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

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Rubi [A]  time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \[ -2 \log \left (\sqrt {4-x}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Log[1 + Sqrt[4 - x]]

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fricas [A]  time = 0.44, 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]

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

[Out]

-2*log(sqrt(-x + 4) + 1)

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giac [A]  time = 0.31, 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]

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

[Out]

-2*log(sqrt(-x + 4) + 1)

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maple [A]  time = 0.01, size = 18, normalized size = 1.29 \[ -2 \arctanh \left (\sqrt {-x +4}\right )-\ln \left (x -3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4-x+(4-x)^(1/2)),x)

[Out]

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

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maxima [A]  time = 0.66, 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]

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

[Out]

-2*log(sqrt(-x + 4) + 1)

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mupad [B]  time = 0.19, size = 12, normalized size = 0.86 \[ -2\,\ln \left (\sqrt {4-x}+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((4 - x)^(1/2) - x + 4),x)

[Out]

-2*log((4 - x)^(1/2) + 1)

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sympy [B]  time = 4.23, size = 32, normalized size = 2.29 \[ \log {\left (2 \sqrt {4 - x} \right )} - \log {\left (2 \sqrt {4 - x} + 2 \right )} - \log {\left (x - \sqrt {4 - x} - 4 \right )} \]

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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