3.72 \(\int \frac{\log (x)}{x^5} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{16 x^4}-\frac{\log (x)}{4 x^4} \]

[Out]

-1/(16*x^4) - Log[x]/(4*x^4)

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Rubi [A]  time = 0.006899, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2304} \[ -\frac{1}{16 x^4}-\frac{\log (x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/x^5,x]

[Out]

-1/(16*x^4) - Log[x]/(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
]

Rubi steps

\begin{align*} \int \frac{\log (x)}{x^5} \, dx &=-\frac{1}{16 x^4}-\frac{\log (x)}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0009253, size = 17, normalized size = 1. \[ -\frac{1}{16 x^4}-\frac{\log (x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x]/x^5,x]

[Out]

-1/(16*x^4) - Log[x]/(4*x^4)

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Maple [A]  time = 0.002, size = 14, normalized size = 0.8 \begin{align*} -{\frac{1}{16\,{x}^{4}}}-{\frac{\ln \left ( x \right ) }{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16/x^4-1/4*ln(x)/x^4

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Maxima [A]  time = 0.934378, size = 18, normalized size = 1.06 \begin{align*} -\frac{\log \left (x\right )}{4 \, x^{4}} - \frac{1}{16 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*log(x)/x^4 - 1/16/x^4

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Fricas [A]  time = 1.67994, size = 35, normalized size = 2.06 \begin{align*} -\frac{4 \, \log \left (x\right ) + 1}{16 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*log(x) + 1)/x^4

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Sympy [A]  time = 0.092494, size = 15, normalized size = 0.88 \begin{align*} - \frac{\log{\left (x \right )}}{4 x^{4}} - \frac{1}{16 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/(4*x**4) - 1/(16*x**4)

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Giac [A]  time = 1.05191, size = 18, normalized size = 1.06 \begin{align*} -\frac{\log \left (x\right )}{4 \, x^{4}} - \frac{1}{16 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*log(x)/x^4 - 1/16/x^4