\(\int \frac {-5+x}{-3+x} \, dx\) [10105]

Optimal result

Integrand size = 9, antiderivative size = 15 \[ \int \frac {-5+x}{-3+x} \, dx=x-2 \log (-3+x)-\frac {10}{\log (\log (2))} \]

[Out]

x-2*ln(-3+x)-10/ln(ln(2))

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {-5+x}{-3+x} \, dx=x-2 \log (3-x) \]

[In]

Int[(-5 + x)/(-3 + x),x]

[Out]

x - 2*Log[3 - 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 (1-\frac {2}{-3+x}\right ) \, dx \\ & = x-2 \log (3-x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.53 \[ \int \frac {-5+x}{-3+x} \, dx=x-2 \log (-3+x) \]

[In]

Integrate[(-5 + x)/(-3 + x),x]

[Out]

x - 2*Log[-3 + x]

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60

method	result	size
default	\(x -2 \ln \left (-3+x \right )\)	\(9\)
norman	\(x -2 \ln \left (-3+x \right )\)	\(9\)
risch	\(x -2 \ln \left (-3+x \right )\)	\(9\)
parallelrisch	\(x -2 \ln \left (-3+x \right )\)	\(9\)
meijerg	\(-2 \ln \left (1-\frac {x}{3}\right )+x\)	\(11\)

[In]

int((-5+x)/(-3+x),x,method=_RETURNVERBOSE)

[Out]

x-2*ln(-3+x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.53 \[ \int \frac {-5+x}{-3+x} \, dx=x - 2 \, \log \left (x - 3\right ) \]

[In]

integrate((-5+x)/(-3+x),x, algorithm="fricas")

[Out]

x - 2*log(x - 3)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.47 \[ \int \frac {-5+x}{-3+x} \, dx=x - 2 \log {\left (x - 3 \right )} \]

[In]

integrate((-5+x)/(-3+x),x)

[Out]

x - 2*log(x - 3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.53 \[ \int \frac {-5+x}{-3+x} \, dx=x - 2 \, \log \left (x - 3\right ) \]

[In]

integrate((-5+x)/(-3+x),x, algorithm="maxima")

[Out]

x - 2*log(x - 3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {-5+x}{-3+x} \, dx=x - 2 \, \log \left ({\left | x - 3 \right |}\right ) \]

[In]

integrate((-5+x)/(-3+x),x, algorithm="giac")

[Out]

x - 2*log(abs(x - 3))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.53 \[ \int \frac {-5+x}{-3+x} \, dx=x-2\,\ln \left (x-3\right ) \]

[In]

int((x - 5)/(x - 3),x)

[Out]

x - 2*log(x - 3)