Optimal. Leaf size=180 \[ -\frac {3 b n x^2}{16 d f}+\frac {1}{16} b n x^4+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {1}{16} b n x^4 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {b n \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2504, 2442, 45,
2423, 2438} \begin {gather*} -\frac {b n \text {PolyLog}\left (2,-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 )}{4 d^2 f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 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 )-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log \left (d f x^2+1\right )}{16 d^2 f^2}-\frac {3 b n x^2}{16 d f}-\frac {1}{16} b n x^4 \log \left (d f x^2+1\right )+\frac {1}{16} b n x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2423
Rule 2438
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int x^3 \left (a+b \log \left (c x^n\right )\right ) \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 )}{4 d f}-\frac {1}{8} x^4 \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}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-(b n) \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 {b n x^2}{8 d f}+\frac {1}{32} b n x^4+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \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}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {1}{4} (b n) \int x^3 \log \left (1+d f x^2\right ) \, dx+\frac {(b n) \int \frac {\log \left (1+d f x^2\right )}{x} \, dx}{4 d^2 f^2}\\ &=-\frac {b n x^2}{8 d f}+\frac {1}{32} b n x^4+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \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}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {b n \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {1}{8} (b n) \text {Subst}\left (\int x \log (1+d f x) \, dx,x,x^2\right )\\ &=-\frac {b n x^2}{8 d f}+\frac {1}{32} b n x^4+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b n x^4 \log \left (1+d f x^2\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}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {b n \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}+\frac {1}{16} (b d f n) \text {Subst}\left (\int \frac {x^2}{1+d f x} \, dx,x,x^2\right )\\ &=-\frac {b n x^2}{8 d f}+\frac {1}{32} b n x^4+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b n x^4 \log \left (1+d f x^2\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}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {b n \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}+\frac {1}{16} (b d f n) \text {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 {3 b n x^2}{16 d f}+\frac {1}{16} b n x^4+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {1}{16} b n x^4 \log \left (1+d f x^2\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}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {b n \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 348, normalized size = 1.93 \begin {gather*} \frac {a x^2}{4 d f}-\frac {a x^4}{8}+\frac {1}{32} b x^4 \left (n-4 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+\frac {b x^2 \left (-n+4 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{16 d f}-\frac {a \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} a x^4 \log \left (1+d f x^2\right )+\frac {b \left (n-4 \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \log \left (1+d f x^2\right )}{16 d^2 f^2}+\frac {1}{16} b x^4 \left (-n+4 n \log (x)+4 \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \log \left (1+d f x^2\right )-\frac {1}{2} b d f n \left (-\frac {-\frac {x^2}{4}+\frac {1}{2} x^2 \log (x)}{d^2 f^2}+\frac {-\frac {x^4}{16}+\frac {1}{4} x^4 \log (x)}{d f}+\frac {\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{2 d^3 f^3}+\frac {\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{2 d^3 f^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.13, size = 840, normalized size = 4.67
method | result | size |
risch | \(-\frac {i \pi b \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{16}-\frac {i \pi b \,x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{16}-\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{4} \ln \left (d f \,x^{2}+1\right )}{8}+\frac {i x^{2} \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8 d f}-\frac {b n \dilog \left (1+x \sqrt {-d f}\right )}{4 d^{2} f^{2}}-\frac {\ln \left (c \right ) b \,x^{4}}{8}-\frac {b n \ln \left (x \right ) \ln \left (1-x \sqrt {-d f}\right )}{4 d^{2} f^{2}}-\frac {b n \ln \left (x \right ) \ln \left (1+x \sqrt {-d f}\right )}{4 d^{2} f^{2}}+\frac {a \,x^{4} \ln \left (d f \,x^{2}+1\right )}{4}+\frac {a \,x^{2}}{4 d f}-\frac {a \ln \left (d f \,x^{2}+1\right )}{4 d^{2} f^{2}}-\frac {i \ln \left (d f \,x^{2}+1\right ) \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8 d^{2} f^{2}}-\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{4} \ln \left (d f \,x^{2}+1\right )}{8}-\frac {i \ln \left (d f \,x^{2}+1\right ) \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8 d^{2} f^{2}}+\frac {i x^{2} \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8 d f}+\frac {b n \,x^{4}}{16}-\frac {a \,x^{4}}{8}-\frac {\ln \left (d f \,x^{2}+1\right ) b \ln \left (c \right )}{4 d^{2} f^{2}}+\frac {x^{2} b \ln \left (c \right )}{4 d f}+\frac {i \pi b \,x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{16}+\left (\frac {x^{4} b \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )}{4}-\frac {b \left (d^{2} f^{2} x^{4}-2 d f \,x^{2}+2 \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )+1\right )}{8 d^{2} f^{2}}\right ) \ln \left (x^{n}\right )+\frac {b n \ln \left (x \right )}{8 d^{2} f^{2}}-\frac {b n \dilog \left (1-x \sqrt {-d f}\right )}{4 d^{2} f^{2}}+\frac {b \ln \left (c \right ) x^{4} \ln \left (d f \,x^{2}+1\right )}{4}-\frac {3 b n \,x^{2}}{16 d f}-\frac {b n \,x^{4} \ln \left (d f \,x^{2}+1\right )}{16}+\frac {i \pi b \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{16}-\frac {i x^{2} \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8 d f}+\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{4} \ln \left (d f \,x^{2}+1\right )}{8}+\frac {i \ln \left (d f \,x^{2}+1\right ) \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8 d^{2} f^{2}}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{4} \ln \left (d f \,x^{2}+1\right )}{8}-\frac {i x^{2} \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8 d f}+\frac {i \ln \left (d f \,x^{2}+1\right ) \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8 d^{2} f^{2}}+\frac {n b \ln \left (x \right ) \ln \left (d f \,x^{2}+1\right )}{4 d^{2} f^{2}}+\frac {b n \ln \left (d f \,x^{2}+1\right )}{16 d^{2} f^{2}}\) | \(840\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \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 ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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