\(\int \frac {-1+2 x}{x} \, dx\) [7985]

Optimal result

Integrand size = 9, antiderivative size = 21 \[ \int \frac {-1+2 x}{x} \, dx=-2+2 x+(i \pi +\log (5))^2-\log (2 x) \]

[Out]

2*x-2-ln(2*x)+(ln(5)+I*Pi)^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.38, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {-1+2 x}{x} \, dx=2 x-\log (x) \]

[In]

Int[(-1 + 2*x)/x,x]

[Out]

2*x - Log[x]

Rule 45

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.38 \[ \int \frac {-1+2 x}{x} \, dx=2 x-\log (x) \]

[In]

Integrate[(-1 + 2*x)/x,x]

[Out]

2*x - Log[x]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.43

method	result	size
default	\(2 x -\ln \left (x \right )\)	\(9\)
norman	\(2 x -\ln \left (x \right )\)	\(9\)
risch	\(2 x -\ln \left (x \right )\)	\(9\)
parallelrisch	\(2 x -\ln \left (x \right )\)	\(9\)

[In]

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

[Out]

2*x-ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.38 \[ \int \frac {-1+2 x}{x} \, dx=2 \, x - \log \left (x\right ) \]

[In]

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

[Out]

2*x - log(x)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.24 \[ \int \frac {-1+2 x}{x} \, dx=2 x - \log {\left (x \right )} \]

[In]

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

[Out]

2*x - log(x)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.38 \[ \int \frac {-1+2 x}{x} \, dx=2 \, x - \log \left (x\right ) \]

[In]

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

[Out]

2*x - log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.43 \[ \int \frac {-1+2 x}{x} \, dx=2 \, x - \log \left ({\left | x \right |}\right ) \]

[In]

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

[Out]

2*x - log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.38 \[ \int \frac {-1+2 x}{x} \, dx=2\,x-\ln \left (x\right ) \]

[In]

int((2*x - 1)/x,x)

[Out]

2*x - log(x)