3.93 \(\int x^2 \cos ^{-1}(x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0215891, 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 = {4628, 266, 43} \[ \frac{1}{3} x^3 \cos ^{-1}(x)+\frac{1}{9} \left (1-x^2\right )^{3/2}-\frac{\sqrt{1-x^2}}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4628

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
s[c*x])^n)/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCos[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 \cos ^{-1}(x) \, dx &=\frac{1}{3} x^3 \cos ^{-1}(x)+\frac{1}{3} \int \frac{x^3}{\sqrt{1-x^2}} \, dx\\ &=\frac{1}{3} x^3 \cos ^{-1}(x)+\frac{1}{6} \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \cos ^{-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{1}{3} \sqrt{1-x^2}+\frac{1}{9} \left (1-x^2\right )^{3/2}+\frac{1}{3} x^3 \cos ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0103854, size = 30, normalized size = 0.75 \[ \frac{1}{3} x^3 \cos ^{-1}(x)-\frac{1}{9} \sqrt{1-x^2} \left (x^2+2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 34, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}\arccos \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*arccos(x),x)

[Out]

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

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Maxima [A]  time = 1.45287, size = 45, normalized size = 1.12 \begin{align*} \frac{1}{3} \, x^{3} \arccos \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*arccos(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.548079, size = 68, normalized size = 1.7 \begin{align*} \frac{1}{3} \, x^{3} \arccos \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*arccos(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.330808, size = 32, normalized size = 0.8 \begin{align*} \frac{x^{3} \operatorname{acos}{\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*acos(x),x)

[Out]

x**3*acos(x)/3 - x**2*sqrt(1 - x**2)/9 - 2*sqrt(1 - x**2)/9

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Giac [A]  time = 1.09278, size = 45, normalized size = 1.12 \begin{align*} \frac{1}{3} \, x^{3} \arccos \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*arccos(x),x, algorithm="giac")

[Out]

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