3.659 \(\int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\)

Optimal. Leaf size=34 \[ \frac {x^2}{4}-\frac {1}{2} \sqrt {1-x^2} x \sin ^{-1}(x)+\frac {1}{4} \sin ^{-1}(x)^2 \]

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Rubi [A]  time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4707, 4641, 30} \[ \frac {x^2}{4}-\frac {1}{2} \sqrt {1-x^2} x \sin ^{-1}(x)+\frac {1}{4} \sin ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[(x^2*ArcSin[x])/Sqrt[1 - x^2],x]

[Out]

x^2/4 - (x*Sqrt[1 - x^2]*ArcSin[x])/2 + ArcSin[x]^2/4

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4641

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

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx &=-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {\int x \, dx}{2}+\frac {1}{2} \int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=\frac {x^2}{4}-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{4} \sin ^{-1}(x)^2\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.82 \[ \frac {1}{4} \left (x^2-2 \sqrt {1-x^2} x \sin ^{-1}(x)+\sin ^{-1}(x)^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*ArcSin[x])/Sqrt[1 - x^2],x]

[Out]

(x^2 - 2*x*Sqrt[1 - x^2]*ArcSin[x] + ArcSin[x]^2)/4

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

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 0.65, size = 26, normalized size = 0.76 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \relax (x) + \frac {1}{4} \, x^{2} + \frac {1}{4} \, \arcsin \relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^2 + 1)*x*arcsin(x) + 1/4*x^2 + 1/4*arcsin(x)^2

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giac [A]  time = 1.01, size = 27, normalized size = 0.79 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \relax (x) + \frac {1}{4} \, x^{2} + \frac {1}{4} \, \arcsin \relax (x)^{2} - \frac {1}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^2 + 1)*x*arcsin(x) + 1/4*x^2 + 1/4*arcsin(x)^2 - 1/8

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maple [A]  time = 0.32, size = 32, normalized size = 0.94

method	result	size
default	\(\frac {\arcsin \relax (x ) \left (-\sqrt {-x^{2}+1}\, x +\arcsin \relax (x )\right )}{2}-\frac {\arcsin \relax (x )^{2}}{4}+\frac {x^{2}}{4}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsin(x)/(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*arcsin(x)*(-(-x^2+1)^(1/2)*x+arcsin(x))-1/4*arcsin(x)^2+1/4*x^2

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maxima [A]  time = 0.97, size = 32, normalized size = 0.94 \[ \frac {1}{4} \, x^{2} - \frac {1}{2} \, {\left (\sqrt {-x^{2} + 1} x - \arcsin \relax (x)\right )} \arcsin \relax (x) - \frac {1}{4} \, \arcsin \relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2 - 1/2*(sqrt(-x^2 + 1)*x - arcsin(x))*arcsin(x) - 1/4*arcsin(x)^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^2\,\mathrm {asin}\relax (x)}{\sqrt {1-x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*asin(x))/(1 - x^2)^(1/2),x)

[Out]

int((x^2*asin(x))/(1 - x^2)^(1/2), x)

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sympy [A]  time = 0.37, size = 26, normalized size = 0.76 \[ \frac {x^{2}}{4} - \frac {x \sqrt {1 - x^{2}} \operatorname {asin}{\relax (x )}}{2} + \frac {\operatorname {asin}^{2}{\relax (x )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/4 - x*sqrt(1 - x**2)*asin(x)/2 + asin(x)**2/4

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