3.84 \(\int \sin ^{-1}(x)^2 \, dx\)

Optimal. Leaf size=25 \[ 2 \sqrt{1-x^2} \sin ^{-1}(x)-2 x+x \sin ^{-1}(x)^2 \]

[Out]

-2*x + 2*Sqrt[1 - x^2]*ArcSin[x] + x*ArcSin[x]^2

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Rubi [A]  time = 0.0323284, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {4619, 4677, 8} \[ 2 \sqrt{1-x^2} \sin ^{-1}(x)-2 x+x \sin ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[x]^2,x]

[Out]

-2*x + 2*Sqrt[1 - x^2]*ArcSin[x] + x*ArcSin[x]^2

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]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0058889, size = 25, normalized size = 1. \[ 2 \sqrt{1-x^2} \sin ^{-1}(x)-2 x+x \sin ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[x]^2,x]

[Out]

-2*x + 2*Sqrt[1 - x^2]*ArcSin[x] + x*ArcSin[x]^2

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Maple [A]  time = 0.019, size = 24, normalized size = 1. \begin{align*} -2\,x+x \left ( \arcsin \left ( x \right ) \right ) ^{2}+2\,\arcsin \left ( x \right ) \sqrt{-{x}^{2}+1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x)^2,x)

[Out]

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

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Maxima [A]  time = 1.42581, size = 31, normalized size = 1.24 \begin{align*} x \arcsin \left (x\right )^{2} + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.8521, size = 68, normalized size = 2.72 \begin{align*} x \arcsin \left (x\right )^{2} + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.177275, size = 22, normalized size = 0.88 \begin{align*} x \operatorname{asin}^{2}{\left (x \right )} - 2 x + 2 \sqrt{1 - x^{2}} \operatorname{asin}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(x)**2,x)

[Out]

x*asin(x)**2 - 2*x + 2*sqrt(1 - x**2)*asin(x)

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Giac [A]  time = 1.05263, size = 31, normalized size = 1.24 \begin{align*} x \arcsin \left (x\right )^{2} + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)^2,x, algorithm="giac")

[Out]

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