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

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Rubi [A]  time = 0.0211592, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, 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 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]

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
]

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*}

Mathematica [A]  time = 0.001618, size = 32, normalized size = 1. $-\frac{1}{32 x^4}-\frac{\log ^2\left (\frac{1}{x}\right )}{4 x^4}+\frac{\log \left (\frac{1}{x}\right )}{8 x^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 27, normalized size = 0.8 \begin{align*} -{\frac{1}{32\,{x}^{4}}}+{\frac{\ln \left ({x}^{-1} \right ) }{8\,{x}^{4}}}-{\frac{ \left ( \ln \left ({x}^{-1} \right ) \right ) ^{2}}{4\,{x}^{4}}} \end{align*}

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 = 0.993019, size = 23, normalized size = 0.72 \begin{align*} -\frac{8 \, \log \left (x\right )^{2} + 4 \, \log \left (x\right ) + 1}{32 \, x^{4}} \end{align*}

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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Fricas [A]  time = 1.75354, size = 58, normalized size = 1.81 \begin{align*} -\frac{8 \, \log \left (\frac{1}{x}\right )^{2} - 4 \, \log \left (\frac{1}{x}\right ) + 1}{32 \, x^{4}} \end{align*}

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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Sympy [A]  time = 0.131458, size = 27, normalized size = 0.84 \begin{align*} - \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}} \end{align*}

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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Giac [A]  time = 1.35236, size = 30, normalized size = 0.94 \begin{align*} -\frac{\log \left (x\right )^{2}}{4 \, x^{4}} - \frac{\log \left (x\right )}{8 \, x^{4}} - \frac{1}{32 \, x^{4}} \end{align*}

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