\(\int \frac {1-\log (x)}{x^2} \, dx\) [237]

Optimal result

Integrand size = 10, antiderivative size = 6 \[ \int \frac {1-\log (x)}{x^2} \, dx=\frac {\log (x)}{x} \]

[Out]

ln(x)/x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2340} \[ \int \frac {1-\log (x)}{x^2} \, dx=\frac {\log (x)}{x} \]

[In]

Int[(1 - Log[x])/x^2,x]

[Out]

Log[x]/x

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{align*} \text {integral}& = \frac {\log (x)}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {1-\log (x)}{x^2} \, dx=\frac {\log (x)}{x} \]

[In]

Integrate[(1 - Log[x])/x^2,x]

[Out]

Log[x]/x

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17

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

[In]

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

[Out]

1/x*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {1-\log (x)}{x^2} \, dx=\frac {\log \left (x\right )}{x} \]

[In]

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

[Out]

log(x)/x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.50 \[ \int \frac {1-\log (x)}{x^2} \, dx=\frac {\log {\left (x \right )}}{x} \]

[In]

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

[Out]

log(x)/x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 2.33 \[ \int \frac {1-\log (x)}{x^2} \, dx=\frac {\log \left (x\right ) + 1}{x} - \frac {1}{x} \]

[In]

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

[Out]

(log(x) + 1)/x - 1/x

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {1-\log (x)}{x^2} \, dx=\frac {\log \left (x\right )}{x} \]

[In]

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

[Out]

log(x)/x

Mupad [B] (verification not implemented)

Time = 1.45 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {1-\log (x)}{x^2} \, dx=\frac {\ln \left (x\right )}{x} \]

[In]

int(-(log(x) - 1)/x^2,x)

[Out]

log(x)/x