3.84.58 \(\int \frac {1-2 \log (x)}{x^3} \, dx\) [8358]

Optimal. Leaf size=8 \[ 3+\frac {\log (x)}{x^2} \]

[Out]

3+ln(x)/x^2

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Rubi [A]
time = 0.01, antiderivative size = 6, normalized size of antiderivative = 0.75, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2340} \begin {gather*} \frac {\log (x)}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*Log[x])/x^3,x]

[Out]

Log[x]/x^2

Rule 2340

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\log (x)}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 6, normalized size = 0.75 \begin {gather*} \frac {\log (x)}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*Log[x])/x^3,x]

[Out]

Log[x]/x^2

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Maple [A]
time = 0.12, size = 7, normalized size = 0.88

method	result	size
default	\(\frac {\ln \left (x \right )}{x^{2}}\)	\(7\)
norman	\(\frac {\ln \left (x \right )}{x^{2}}\)	\(7\)
risch	\(\frac {\ln \left (x \right )}{x^{2}}\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(x)+1)/x^3,x,method=_RETURNVERBOSE)

[Out]

ln(x)/x^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (8) = 16\).
time = 0.28, size = 17, normalized size = 2.12 \begin {gather*} \frac {2 \, \log \left (x\right ) + 1}{2 \, x^{2}} - \frac {1}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)+1)/x^3,x, algorithm="maxima")

[Out]

1/2*(2*log(x) + 1)/x^2 - 1/2/x^2

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Fricas [A]
time = 0.45, size = 6, normalized size = 0.75 \begin {gather*} \frac {\log \left (x\right )}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)+1)/x^3,x, algorithm="fricas")

[Out]

log(x)/x^2

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Sympy [A]
time = 0.03, size = 5, normalized size = 0.62 \begin {gather*} \frac {\log {\left (x \right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(x)+1)/x**3,x)

[Out]

log(x)/x**2

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Giac [A]
time = 0.40, size = 6, normalized size = 0.75 \begin {gather*} \frac {\log \left (x\right )}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)+1)/x^3,x, algorithm="giac")

[Out]

log(x)/x^2

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Mupad [B]
time = 5.15, size = 6, normalized size = 0.75 \begin {gather*} \frac {\ln \left (x\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*log(x) - 1)/x^3,x)

[Out]

log(x)/x^2

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