3.262 \(\int \frac {\log ^2(\frac {1}{x})}{x^5} \, dx\)

Optimal. Leaf size=32 \[ -\frac {1}{32 x^4}-\frac {\log ^2\left (\frac {1}{x}\right )}{4 x^4}+\frac {\log \left (\frac {1}{x}\right )}{8 x^4} \]

[Out]

-1/32/x^4+1/8*ln(1/x)/x^4-1/4*ln(1/x)^2/x^4

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Rubi [A]  time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2305, 2304} \[ -\frac {1}{32 x^4}-\frac {\log ^2\left (\frac {1}{x}\right )}{4 x^4}+\frac {\log \left (\frac {1}{x}\right )}{8 x^4} \]

Antiderivative was successfully verified.

[In]

Int[Log[x^(-1)]^2/x^5,x]

[Out]

-1/(32*x^4) + Log[x^(-1)]/(8*x^4) - Log[x^(-1)]^2/(4*x^4)

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\log ^2\left (\frac {1}{x}\right )}{x^5} \, dx &=-\frac {\log ^2\left (\frac {1}{x}\right )}{4 x^4}-\frac {1}{2} \int \frac {\log \left (\frac {1}{x}\right )}{x^5} \, dx\\ &=-\frac {1}{32 x^4}+\frac {\log \left (\frac {1}{x}\right )}{8 x^4}-\frac {\log ^2\left (\frac {1}{x}\right )}{4 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 32, normalized size = 1.00 \[ -\frac {1}{32 x^4}-\frac {\log ^2\left (\frac {1}{x}\right )}{4 x^4}+\frac {\log \left (\frac {1}{x}\right )}{8 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x^(-1)]^2/x^5,x]

[Out]

-1/32*1/x^4 + Log[x^(-1)]/(8*x^4) - Log[x^(-1)]^2/(4*x^4)

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fricas [A]  time = 0.44, size = 21, normalized size = 0.66 \[ -\frac {8 \, \log \left (\frac {1}{x}\right )^{2} - 4 \, \log \left (\frac {1}{x}\right ) + 1}{32 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1/x)^2/x^5,x, algorithm="fricas")

[Out]

-1/32*(8*log(1/x)^2 - 4*log(1/x) + 1)/x^4

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giac [A]  time = 0.16, size = 22, normalized size = 0.69 \[ -\frac {\log \relax (x)^{2}}{4 \, x^{4}} - \frac {\log \relax (x)}{8 \, x^{4}} - \frac {1}{32 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1/x)^2/x^5,x, algorithm="giac")

[Out]

-1/4*log(x)^2/x^4 - 1/8*log(x)/x^4 - 1/32/x^4

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maple [A]  time = 0.07, size = 27, normalized size = 0.84 \[ -\frac {\ln \left (\frac {1}{x}\right )^{2}}{4 x^{4}}+\frac {\ln \left (\frac {1}{x}\right )}{8 x^{4}}-\frac {1}{32 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(1/x)^2/x^5,x)

[Out]

-1/32/x^4+1/8*ln(1/x)/x^4-1/4*ln(1/x)^2/x^4

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maxima [A]  time = 0.68, size = 17, normalized size = 0.53 \[ -\frac {8 \, \log \relax (x)^{2} + 4 \, \log \relax (x) + 1}{32 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1/x)^2/x^5,x, algorithm="maxima")

[Out]

-1/32*(8*log(x)^2 + 4*log(x) + 1)/x^4

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mupad [B]  time = 0.41, size = 21, normalized size = 0.66 \[ -\frac {\frac {{\ln \left (\frac {1}{x}\right )}^2}{4}-\frac {\ln \left (\frac {1}{x}\right )}{8}+\frac {1}{32}}{x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(1/x)^2/x^5,x)

[Out]

-(log(1/x)^2/4 - log(1/x)/8 + 1/32)/x^4

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sympy [A]  time = 0.13, size = 27, normalized size = 0.84 \[ - \frac {\log {\left (\frac {1}{x} \right )}^{2}}{4 x^{4}} + \frac {\log {\left (\frac {1}{x} \right )}}{8 x^{4}} - \frac {1}{32 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(1/x)**2/x**5,x)

[Out]

-log(1/x)**2/(4*x**4) + log(1/x)/(8*x**4) - 1/(32*x**4)

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