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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0241101, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 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]

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 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])

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*}

Mathematica [A]  time = 0.0117271, size = 29, normalized size = 0.72 \[ \frac{1}{9} \left (\sqrt{1-x^2} \left (x^2+2\right )+3 x^3 \sin ^{-1}(x)\right ) \]

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

________________________________________________________________________________________

Maple [A]  time = 0., size = 34, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}\arcsin \left ( x \right ) }{3}}+{\frac{{x}^{2}}{9}\sqrt{-{x}^{2}+1}}+{\frac{2}{9}\sqrt{-{x}^{2}+1}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.40179, size = 45, normalized size = 1.12 \begin{align*} \frac{1}{3} \, x^{3} \arcsin \left (x\right ) + \frac{1}{9} \, \sqrt{-x^{2} + 1} x^{2} + \frac{2}{9} \, \sqrt{-x^{2} + 1} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 2.04512, size = 68, normalized size = 1.7 \begin{align*} \frac{1}{3} \, x^{3} \arcsin \left (x\right ) + \frac{1}{9} \,{\left (x^{2} + 2\right )} \sqrt{-x^{2} + 1} \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.320597, size = 32, normalized size = 0.8 \begin{align*} \frac{x^{3} \operatorname{asin}{\left (x \right )}}{3} + \frac{x^{2} \sqrt{1 - x^{2}}}{9} + \frac{2 \sqrt{1 - x^{2}}}{9} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.07506, size = 51, normalized size = 1.27 \begin{align*} \frac{1}{3} \,{\left (x^{2} - 1\right )} x \arcsin \left (x\right ) + \frac{1}{3} \, x \arcsin \left (x\right ) - \frac{1}{9} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} + \frac{1}{3} \, \sqrt{-x^{2} + 1} \end{align*}

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)