3.615 \(\int \frac{1}{x^3 \log ^4(x)} \, dx\)

Optimal. Leaf size=43 \[ -\frac{4}{3} \text{Ei}(-2 \log (x))-\frac{1}{3 x^2 \log ^3(x)}+\frac{1}{3 x^2 \log ^2(x)}-\frac{2}{3 x^2 \log (x)} \]

[Out]

(-4*ExpIntegralEi[-2*Log[x]])/3 - 1/(3*x^2*Log[x]^3) + 1/(3*x^2*Log[x]^2) - 2/(3
*x^2*Log[x])

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Rubi [A]  time = 0.0905949, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{4}{3} \text{Ei}(-2 \log (x))-\frac{1}{3 x^2 \log ^3(x)}+\frac{1}{3 x^2 \log ^2(x)}-\frac{2}{3 x^2 \log (x)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Log[x]^4),x]

[Out]

(-4*ExpIntegralEi[-2*Log[x]])/3 - 1/(3*x^2*Log[x]^3) + 1/(3*x^2*Log[x]^2) - 2/(3
*x^2*Log[x])

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Rubi in Sympy [A]  time = 4.31623, size = 44, normalized size = 1.02 \[ - \frac{4 \operatorname{Ei}{\left (- 2 \log{\left (x \right )} \right )}}{3} - \frac{2}{3 x^{2} \log{\left (x \right )}} + \frac{1}{3 x^{2} \log{\left (x \right )}^{2}} - \frac{1}{3 x^{2} \log{\left (x \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/ln(x)**4,x)

[Out]

-4*Ei(-2*log(x))/3 - 2/(3*x**2*log(x)) + 1/(3*x**2*log(x)**2) - 1/(3*x**2*log(x)
**3)

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Mathematica [A]  time = 0.00837652, size = 43, normalized size = 1. \[ -\frac{4}{3} \text{Ei}(-2 \log (x))-\frac{1}{3 x^2 \log ^3(x)}+\frac{1}{3 x^2 \log ^2(x)}-\frac{2}{3 x^2 \log (x)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Log[x]^4),x]

[Out]

(-4*ExpIntegralEi[-2*Log[x]])/3 - 1/(3*x^2*Log[x]^3) + 1/(3*x^2*Log[x]^2) - 2/(3
*x^2*Log[x])

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Maple [A]  time = 0.01, size = 37, normalized size = 0.9 \[ -{\frac{1}{3\,{x}^{2} \left ( \ln \left ( x \right ) \right ) ^{3}}}+{\frac{1}{3\,{x}^{2} \left ( \ln \left ( x \right ) \right ) ^{2}}}-{\frac{2}{3\,{x}^{2}\ln \left ( x \right ) }}+{\frac{4\,{\it Ei} \left ( 1,2\,\ln \left ( x \right ) \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/ln(x)^4,x)

[Out]

-1/3/x^2/ln(x)^3+1/3/x^2/ln(x)^2-2/3/x^2/ln(x)+4/3*Ei(1,2*ln(x))

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Maxima [A]  time = 1.79426, size = 11, normalized size = 0.26 \[ -8 \, \Gamma \left (-3, 2 \, \log \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^3*log(x)^4),x, algorithm="maxima")

[Out]

-8*gamma(-3, 2*log(x))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{4 \, x^{2} \log \left (x\right )^{3} log_integral\left (\frac{1}{x^{2}}\right ) + 2 \, \log \left (x\right )^{2} - \log \left (x\right ) + 1}{3 \, x^{2} \log \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^3*log(x)^4),x, algorithm="fricas")

[Out]

-1/3*(4*x^2*log(x)^3*log_integral(x^(-2)) + 2*log(x)^2 - log(x) + 1)/(x^2*log(x)
^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4 \int \frac{1}{x^{3} \log{\left (x \right )}}\, dx}{3} + \frac{- 2 \log{\left (x \right )}^{2} + \log{\left (x \right )} - 1}{3 x^{2} \log{\left (x \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/ln(x)**4,x)

[Out]

-4*Integral(1/(x**3*log(x)), x)/3 + (-2*log(x)**2 + log(x) - 1)/(3*x**2*log(x)**
3)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \log \left (x\right )^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^3*log(x)^4),x, algorithm="giac")

[Out]

integrate(1/(x^3*log(x)^4), x)