3.539 \(\int \frac{1}{4+\sqrt{4-x}-x} \, dx\)

Optimal. Leaf size=14 \[ -2 \log \left (\sqrt{4-x}+1\right ) \]

[Out]

-2*Log[1 + Sqrt[4 - x]]

_______________________________________________________________________________________

Rubi [A]  time = 0.0342119, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -2 \log \left (\sqrt{4-x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(4 + Sqrt[4 - x] - x)^(-1),x]

[Out]

-2*Log[1 + Sqrt[4 - x]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.25729, size = 12, normalized size = 0.86 \[ - 2 \log{\left (\sqrt{- x + 4} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4-x+(4-x)**(1/2)),x)

[Out]

-2*log(sqrt(-x + 4) + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00647614, size = 14, normalized size = 1. \[ -2 \log \left (\sqrt{4-x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(4 + Sqrt[4 - x] - x)^(-1),x]

[Out]

-2*Log[1 + Sqrt[4 - x]]

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 18, normalized size = 1.3 \[ -\ln \left ( -3+x \right ) -2\,{\it Artanh} \left ( \sqrt{4-x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4-x+(4-x)^(1/2)),x)

[Out]

-ln(-3+x)-2*arctanh((4-x)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.719177, size = 16, normalized size = 1.14 \[ -2 \, \log \left (\sqrt{-x + 4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(x - sqrt(-x + 4) - 4),x, algorithm="maxima")

[Out]

-2*log(sqrt(-x + 4) + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.262294, size = 16, normalized size = 1.14 \[ -2 \, \log \left (\sqrt{-x + 4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(x - sqrt(-x + 4) - 4),x, algorithm="fricas")

[Out]

-2*log(sqrt(-x + 4) + 1)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{x - \sqrt{- x + 4} - 4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4-x+(4-x)**(1/2)),x)

[Out]

-Integral(1/(x - sqrt(-x + 4) - 4), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.285926, size = 16, normalized size = 1.14 \[ -2 \,{\rm ln}\left (\sqrt{-x + 4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(x - sqrt(-x + 4) - 4),x, algorithm="giac")

[Out]

-2*ln(sqrt(-x + 4) + 1)