Optimal. Leaf size=257 \[ \frac {1}{2} b d f n \text {Li}_2\left (-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b d f n \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{2} d f \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac {1}{4} b^2 d f n^2 \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {1}{4} b^2 d f n^2 \text {Li}_3\left (-\frac {1}{d f x^2}\right )-\frac {1}{4} b^2 d f n^2 \log \left (d f x^2+1\right )-\frac {b^2 n^2 \log \left (d f x^2+1\right )}{4 x^2}+\frac {1}{2} b^2 d f n^2 \log (x) \]
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Rubi [A] time = 0.34, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {2305, 2304, 2378, 266, 36, 29, 31, 2345, 2391, 2374, 6589} \[ \frac {1}{2} b d f n \text {PolyLog}\left (2,-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b^2 d f n^2 \text {PolyLog}\left (2,-\frac {1}{d f x^2}\right )+\frac {1}{4} b^2 d f n^2 \text {PolyLog}\left (3,-\frac {1}{d f x^2}\right )-\frac {1}{2} b d f n \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{2} d f \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {1}{4} b^2 d f n^2 \log \left (d f x^2+1\right )-\frac {b^2 n^2 \log \left (d f x^2+1\right )}{4 x^2}+\frac {1}{2} b^2 d f n^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 2304
Rule 2305
Rule 2345
Rule 2374
Rule 2378
Rule 2391
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^3} \, dx &=-\frac {b^2 n^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 x^2}-(2 f) \int \left (-\frac {b^2 d n^2}{4 x \left (1+d f x^2\right )}-\frac {b d n \left (a+b \log \left (c x^n\right )\right )}{2 x \left (1+d f x^2\right )}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 x \left (1+d f x^2\right )}\right ) \, dx\\ &=-\frac {b^2 n^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 x^2}+(d f) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (1+d f x^2\right )} \, dx+(b d f n) \int \frac {a+b \log \left (c x^n\right )}{x \left (1+d f x^2\right )} \, dx+\frac {1}{2} \left (b^2 d f n^2\right ) \int \frac {1}{x \left (1+d f x^2\right )} \, dx\\ &=-\frac {1}{2} b d f n \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} d f \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 n^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 x^2}+(b d f n) \int \frac {\log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx+\frac {1}{4} \left (b^2 d f n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x (1+d f x)} \, dx,x,x^2\right )+\frac {1}{2} \left (b^2 d f n^2\right ) \int \frac {\log \left (1+\frac {1}{d f x^2}\right )}{x} \, dx\\ &=-\frac {1}{2} b d f n \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} d f \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 n^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 x^2}+\frac {1}{4} b^2 d f n^2 \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {1}{2} b d f n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {1}{4} \left (b^2 d f n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (b^2 d f n^2\right ) \int \frac {\text {Li}_2\left (-\frac {1}{d f x^2}\right )}{x} \, dx-\frac {1}{4} \left (b^2 d^2 f^2 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+d f x} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^2 d f n^2 \log (x)-\frac {1}{2} b d f n \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} d f \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{4} b^2 d f n^2 \log \left (1+d f x^2\right )-\frac {b^2 n^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 x^2}+\frac {1}{4} b^2 d f n^2 \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {1}{2} b d f n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {1}{4} b^2 d f n^2 \text {Li}_3\left (-\frac {1}{d f x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.36, size = 488, normalized size = 1.90 \[ \frac {1}{4} \left (2 d f \log (x) \left (2 a^2+4 a b \left (\log \left (c x^n\right )-n \log (x)\right )+2 a b n+2 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+2 b^2 n \left (\log \left (c x^n\right )-n \log (x)\right )+b^2 n^2\right )-\frac {\log \left (d f x^2+1\right ) \left (2 a^2+2 b (2 a+b n) \log \left (c x^n\right )+2 a b n+2 b^2 \log ^2\left (c x^n\right )+b^2 n^2\right )}{x^2}-d f \log \left (d f x^2+1\right ) \left (2 a^2+4 a b \left (\log \left (c x^n\right )-n \log (x)\right )+2 a b n+2 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+2 b^2 n \left (\log \left (c x^n\right )-n \log (x)\right )+b^2 n^2\right )-2 b d f n \left (-\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log (x) \left (-\log \left (1-i \sqrt {d} \sqrt {f} x\right )-\log \left (1+i \sqrt {d} \sqrt {f} x\right )+\log (x)\right )\right ) \left (-2 a-2 b \log \left (c x^n\right )+2 b n \log (x)-b n\right )+\frac {2}{3} b^2 d f n^2 \left (6 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-3 \log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-3 \log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+2 \log ^3(x)\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{2} \log \left (d f x^{2} + 1\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )}{x^{3}}\, dx \]
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 \[ -\frac {{\left (2 \, b^{2} \log \left (x^{n}\right )^{2} + {\left (n^{2} + 2 \, n \log \relax (c) + 2 \, \log \relax (c)^{2}\right )} b^{2} + 2 \, a b {\left (n + 2 \, \log \relax (c)\right )} + 2 \, a^{2} + 2 \, {\left (b^{2} {\left (n + 2 \, \log \relax (c)\right )} + 2 \, a b\right )} \log \left (x^{n}\right )\right )} \log \left (d f x^{2} + 1\right )}{4 \, x^{2}} + \int \frac {2 \, b^{2} d f \log \left (x^{n}\right )^{2} + 2 \, a^{2} d f + 2 \, {\left (d f n + 2 \, d f \log \relax (c)\right )} a b + {\left (d f n^{2} + 2 \, d f n \log \relax (c) + 2 \, d f \log \relax (c)^{2}\right )} b^{2} + 2 \, {\left (2 \, a b d f + {\left (d f n + 2 \, d f \log \relax (c)\right )} b^{2}\right )} \log \left (x^{n}\right )}{2 \, {\left (d f x^{3} + x\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3} \,d x \]
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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