3.6.86 \(\int \frac {1+x}{\sqrt {4-x^2}} \, dx\) [586]

Optimal. Leaf size=20 \[ -\sqrt {4-x^2}+\sin ^{-1}\left (\frac {x}{2}\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {655, 222} \begin {gather*} \text {ArcSin}\left (\frac {x}{2}\right )-\sqrt {4-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)/Sqrt[4 - x^2],x]

[Out]

-Sqrt[4 - x^2] + ArcSin[x/2]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 34, normalized size = 1.70 \begin {gather*} -\sqrt {4-x^2}-2 \tan ^{-1}\left (\frac {\sqrt {4-x^2}}{2+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/Sqrt[4 - x^2],x]

[Out]

-Sqrt[4 - x^2] - 2*ArcTan[Sqrt[4 - x^2]/(2 + x)]

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Maple [A]
time = 0.42, size = 17, normalized size = 0.85

method	result	size
default	\(\arcsin \left (\frac {x}{2}\right )-\sqrt {-x^{2}+4}\)	\(17\)
risch	\(\frac {x^{2}-4}{\sqrt {-x^{2}+4}}+\arcsin \left (\frac {x}{2}\right )\)	\(21\)
meijerg	\(\arcsin \left (\frac {x}{2}\right )-\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-\frac {x^{2}}{4}+1}}{\sqrt {\pi }}\)	\(31\)
trager	\(-\sqrt {-x^{2}+4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+4}+x \right )\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.57, size = 16, normalized size = 0.80 \begin {gather*} -\sqrt {-x^{2} + 4} + \arcsin \left (\frac {1}{2} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 4) + arcsin(1/2*x)

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Fricas [A]
time = 3.06, size = 30, normalized size = 1.50 \begin {gather*} -\sqrt {-x^{2} + 4} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 4} - 2}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 4) - 2*arctan((sqrt(-x^2 + 4) - 2)/x)

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Sympy [A]
time = 0.06, size = 12, normalized size = 0.60 \begin {gather*} - \sqrt {4 - x^{2}} + \operatorname {asin}{\left (\frac {x}{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(4 - x**2) + asin(x/2)

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Giac [A]
time = 2.44, size = 16, normalized size = 0.80 \begin {gather*} -\sqrt {-x^{2} + 4} + \arcsin \left (\frac {1}{2} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 4) + arcsin(1/2*x)

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Mupad [B]
time = 0.03, size = 16, normalized size = 0.80 \begin {gather*} \mathrm {asin}\left (\frac {x}{2}\right )-\sqrt {4-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(4 - x^2)^(1/2),x)

[Out]

asin(x/2) - (4 - x^2)^(1/2)

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