3.263 \(\int \sin ^{-1}(x) \, dx\)

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

[Out]

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

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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 = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4619, 261} \[ \sqrt {1-x^2}+x \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[x],x]

[Out]

Sqrt[1 - x^2] + x*ArcSin[x]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4619

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

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ \sqrt {1-x^2}+x \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[x],x]

[Out]

Sqrt[1 - x^2] + x*ArcSin[x]

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fricas [A]  time = 0.44, size = 14, normalized size = 0.88 \[ x \arcsin \relax (x) + \sqrt {-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x),x, algorithm="fricas")

[Out]

x*arcsin(x) + sqrt(-x^2 + 1)

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giac [A]  time = 1.07, size = 14, normalized size = 0.88 \[ x \arcsin \relax (x) + \sqrt {-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x),x, algorithm="giac")

[Out]

x*arcsin(x) + sqrt(-x^2 + 1)

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maple [A]  time = 0.00, size = 15, normalized size = 0.94 \[ x \arcsin \relax (x )+\sqrt {-x^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x),x)

[Out]

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

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maxima [A]  time = 0.95, size = 14, normalized size = 0.88 \[ x \arcsin \relax (x) + \sqrt {-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x),x, algorithm="maxima")

[Out]

x*arcsin(x) + sqrt(-x^2 + 1)

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mupad [B]  time = 0.22, size = 14, normalized size = 0.88 \[ x\,\mathrm {asin}\relax (x)+\sqrt {1-x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x),x)

[Out]

x*asin(x) + (1 - x^2)^(1/2)

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sympy [A]  time = 0.13, size = 12, normalized size = 0.75 \[ x \operatorname {asin}{\relax (x )} + \sqrt {1 - x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(x),x)

[Out]

x*asin(x) + sqrt(1 - x**2)

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