3.1.27 \(\int \sin ^{-1}(x) \, dx\) [27]

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 = {4715, 267} \begin {gather*} x \text {ArcSin}(x)+\sqrt {1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSin[x],x]

[Out]

Sqrt[1 - x^2] + x*ArcSin[x]

Rule 267

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 4715

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 \begin {gather*} \sqrt {1-x^2}+x \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[x],x]

[Out]

Sqrt[1 - x^2] + x*ArcSin[x]

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Maple [A]
time = 0.00, size = 15, normalized size = 0.94

method	result	size
lookup	\(\arcsin \left (x \right ) x +\sqrt {-x^{2}+1}\)	\(15\)
default	\(\arcsin \left (x \right ) x +\sqrt {-x^{2}+1}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 3.10, size = 14, normalized size = 0.88 \begin {gather*} x \arcsin \left (x\right ) + \sqrt {-x^{2} + 1} \end {gather*}

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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Fricas [A]
time = 1.11, size = 14, normalized size = 0.88 \begin {gather*} x \arcsin \left (x\right ) + \sqrt {-x^{2} + 1} \end {gather*}

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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Sympy [A]
time = 0.04, size = 12, normalized size = 0.75 \begin {gather*} x \operatorname {asin}{\left (x \right )} + \sqrt {1 - x^{2}} \end {gather*}

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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Giac [A]
time = 0.59, size = 14, normalized size = 0.88 \begin {gather*} x \arcsin \left (x\right ) + \sqrt {-x^{2} + 1} \end {gather*}

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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Mupad [B]
time = 0.00, size = 14, normalized size = 0.88 \begin {gather*} x\,\mathrm {asin}\left (x\right )+\sqrt {1-x^2} \end {gather*}

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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