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

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

[Out]

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

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Rubi [A]  time = 0.0597781, antiderivative size = 34, normalized size of antiderivative = 1., 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 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]

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 30

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

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*}

Mathematica [A]  time = 0.0112583, 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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Maple [A]  time = 0.035, size = 32, normalized size = 0.9 \begin{align*}{\frac{\arcsin \left ( x \right ) }{2} \left ( -x\sqrt{-{x}^{2}+1}+\arcsin \left ( x \right ) \right ) }-{\frac{ \left ( \arcsin \left ( x \right ) \right ) ^{2}}{4}}+{\frac{{x}^{2}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.40747, size = 43, normalized size = 1.26 \begin{align*} \frac{1}{4} \, x^{2} - \frac{1}{2} \,{\left (\sqrt{-x^{2} + 1} x - \arcsin \left (x\right )\right )} \arcsin \left (x\right ) - \frac{1}{4} \, \arcsin \left (x\right )^{2} \end{align*}

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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Fricas [A]  time = 2.59129, size = 82, normalized size = 2.41 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 1} x \arcsin \left (x\right ) + \frac{1}{4} \, x^{2} + \frac{1}{4} \, \arcsin \left (x\right )^{2} \end{align*}

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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Sympy [A]  time = 0.332589, size = 26, normalized size = 0.76 \begin{align*} \frac{x^{2}}{4} - \frac{x \sqrt{1 - x^{2}} \operatorname{asin}{\left (x \right )}}{2} + \frac{\operatorname{asin}^{2}{\left (x \right )}}{4} \end{align*}

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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Giac [A]  time = 1.07644, size = 36, normalized size = 1.06 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 1} x \arcsin \left (x\right ) + \frac{1}{4} \, x^{2} + \frac{1}{4} \, \arcsin \left (x\right )^{2} - \frac{1}{8} \end{align*}

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