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

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

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Rubi [A]  time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, 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 8

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

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]

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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ 2 \sqrt {1-x^2} \sin ^{-1}(x)-2 x+x \sin ^{-1}(x)^2 \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin ^{-1}(x)^2 \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[ArcSin[x]^2,x]

[Out]

Could not integrate

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fricas [A]  time = 0.95, size = 23, normalized size = 0.92 \[ x \arcsin \relax (x)^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \relax (x) - 2 \, x \]

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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giac [A]  time = 0.96, size = 23, normalized size = 0.92 \[ x \arcsin \relax (x)^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \relax (x) - 2 \, x \]

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

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maple [A]  time = 0.05, size = 24, normalized size = 0.96

method	result	size
default	\(-2 x +x \arcsin \relax (x )^{2}+2 \arcsin \relax (x ) \sqrt {-x^{2}+1}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.97, size = 23, normalized size = 0.92 \[ x \arcsin \relax (x)^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \relax (x) - 2 \, x \]

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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mupad [B]  time = 0.05, size = 22, normalized size = 0.88 \[ 2\,\mathrm {asin}\relax (x)\,\sqrt {1-x^2}+x\,\left ({\mathrm {asin}\relax (x)}^2-2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x)^2,x)

[Out]

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

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sympy [A]  time = 0.20, size = 22, normalized size = 0.88 \[ x \operatorname {asin}^{2}{\relax (x )} - 2 x + 2 \sqrt {1 - x^{2}} \operatorname {asin}{\relax (x )} \]

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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