3.67 \(\int x^3 \log ^3(x) \, dx\)

Optimal. Leaf size=39 \[ -\frac{3 x^4}{128}+\frac{1}{4} x^4 \log ^3(x)-\frac{3}{16} x^4 \log ^2(x)+\frac{3}{32} x^4 \log (x) \]

[Out]

(-3*x^4)/128 + (3*x^4*Log[x])/32 - (3*x^4*Log[x]^2)/16 + (x^4*Log[x]^3)/4

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Rubi [A]  time = 0.0485654, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 x^4}{128}+\frac{1}{4} x^4 \log ^3(x)-\frac{3}{16} x^4 \log ^2(x)+\frac{3}{32} x^4 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[x^3*Log[x]^3,x]

[Out]

(-3*x^4)/128 + (3*x^4*Log[x])/32 - (3*x^4*Log[x]^2)/16 + (x^4*Log[x]^3)/4

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Rubi in Sympy [A]  time = 2.80623, size = 37, normalized size = 0.95 \[ \frac{x^{4} \log{\left (x \right )}^{3}}{4} - \frac{3 x^{4} \log{\left (x \right )}^{2}}{16} + \frac{3 x^{4} \log{\left (x \right )}}{32} - \frac{3 x^{4}}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*ln(x)**3,x)

[Out]

x**4*log(x)**3/4 - 3*x**4*log(x)**2/16 + 3*x**4*log(x)/32 - 3*x**4/128

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Mathematica [A]  time = 0.00378348, size = 39, normalized size = 1. \[ -\frac{3 x^4}{128}+\frac{1}{4} x^4 \log ^3(x)-\frac{3}{16} x^4 \log ^2(x)+\frac{3}{32} x^4 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Log[x]^3,x]

[Out]

(-3*x^4)/128 + (3*x^4*Log[x])/32 - (3*x^4*Log[x]^2)/16 + (x^4*Log[x]^3)/4

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Maple [A]  time = 0.003, size = 32, normalized size = 0.8 \[ -{\frac{3\,{x}^{4}}{128}}+{\frac{3\,{x}^{4}\ln \left ( x \right ) }{32}}-{\frac{3\,{x}^{4} \left ( \ln \left ( x \right ) \right ) ^{2}}{16}}+{\frac{{x}^{4} \left ( \ln \left ( x \right ) \right ) ^{3}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*ln(x)^3,x)

[Out]

-3/128*x^4+3/32*x^4*ln(x)-3/16*x^4*ln(x)^2+1/4*x^4*ln(x)^3

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Maxima [A]  time = 1.56424, size = 31, normalized size = 0.79 \[ \frac{1}{128} \,{\left (32 \, \log \left (x\right )^{3} - 24 \, \log \left (x\right )^{2} + 12 \, \log \left (x\right ) - 3\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*log(x)^3,x, algorithm="maxima")

[Out]

1/128*(32*log(x)^3 - 24*log(x)^2 + 12*log(x) - 3)*x^4

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Fricas [A]  time = 0.204995, size = 42, normalized size = 1.08 \[ \frac{1}{4} \, x^{4} \log \left (x\right )^{3} - \frac{3}{16} \, x^{4} \log \left (x\right )^{2} + \frac{3}{32} \, x^{4} \log \left (x\right ) - \frac{3}{128} \, x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*log(x)^3,x, algorithm="fricas")

[Out]

1/4*x^4*log(x)^3 - 3/16*x^4*log(x)^2 + 3/32*x^4*log(x) - 3/128*x^4

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Sympy [A]  time = 0.113336, size = 37, normalized size = 0.95 \[ \frac{x^{4} \log{\left (x \right )}^{3}}{4} - \frac{3 x^{4} \log{\left (x \right )}^{2}}{16} + \frac{3 x^{4} \log{\left (x \right )}}{32} - \frac{3 x^{4}}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*ln(x)**3,x)

[Out]

x**4*log(x)**3/4 - 3*x**4*log(x)**2/16 + 3*x**4*log(x)/32 - 3*x**4/128

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GIAC/XCAS [A]  time = 0.215188, size = 42, normalized size = 1.08 \[ \frac{1}{4} \, x^{4}{\rm ln}\left (x\right )^{3} - \frac{3}{16} \, x^{4}{\rm ln}\left (x\right )^{2} + \frac{3}{32} \, x^{4}{\rm ln}\left (x\right ) - \frac{3}{128} \, x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*log(x)^3,x, algorithm="giac")

[Out]

1/4*x^4*ln(x)^3 - 3/16*x^4*ln(x)^2 + 3/32*x^4*ln(x) - 3/128*x^4