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

Optimal. Leaf size=16 \[ \sin ^{-1}(x)-\sqrt {1-x^2} \]

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Rubi [A]  time = 0.00, antiderivative size = 16, 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}(x)-\sqrt {1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2] + ArcSin[x]

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \sin ^{-1}(x)-\sqrt {1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2] + ArcSin[x]

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IntegrateAlgebraic [B]  time = 0.13, size = 34, normalized size = 2.12 \begin {gather*} -\sqrt {1-x^2}-2 \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[1 - x^2] - 2*ArcTan[Sqrt[1 - x^2]/(1 + x)]

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fricas [B]  time = 0.40, size = 30, normalized size = 1.88 \begin {gather*} -\sqrt {-x^{2} + 1} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1) - 2*arctan((sqrt(-x^2 + 1) - 1)/x)

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giac [A]  time = 0.18, size = 14, normalized size = 0.88 \begin {gather*} -\sqrt {-x^{2} + 1} + \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1) + arcsin(x)

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maple [A]  time = 0.05, size = 15, normalized size = 0.94 \begin {gather*} \arcsin \relax (x )-\sqrt {-x^{2}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.96, size = 14, normalized size = 0.88 \begin {gather*} -\sqrt {-x^{2} + 1} + \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1) + arcsin(x)

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mupad [B]  time = 0.04, size = 14, normalized size = 0.88 \begin {gather*} \mathrm {asin}\relax (x)-\sqrt {1-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(1 - x^2)^(1/2),x)

[Out]

asin(x) - (1 - x^2)^(1/2)

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sympy [A]  time = 0.15, size = 10, normalized size = 0.62 \begin {gather*} - \sqrt {1 - x^{2}} + \operatorname {asin}{\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1 - x**2) + asin(x)

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