3.6.6 \(\int \frac {1+x}{\sqrt {4-x^2}} \, dx\)

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

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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 = {641, 216} \begin {gather*} \sin ^{-1}\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 216

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 641

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 = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \sin ^{-1}\left (\frac {x}{2}\right )-\sqrt {4-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/Sqrt[4 - x^2],x]

[Out]

-Sqrt[4 - x^2] + ArcSin[x/2]

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IntegrateAlgebraic [A]  time = 0.16, size = 34, normalized size = 1.70 \begin {gather*} -\sqrt {4-x^2}-2 \tan ^{-1}\left (\frac {\sqrt {4-x^2}}{x+2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)/Sqrt[4 - x^2],x]

[Out]

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

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fricas [A]  time = 0.40, 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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giac [A]  time = 0.21, 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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maple [A]  time = 0.05, size = 17, normalized size = 0.85 \begin {gather*} \arcsin \left (\frac {x}{2}\right )-\sqrt {-x^{2}+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.01, 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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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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sympy [A]  time = 0.15, 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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