3.52.100 \(\int \frac {-2+x}{x} \, dx\)

Optimal. Leaf size=14 \[ -1+\frac {1}{16+e^2}+x+\log \left (\frac {1}{x^2}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 6, normalized size of antiderivative = 0.43, number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {43} \begin {gather*} x-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + x)/x,x]

[Out]

x - 2*Log[x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {2}{x}\right ) \, dx\\ &=x-2 \log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 6, normalized size = 0.43 \begin {gather*} x-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + x)/x,x]

[Out]

x - 2*Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 6, normalized size = 0.43 \begin {gather*} x - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(x)

________________________________________________________________________________________

giac [A]  time = 0.14, size = 7, normalized size = 0.50 \begin {gather*} x - 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 7, normalized size = 0.50

method	result	size
default	\(x -2 \ln \relax (x )\)	\(7\)
norman	\(x -2 \ln \relax (x )\)	\(7\)
risch	\(x -2 \ln \relax (x )\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-2)/x,x,method=_RETURNVERBOSE)

[Out]

x-2*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 6, normalized size = 0.43 \begin {gather*} x - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 6, normalized size = 0.43 \begin {gather*} x-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 2)/x,x)

[Out]

x - 2*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 5, normalized size = 0.36 \begin {gather*} x - 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-2)/x,x)

[Out]

x - 2*log(x)

________________________________________________________________________________________