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

Optimal. Leaf size=40 \[ \frac {1}{3} x^3 \sin ^{-1}(x)-\frac {1}{9} \left (1-x^2\right )^{3/2}+\frac {\sqrt {1-x^2}}{3} \]

[Out]

-1/9*(-x^2+1)^(3/2)+1/3*x^3*arcsin(x)+1/3*(-x^2+1)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4627, 266, 43} \[ -\frac {1}{9} \left (1-x^2\right )^{3/2}+\frac {\sqrt {1-x^2}}{3}+\frac {1}{3} x^3 \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcSin[x],x]

[Out]

Sqrt[1 - x^2]/3 - (1 - x^2)^(3/2)/9 + (x^3*ArcSin[x])/3

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

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

Rule 4627

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

Rubi steps

\begin {align*} \int x^2 \sin ^{-1}(x) \, dx &=\frac {1}{3} x^3 \sin ^{-1}(x)-\frac {1}{3} \int \frac {x^3}{\sqrt {1-x^2}} \, dx\\ &=\frac {1}{3} x^3 \sin ^{-1}(x)-\frac {1}{6} \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \sin ^{-1}(x)-\frac {1}{6} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx,x,x^2\right )\\ &=\frac {\sqrt {1-x^2}}{3}-\frac {1}{9} \left (1-x^2\right )^{3/2}+\frac {1}{3} x^3 \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 0.72 \[ \frac {1}{9} \left (3 x^3 \sin ^{-1}(x)+\sqrt {1-x^2} \left (x^2+2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcSin[x],x]

[Out]

(Sqrt[1 - x^2]*(2 + x^2) + 3*x^3*ArcSin[x])/9

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fricas [A]  time = 0.44, size = 24, normalized size = 0.60 \[ \frac {1}{3} \, x^{3} \arcsin \relax (x) + \frac {1}{9} \, {\left (x^{2} + 2\right )} \sqrt {-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arcsin(x) + 1/9*(x^2 + 2)*sqrt(-x^2 + 1)

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giac [A]  time = 1.09, size = 38, normalized size = 0.95 \[ \frac {1}{3} \, {\left (x^{2} - 1\right )} x \arcsin \relax (x) + \frac {1}{3} \, x \arcsin \relax (x) - \frac {1}{9} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + \frac {1}{3} \, \sqrt {-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^2 - 1)*x*arcsin(x) + 1/3*x*arcsin(x) - 1/9*(-x^2 + 1)^(3/2) + 1/3*sqrt(-x^2 + 1)

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maple [A]  time = 0.00, size = 34, normalized size = 0.85 \[ \frac {x^{3} \arcsin \relax (x )}{3}+\frac {\sqrt {-x^{2}+1}\, x^{2}}{9}+\frac {2 \sqrt {-x^{2}+1}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsin(x),x)

[Out]

1/3*x^3*arcsin(x)+1/9*(-x^2+1)^(1/2)*x^2+2/9*(-x^2+1)^(1/2)

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maxima [A]  time = 1.29, size = 33, normalized size = 0.82 \[ \frac {1}{3} \, x^{3} \arcsin \relax (x) + \frac {1}{9} \, \sqrt {-x^{2} + 1} x^{2} + \frac {2}{9} \, \sqrt {-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arcsin(x) + 1/9*sqrt(-x^2 + 1)*x^2 + 2/9*sqrt(-x^2 + 1)

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mupad [B]  time = 0.00, size = 24, normalized size = 0.60 \[ \frac {x^3\,\mathrm {asin}\relax (x)}{3}+\frac {\sqrt {1-x^2}\,\left (x^2+2\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*asin(x),x)

[Out]

(x^3*asin(x))/3 + ((1 - x^2)^(1/2)*(x^2 + 2))/9

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sympy [A]  time = 0.34, size = 32, normalized size = 0.80 \[ \frac {x^{3} \operatorname {asin}{\relax (x )}}{3} + \frac {x^{2} \sqrt {1 - x^{2}}}{9} + \frac {2 \sqrt {1 - x^{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*asin(x)/3 + x**2*sqrt(1 - x**2)/9 + 2*sqrt(1 - x**2)/9

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