Optimal. Leaf size=459 \[ -2 b \sqrt {-d} \sqrt {f} n \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+2 b \sqrt {-d} \sqrt {f} n \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+\sqrt {-d} \sqrt {f} \log \left (1-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2-\sqrt {-d} \sqrt {f} \log \left (\sqrt {-d} \sqrt {f} x+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+4 b \sqrt {d} \sqrt {f} n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-2 i b^2 \sqrt {d} \sqrt {f} n^2 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+2 i b^2 \sqrt {d} \sqrt {f} n^2 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+2 b^2 \sqrt {-d} \sqrt {f} n^2 \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )-2 b^2 \sqrt {-d} \sqrt {f} n^2 \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )-\frac {2 b^2 n^2 \log \left (d f x^2+1\right )}{x}+4 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \]
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Rubi [A] time = 0.56, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2305, 2304, 2378, 203, 2324, 12, 4848, 2391, 2330, 2317, 2374, 6589} \[ -2 b \sqrt {-d} \sqrt {f} n \text {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+2 b \sqrt {-d} \sqrt {f} n \text {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-2 i b^2 \sqrt {d} \sqrt {f} n^2 \text {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+2 i b^2 \sqrt {d} \sqrt {f} n^2 \text {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )+2 b^2 \sqrt {-d} \sqrt {f} n^2 \text {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )-2 b^2 \sqrt {-d} \sqrt {f} n^2 \text {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right )-\frac {2 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+\sqrt {-d} \sqrt {f} \log \left (1-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2-\sqrt {-d} \sqrt {f} \log \left (\sqrt {-d} \sqrt {f} x+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b \sqrt {d} \sqrt {f} n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b^2 n^2 \log \left (d f x^2+1\right )}{x}+4 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 2304
Rule 2305
Rule 2317
Rule 2324
Rule 2330
Rule 2374
Rule 2378
Rule 2391
Rule 4848
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^2} \, dx &=-\frac {2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-(2 f) \int \left (-\frac {2 b^2 d n^2}{1+d f x^2}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right )}{1+d f x^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2}\right ) \, dx\\ &=-\frac {2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}+(2 d f) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2} \, dx+(4 b d f n) \int \frac {a+b \log \left (c x^n\right )}{1+d f x^2} \, dx+\left (4 b^2 d f n^2\right ) \int \frac {1}{1+d f x^2} \, dx\\ &=4 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+4 b \sqrt {d} \sqrt {f} n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}+(2 d f) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1-\sqrt {-d} \sqrt {f} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1+\sqrt {-d} \sqrt {f} x\right )}\right ) \, dx-\left (4 b^2 d f n^2\right ) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f} x} \, dx\\ &=4 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+4 b \sqrt {d} \sqrt {f} n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}+(d f) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1-\sqrt {-d} \sqrt {f} x} \, dx+(d f) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1+\sqrt {-d} \sqrt {f} x} \, dx-\left (4 b^2 \sqrt {d} \sqrt {f} n^2\right ) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{x} \, dx\\ &=4 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+4 b \sqrt {d} \sqrt {f} n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )-\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\frac {2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\left (2 b \sqrt {-d} \sqrt {f} n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx+\left (2 b \sqrt {-d} \sqrt {f} n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{x} \, dx-\left (2 i b^2 \sqrt {d} \sqrt {f} n^2\right ) \int \frac {\log \left (1-i \sqrt {d} \sqrt {f} x\right )}{x} \, dx+\left (2 i b^2 \sqrt {d} \sqrt {f} n^2\right ) \int \frac {\log \left (1+i \sqrt {d} \sqrt {f} x\right )}{x} \, dx\\ &=4 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+4 b \sqrt {d} \sqrt {f} n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )-\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\frac {2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-2 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )+2 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )-2 i b^2 \sqrt {d} \sqrt {f} n^2 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+2 i b^2 \sqrt {d} \sqrt {f} n^2 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\left (2 b^2 \sqrt {-d} \sqrt {f} n^2\right ) \int \frac {\text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx-\left (2 b^2 \sqrt {-d} \sqrt {f} n^2\right ) \int \frac {\text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{x} \, dx\\ &=4 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+4 b \sqrt {d} \sqrt {f} n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )-\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\frac {2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-2 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )+2 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )-2 i b^2 \sqrt {d} \sqrt {f} n^2 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+2 i b^2 \sqrt {d} \sqrt {f} n^2 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+2 b^2 \sqrt {-d} \sqrt {f} n^2 \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )-2 b^2 \sqrt {-d} \sqrt {f} n^2 \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )\\ \end {align*}
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Mathematica [A] time = 0.31, size = 414, normalized size = 0.90 \[ 2 \sqrt {d} \sqrt {f} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a^2+2 a b \left (\log \left (c x^n\right )-n \log (x)\right )+2 a b n+b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+2 b^2 n \left (\log \left (c x^n\right )-n \log (x)\right )+2 b^2 n^2\right )-\frac {\log \left (d f x^2+1\right ) \left (a^2+2 b (a+b n) \log \left (c x^n\right )+2 a b n+b^2 \log ^2\left (c x^n\right )+2 b^2 n^2\right )}{x}+2 i b \sqrt {d} \sqrt {f} n \left (-\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log (x) \left (\log \left (1-i \sqrt {d} \sqrt {f} x\right )-\log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)+b n\right )+i b^2 \sqrt {d} \sqrt {f} n^2 \left (2 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )-2 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )-2 \log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{2} \log \left (d f x^{2} + 1\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )}{x^{2}}\, dx \]
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 \[ -\frac {{\left (b^{2} \log \left (x^{n}\right )^{2} + {\left (2 \, n^{2} + 2 \, n \log \relax (c) + \log \relax (c)^{2}\right )} b^{2} + 2 \, a b {\left (n + \log \relax (c)\right )} + a^{2} + 2 \, {\left (b^{2} {\left (n + \log \relax (c)\right )} + a b\right )} \log \left (x^{n}\right )\right )} \log \left (d f x^{2} + 1\right )}{x} + \int \frac {2 \, {\left (b^{2} d f \log \left (x^{n}\right )^{2} + a^{2} d f + 2 \, {\left (d f n + d f \log \relax (c)\right )} a b + {\left (2 \, d f n^{2} + 2 \, d f n \log \relax (c) + d f \log \relax (c)^{2}\right )} b^{2} + 2 \, {\left (a b d f + {\left (d f n + d f \log \relax (c)\right )} b^{2}\right )} \log \left (x^{n}\right )\right )}}{d f x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2} \,d x \]
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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