3.102.5 \(\int \frac {-5+x}{-3+x} \, dx\)

Optimal. Leaf size=15 \[ x-2 \log (-3+x)-\frac {10}{\log (\log (2))} \]

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \begin {gather*} x-2 \log (3-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + x)/(-3 + x),x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.53 \begin {gather*} x-2 \log (-3+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + x)/(-3 + x),x]

[Out]

x - 2*Log[-3 + x]

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fricas [A]  time = 0.69, size = 8, normalized size = 0.53 \begin {gather*} x - 2 \, \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(x - 3)

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giac [A]  time = 0.13, size = 9, normalized size = 0.60 \begin {gather*} x - 2 \, \log \left ({\left | x - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(abs(x - 3))

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maple [A]  time = 0.14, size = 9, normalized size = 0.60

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-2*ln(x-3)

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maxima [A]  time = 0.34, size = 8, normalized size = 0.53 \begin {gather*} x - 2 \, \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(x - 3)

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mupad [B]  time = 0.04, size = 8, normalized size = 0.53 \begin {gather*} x-2\,\ln \left (x-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 5)/(x - 3),x)

[Out]

x - 2*log(x - 3)

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sympy [A]  time = 0.07, size = 7, normalized size = 0.47 \begin {gather*} x - 2 \log {\left (x - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(x - 3)

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