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

Optimal. Leaf size=20 \[ \frac {\frac {1}{3} x (1+2 \log (3))-\log (x)}{x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 7, normalized size of antiderivative = 0.35, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2303} \begin {gather*} -\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + Log[x])/x^2,x]

[Out]

-(Log[x]/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 {\log (x)}{x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 7, normalized size = 0.35 \begin {gather*} -\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Log[x])/x^2,x]

[Out]

-(Log[x]/x)

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 7, normalized size = 0.35 \begin {gather*} -\frac {\log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/x

________________________________________________________________________________________

giac [A]  time = 0.13, size = 7, normalized size = 0.35 \begin {gather*} -\frac {\log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/x

________________________________________________________________________________________

maple [A]  time = 0.01, size = 8, normalized size = 0.40

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(x)/x

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 13, normalized size = 0.65 \begin {gather*} -\frac {\log \relax (x) + 1}{x} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(x) + 1)/x + 1/x

________________________________________________________________________________________

mupad [B]  time = 2.96, size = 7, normalized size = 0.35 \begin {gather*} -\frac {\ln \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x) - 1)/x^2,x)

[Out]

-log(x)/x

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 5, normalized size = 0.25 \begin {gather*} - \frac {\log {\relax (x )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/x

________________________________________________________________________________________