Optimal. Leaf size=211 \[ -\frac {2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac {1}{3} i b d^{3/2} f^{3/2} n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-\frac {1}{3} i b d^{3/2} f^{3/2} n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-\frac {2}{9} b d^{3/2} f^{3/2} n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )-\frac {b n \log \left (d f x^2+1\right )}{9 x^3}-\frac {8 b d f n}{9 x} \]
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Rubi [A] time = 0.14, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2455, 325, 205, 2376, 4848, 2391, 203} \[ \frac {1}{3} i b d^{3/2} f^{3/2} n \text {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )-\frac {1}{3} i b d^{3/2} f^{3/2} n \text {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-\frac {2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2}{9} b d^{3/2} f^{3/2} n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )-\frac {b n \log \left (d f x^2+1\right )}{9 x^3}-\frac {8 b d f n}{9 x} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 325
Rule 2376
Rule 2391
Rule 2455
Rule 4848
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^4} \, dx &=-\frac {2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac {2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{3 x^3}-(b n) \int \left (-\frac {2 d f}{3 x^2}-\frac {2 d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{3 x}-\frac {\log \left (1+d f x^2\right )}{3 x^4}\right ) \, dx\\ &=-\frac {2 b d f n}{3 x}-\frac {2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac {2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{3 x^3}+\frac {1}{3} (b n) \int \frac {\log \left (1+d f x^2\right )}{x^4} \, dx+\frac {1}{3} \left (2 b d^{3/2} f^{3/2} n\right ) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{x} \, dx\\ &=-\frac {2 b d f n}{3 x}-\frac {2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac {2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f x^2\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{3 x^3}+\frac {1}{9} (2 b d f n) \int \frac {1}{x^2 \left (1+d f x^2\right )} \, dx+\frac {1}{3} \left (i b d^{3/2} f^{3/2} n\right ) \int \frac {\log \left (1-i \sqrt {d} \sqrt {f} x\right )}{x} \, dx-\frac {1}{3} \left (i b d^{3/2} f^{3/2} n\right ) \int \frac {\log \left (1+i \sqrt {d} \sqrt {f} x\right )}{x} \, dx\\ &=-\frac {8 b d f n}{9 x}-\frac {2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac {2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f x^2\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{3 x^3}+\frac {1}{3} i b d^{3/2} f^{3/2} n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-\frac {1}{3} i b d^{3/2} f^{3/2} n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-\frac {1}{9} \left (2 b d^2 f^2 n\right ) \int \frac {1}{1+d f x^2} \, dx\\ &=-\frac {8 b d f n}{9 x}-\frac {2}{9} b d^{3/2} f^{3/2} n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )-\frac {2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac {2}{3} d^{3/2} f^{3/2} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f x^2\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{3 x^3}+\frac {1}{3} i b d^{3/2} f^{3/2} n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-\frac {1}{3} i b d^{3/2} f^{3/2} n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\\ \end {align*}
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Mathematica [C] time = 0.19, size = 285, normalized size = 1.35 \[ -\frac {2 a d f \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-d f x^2\right )}{3 x}-\frac {a \log \left (d f x^2+1\right )}{3 x^3}-\frac {2}{9} b d^{3/2} f^{3/2} \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )+n\right ) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )-\frac {2 b \left (3 d f \left (\log \left (c x^n\right )-n \log (x)\right )+d f n\right )}{9 x}-\frac {b \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )+3 n \log (x)+n\right ) \log \left (d f x^2+1\right )}{9 x^3}+\frac {2}{3} b d f n \left (\frac {1}{2} i \sqrt {d} \sqrt {f} \left (\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )-\frac {1}{2} i \sqrt {d} \sqrt {f} \left (\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )\right )-\frac {1}{x}-\frac {\log (x)}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a \log \left (d f x^{2} + 1\right )}{x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )}{x^{4}}\, dx \]
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 \[ -\frac {{\left (b {\left (n + 3 \, \log \relax (c)\right )} + 3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} \log \left (d f x^{2} + 1\right )}{9 \, x^{3}} + \int \frac {2 \, {\left (3 \, b d f \log \left (x^{n}\right ) + 3 \, a d f + {\left (d f n + 3 \, d f \log \relax (c)\right )} b\right )}}{9 \, {\left (d f x^{4} + x^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]
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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