3.7.95 \(\int \frac {1}{4+\sqrt {4-x}-x} \, dx\) [695]

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

[Out]

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

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Rubi [A]
time = 0.01, 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} \begin {gather*} -2 \log \left (\sqrt {4-x}+1\right ) \end {gather*}

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 \text {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 \begin {gather*} -2 \log \left (1+\sqrt {4-x}\right ) \end {gather*}

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 = 0.05, size = 18, normalized size = 1.29

method	result	size
derivativedivides	\(-2 \ln \left (1+\sqrt {4-x}\right )\)	\(13\)
default	\(-\ln \left (x -3\right )-2 \arctanh \left (\sqrt {4-x}\right )\)	\(18\)
trager	\(-\ln \left (2 \sqrt {4-x}+5-x \right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 12, normalized size = 0.86 \begin {gather*} -2 \, \log \left (\sqrt {-x + 4} + 1\right ) \end {gather*}

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 = 0.37, size = 12, normalized size = 0.86 \begin {gather*} -2 \, \log \left (\sqrt {-x + 4} + 1\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (12) = 24\).
time = 1.83, size = 32, normalized size = 2.29 \begin {gather*} \log {\left (2 \sqrt {4 - x} \right )} - \log {\left (2 \sqrt {4 - x} + 2 \right )} - \log {\left (x - \sqrt {4 - x} - 4 \right )} \end {gather*}

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 = 1.74, size = 12, normalized size = 0.86 \begin {gather*} -2 \, \log \left (\sqrt {-x + 4} + 1\right ) \end {gather*}

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

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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