3.59.60 \(\int \frac {-1+\log (x)}{2 x^2} \, dx\)

Optimal. Leaf size=9 \[ -\frac {\log (x)}{2 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2303} \begin {gather*} -\frac {\log (x)}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + Log[x])/(2*x^2),x]

[Out]

-1/2*Log[x]/x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2303

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 {1}{2} \int \frac {-1+\log (x)}{x^2} \, dx\\ &=-\frac {\log (x)}{2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} -\frac {\log (x)}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Log[x])/(2*x^2),x]

[Out]

-1/2*Log[x]/x

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fricas [A]  time = 1.05, size = 7, normalized size = 0.78 \begin {gather*} -\frac {\log \relax (x)}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(x)/x

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giac [A]  time = 0.13, size = 7, normalized size = 0.78 \begin {gather*} -\frac {\log \relax (x)}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(x)/x

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maple [A]  time = 0.02, size = 8, normalized size = 0.89

method	result	size
default	\(-\frac {\ln \relax (x )}{2 x}\)	\(8\)
norman	\(-\frac {\ln \relax (x )}{2 x}\)	\(8\)
risch	\(-\frac {\ln \relax (x )}{2 x}\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*ln(x)/x

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maxima [B]  time = 0.35, size = 15, normalized size = 1.67 \begin {gather*} -\frac {\log \relax (x) + 1}{2 \, x} + \frac {1}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(log(x) + 1)/x + 1/2/x

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mupad [B]  time = 4.03, size = 7, normalized size = 0.78 \begin {gather*} -\frac {\ln \relax (x)}{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)/2 - 1/2)/x^2,x)

[Out]

-log(x)/(2*x)

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sympy [A]  time = 0.08, size = 7, normalized size = 0.78 \begin {gather*} - \frac {\log {\relax (x )}}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/(2*x)

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