Optimal. Leaf size=367 \[ -\frac {b n \text {Li}_2\left (-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac {1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} b n x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b^2 n^2 \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}+\frac {b^2 n^2 \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {b^2 n^2 \log \left (d f x^2+1\right )}{32 d^2 f^2}+\frac {7 b^2 n^2 x^2}{32 d f}+\frac {1}{32} b^2 n^2 x^4 \log \left (d f x^2+1\right )-\frac {3}{64} b^2 n^2 x^4 \]
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Rubi [A] time = 0.36, antiderivative size = 367, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {2454, 2395, 43, 2377, 2304, 2374, 6589, 2376, 2391} \[ -\frac {b n \text {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {b^2 n^2 \text {PolyLog}\left (2,-d f x^2\right )}{16 d^2 f^2}+\frac {b^2 n^2 \text {PolyLog}\left (3,-d f x^2\right )}{8 d^2 f^2}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} b n x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {b^2 n^2 \log \left (d f x^2+1\right )}{32 d^2 f^2}+\frac {7 b^2 n^2 x^2}{32 d f}+\frac {1}{32} b^2 n^2 x^4 \log \left (d f x^2+1\right )-\frac {3}{64} b^2 n^2 x^4 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2304
Rule 2374
Rule 2376
Rule 2377
Rule 2391
Rule 2395
Rule 2454
Rule 6589
Rubi steps
\begin {align*} \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx &=\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-(2 b n) \int \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^3 \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 )}{4 d^2 f^2 x}+\frac {1}{4} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )\right ) \, dx\\ &=\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac {1}{4} (b n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {1}{2} (b n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right ) \, dx+\frac {(b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x} \, dx}{2 d^2 f^2}-\frac {(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 d f}\\ &=\frac {b^2 n^2 x^2}{8 d f}-\frac {1}{64} b^2 n^2 x^4-\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac {1}{2} \left (b^2 n^2\right ) \int \left (\frac {x}{4 d f}-\frac {x^3}{8}-\frac {\log \left (1+d f x^2\right )}{4 d^2 f^2 x}+\frac {1}{4} x^3 \log \left (1+d f x^2\right )\right ) \, dx+\frac {\left (b^2 n^2\right ) \int \frac {\text {Li}_2\left (-d f x^2\right )}{x} \, dx}{4 d^2 f^2}\\ &=\frac {3 b^2 n^2 x^2}{16 d f}-\frac {1}{32} b^2 n^2 x^4-\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac {b^2 n^2 \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}+\frac {1}{8} \left (b^2 n^2\right ) \int x^3 \log \left (1+d f x^2\right ) \, dx-\frac {\left (b^2 n^2\right ) \int \frac {\log \left (1+d f x^2\right )}{x} \, dx}{8 d^2 f^2}\\ &=\frac {3 b^2 n^2 x^2}{16 d f}-\frac {1}{32} b^2 n^2 x^4-\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac {b^2 n^2 \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac {b^2 n^2 \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}+\frac {1}{16} \left (b^2 n^2\right ) \operatorname {Subst}\left (\int x \log (1+d f x) \, dx,x,x^2\right )\\ &=\frac {3 b^2 n^2 x^2}{16 d f}-\frac {1}{32} b^2 n^2 x^4-\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{32} b^2 n^2 x^4 \log \left (1+d f x^2\right )+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac {b^2 n^2 \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac {b^2 n^2 \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {1}{32} \left (b^2 d f n^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+d f x} \, dx,x,x^2\right )\\ &=\frac {3 b^2 n^2 x^2}{16 d f}-\frac {1}{32} b^2 n^2 x^4-\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{32} b^2 n^2 x^4 \log \left (1+d f x^2\right )+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac {b^2 n^2 \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac {b^2 n^2 \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {1}{32} \left (b^2 d f n^2\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{d^2 f^2}+\frac {x}{d f}+\frac {1}{d^2 f^2 (1+d f x)}\right ) \, dx,x,x^2\right )\\ &=\frac {7 b^2 n^2 x^2}{32 d f}-\frac {3}{64} b^2 n^2 x^4-\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 n^2 \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {1}{32} b^2 n^2 x^4 \log \left (1+d f x^2\right )+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac {b^2 n^2 \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac {b^2 n^2 \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}\\ \end {align*}
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Mathematica [C] time = 0.36, size = 654, normalized size = 1.78 \[ \frac {-d^2 f^2 x^4 \left (8 a^2+16 a b \left (\log \left (c x^n\right )-n \log (x)\right )-4 a b n+8 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+4 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+b^2 n^2\right )+2 d^2 f^2 x^4 \log \left (d f x^2+1\right ) \left (8 a^2-4 b (b n-4 a) \log \left (c x^n\right )-4 a b n+8 b^2 \log ^2\left (c x^n\right )+b^2 n^2\right )+2 d f x^2 \left (8 a^2+16 a b \left (\log \left (c x^n\right )-n \log (x)\right )-4 a b n+8 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+4 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+b^2 n^2\right )-2 \log \left (d f x^2+1\right ) \left (8 a^2+16 a b \left (\log \left (c x^n\right )-n \log (x)\right )-4 a b n+8 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+4 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+b^2 n^2\right )+b n \left (-d^2 f^2 x^4+4 d^2 f^2 x^4 \log (x)+8 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+8 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+4 d f x^2-8 d f x^2 \log (x)+8 \log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+8 \log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )\right ) \left (-4 a-4 b \log \left (c x^n\right )+4 b n \log (x)+b n\right )+32 b^2 n^2 \left (-\frac {1}{32} d^2 f^2 x^4 \left (8 \log ^2(x)-4 \log (x)+1\right )+\text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )-\log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-\log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\frac {1}{4} d f x^2 \left (2 \log ^2(x)-2 \log (x)+1\right )-\frac {1}{2} \log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-\frac {1}{2} \log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )}{64 d^2 f^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 2 \, a b x^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{2} x^{3} \log \left (d f x^{2} + 1\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{3} \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )\, dx \]
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 \[ \frac {1}{32} \, {\left (8 \, b^{2} x^{4} \log \left (x^{n}\right )^{2} - 4 \, {\left (b^{2} {\left (n - 4 \, \log \relax (c)\right )} - 4 \, a b\right )} x^{4} \log \left (x^{n}\right ) + {\left ({\left (n^{2} - 4 \, n \log \relax (c) + 8 \, \log \relax (c)^{2}\right )} b^{2} - 4 \, a b {\left (n - 4 \, \log \relax (c)\right )} + 8 \, a^{2}\right )} x^{4}\right )} \log \left (d f x^{2} + 1\right ) - \int \frac {8 \, b^{2} d f x^{5} \log \left (x^{n}\right )^{2} + 4 \, {\left (4 \, a b d f - {\left (d f n - 4 \, d f \log \relax (c)\right )} b^{2}\right )} x^{5} \log \left (x^{n}\right ) + {\left (8 \, a^{2} d f - 4 \, {\left (d f n - 4 \, d f \log \relax (c)\right )} a b + {\left (d f n^{2} - 4 \, d f n \log \relax (c) + 8 \, d f \log \relax (c)^{2}\right )} b^{2}\right )} x^{5}}{16 \, {\left (d f x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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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