\(\int \frac {4-4 x}{x} \, dx\) [5752]

Optimal result

Integrand size = 9, antiderivative size = 11 \[ \int \frac {4-4 x}{x} \, dx=-1+4 (1-x+\log (x)) \]

[Out]

3+4*ln(x)-4*x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {4-4 x}{x} \, dx=4 \log (x)-4 x \]

[In]

Int[(4 - 4*x)/x,x]

[Out]

-4*x + 4*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 (-4+\frac {4}{x}\right ) \, dx \\ & = -4 x+4 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {4-4 x}{x} \, dx=-4 x+4 \log (x) \]

[In]

Integrate[(4 - 4*x)/x,x]

[Out]

-4*x + 4*Log[x]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82

method	result	size
default	\(4 \ln \left (x \right )-4 x\)	\(9\)
norman	\(4 \ln \left (x \right )-4 x\)	\(9\)
risch	\(4 \ln \left (x \right )-4 x\)	\(9\)
parallelrisch	\(4 \ln \left (x \right )-4 x\)	\(9\)

[In]

int((4-4*x)/x,x,method=_RETURNVERBOSE)

[Out]

4*ln(x)-4*x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {4-4 x}{x} \, dx=-4 \, x + 4 \, \log \left (x\right ) \]

[In]

integrate((4-4*x)/x,x, algorithm="fricas")

[Out]

-4*x + 4*log(x)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int \frac {4-4 x}{x} \, dx=- 4 x + 4 \log {\left (x \right )} \]

[In]

integrate((4-4*x)/x,x)

[Out]

-4*x + 4*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {4-4 x}{x} \, dx=-4 \, x + 4 \, \log \left (x\right ) \]

[In]

integrate((4-4*x)/x,x, algorithm="maxima")

[Out]

-4*x + 4*log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {4-4 x}{x} \, dx=-4 \, x + 4 \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate((4-4*x)/x,x, algorithm="giac")

[Out]

-4*x + 4*log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {4-4 x}{x} \, dx=4\,\ln \left (x\right )-4\,x \]

[In]

int(-(4*x - 4)/x,x)

[Out]

4*log(x) - 4*x