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

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {1}{16 x^4}-\frac {\log (x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*1/x^4 - Log[x]/(4*x^4)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log (x)}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 0.86, size = 11, normalized size = 0.65 \[ -\frac {4 \, \log \relax (x) + 1}{16 \, x^{4}} \]

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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giac [A]  time = 0.79, size = 13, normalized size = 0.76 \[ -\frac {\log \relax (x)}{4 \, x^{4}} - \frac {1}{16 \, x^{4}} \]

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

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maple [A]  time = 0.02, size = 11, normalized size = 0.65

method	result	size
norman	\(\frac {-\frac {1}{16}-\frac {\ln \relax (x )}{4}}{x^{4}}\)	\(11\)
default	\(-\frac {1}{16 x^{4}}-\frac {\ln \relax (x )}{4 x^{4}}\)	\(14\)
risch	\(-\frac {1}{16 x^{4}}-\frac {\ln \relax (x )}{4 x^{4}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 13, normalized size = 0.76 \[ -\frac {\log \relax (x)}{4 \, x^{4}} - \frac {1}{16 \, x^{4}} \]

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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mupad [B]  time = 0.17, size = 9, normalized size = 0.53 \[ -\frac {\ln \relax (x)+\frac {1}{4}}{4\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(x) + 1/4)/(4*x^4)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.88 \[ - \frac {\log {\relax (x )}}{4 x^{4}} - \frac {1}{16 x^{4}} \]

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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