b^{2} d f \log \relax (c) + a b d f\right )} x \log \relax (x)\right )} \log \left (x^{n}\right )\right )}}{3 \, {\left (d f x^{2} + 1\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \]
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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