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

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

[Out]

-2*Log[1 + Sqrt[4 - x]]

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Rubi [A]  time = 0.0185971, 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, Rules used = {31} \[ -2 \log \left (\sqrt{4-x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(4 + Sqrt[4 - x] - x)^(-1),x]

[Out]

-2*Log[1 + Sqrt[4 - x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{4+\sqrt{4-x}-x} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sqrt{4-x}\right )\right )\\ &=-2 \log \left (1+\sqrt{4-x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0052837, 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]]

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Maple [A]  time = 0.01, size = 18, normalized size = 1.3 \begin{align*} -\ln \left ( -3+x \right ) -2\,{\it Artanh} \left ( \sqrt{4-x} \right ) \end{align*}

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))

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Maxima [A]  time = 0.950734, size = 16, normalized size = 1.14 \begin{align*} -2 \, \log \left (\sqrt{-x + 4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-x+(4-x)^(1/2)),x, algorithm="maxima")

[Out]

-2*log(sqrt(-x + 4) + 1)

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Fricas [A]  time = 1.40136, size = 35, normalized size = 2.5 \begin{align*} -2 \, \log \left (\sqrt{-x + 4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-x+(4-x)^(1/2)),x, algorithm="fricas")

[Out]

-2*log(sqrt(-x + 4) + 1)

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Sympy [B]  time = 2.10782, size = 32, normalized size = 2.29 \begin{align*} \log{\left (2 \sqrt{4 - x} \right )} - \log{\left (2 \sqrt{4 - x} + 2 \right )} - \log{\left (x - \sqrt{4 - x} - 4 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*sqrt(4 - x)) - log(2*sqrt(4 - x) + 2) - log(x - sqrt(4 - x) - 4)

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Giac [A]  time = 1.14667, size = 16, normalized size = 1.14 \begin{align*} -2 \, \log \left (\sqrt{-x + 4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-x+(4-x)^(1/2)),x, algorithm="giac")

[Out]

-2*log(sqrt(-x + 4) + 1)