3.6.2 \(\int \frac {6-x}{-5+x} \, dx\)

Optimal. Leaf size=14 \[ -8+e^4-x+\log (2)+\log (-5+x) \]

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 0.71, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} \log (5-x)-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 - x)/(-5 + x),x]

[Out]

-x + Log[5 - 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 {1}{-5+x}\right ) \, dx\\ &=-x+\log (5-x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.57 \begin {gather*} -x+\log (-5+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 - x)/(-5 + x),x]

[Out]

-x + Log[-5 + x]

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fricas [A]  time = 0.94, size = 8, normalized size = 0.57 \begin {gather*} -x + \log \left (x - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x+6)/(x-5),x, algorithm="fricas")

[Out]

-x + log(x - 5)

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giac [A]  time = 0.31, size = 9, normalized size = 0.64 \begin {gather*} -x + \log \left ({\left | x - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x+6)/(x-5),x, algorithm="giac")

[Out]

-x + log(abs(x - 5))

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maple [A]  time = 0.29, size = 9, normalized size = 0.64

method	result	size
default	\(-x +\ln \left (x -5\right )\)	\(9\)
norman	\(-x +\ln \left (x -5\right )\)	\(9\)
risch	\(-x +\ln \left (x -5\right )\)	\(9\)
meijerg	\(\ln \left (1-\frac {x}{5}\right )-x\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x+6)/(x-5),x,method=_RETURNVERBOSE)

[Out]

-x+ln(x-5)

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maxima [A]  time = 0.52, size = 8, normalized size = 0.57 \begin {gather*} -x + \log \left (x - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x+6)/(x-5),x, algorithm="maxima")

[Out]

-x + log(x - 5)

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mupad [B]  time = 0.03, size = 8, normalized size = 0.57 \begin {gather*} \ln \left (x-5\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 6)/(x - 5),x)

[Out]

log(x - 5) - x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x+6)/(x-5),x)

[Out]

-x + log(x - 5)

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