Optimal. Leaf size=182 \[ \frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {f}}+x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-2 x \left (a+b \log \left (c x^n\right )\right )-\frac {i b n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+\frac {i b n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-b n x \log \left (d f x^2+1\right )-\frac {2 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+4 b n x \]
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Rubi [A] time = 0.11, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2448, 321, 205, 2370, 4848, 2391, 203} \[ -\frac {i b n \text {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+\frac {i b n \text {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {f}}-2 x \left (a+b \log \left (c x^n\right )\right )-b n x \log \left (d f x^2+1\right )-\frac {2 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+4 b n x \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 321
Rule 2370
Rule 2391
Rule 2448
Rule 4848
Rubi steps
\begin {align*} \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx &=-2 x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-(b n) \int \left (-2+\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f} x}+\log \left (1+d f x^2\right )\right ) \, dx\\ &=2 b n x-2 x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-(b n) \int \log \left (1+d f x^2\right ) \, dx-\frac {(2 b n) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {d} \sqrt {f}}\\ &=2 b n x-2 x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {f}}-b n x \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {(i b n) \int \frac {\log \left (1-i \sqrt {d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {d} \sqrt {f}}+\frac {(i b n) \int \frac {\log \left (1+i \sqrt {d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {d} \sqrt {f}}+(2 b d f n) \int \frac {x^2}{1+d f x^2} \, dx\\ &=4 b n x-2 x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {f}}-b n x \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {i b n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+\frac {i b n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-(2 b n) \int \frac {1}{1+d f x^2} \, dx\\ &=4 b n x-\frac {2 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-2 x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {f}}-b n x \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {i b n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+\frac {i b n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 254, normalized size = 1.40 \[ a x \log \left (d f x^2+1\right )+\frac {2 a \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-2 a x+\frac {2 b \left (\log \left (c x^n\right )+n (-\log (x))-n\right ) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+b x \left (\log \left (c x^n\right )-n\right ) \log \left (d f x^2+1\right )-2 b x \left (\log \left (c x^n\right )+n (-\log (x))-n\right )-2 b d f n \left (\frac {i \left (\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )}{2 d^{3/2} f^{3/2}}-\frac {i \left (\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )\right )}{2 d^{3/2} f^{3/2}}+\frac {x (\log (x)-1)}{d f}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a \log \left (d f x^{2} + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )\, dx \]
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 \[ {\left (b x \log \left (x^{n}\right ) - {\left (b {\left (n - \log \relax (c)\right )} - a\right )} x\right )} \log \left (d f x^{2} + 1\right ) - \int \frac {2 \, {\left (b d f x^{2} \log \left (x^{n}\right ) + {\left (a d f - {\left (d f n - d f \log \relax (c)\right )} b\right )} x^{2}\right )}}{d f x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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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