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

Optimal. Leaf size=30 \[ \frac {x^3}{9}-\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-\frac {x}{3} \]

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Rubi [A]  time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4678} \[ \frac {x^3}{9}-\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-\frac {x}{3} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

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

Rule 4678

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 26, normalized size = 0.87 \[ \frac {1}{9} \left (x^3-3 \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-3 x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

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

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

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[x*Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

Could not integrate

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fricas [A]  time = 1.12, size = 27, normalized size = 0.90 \[ \frac {1}{9} \, x^{3} + \frac {1}{3} \, {\left (x^{2} - 1\right )} \sqrt {-x^{2} + 1} \arccos \relax (x) - \frac {1}{3} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.16, size = 22, normalized size = 0.73 \[ \frac {1}{9} \, x^{3} - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \relax (x) - \frac {1}{3} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.32, size = 134, normalized size = 4.47

method	result	size
default	\(-\frac {\left (i+3 \arccos \relax (x )\right ) \left (4 i x^{3}-4 \sqrt {-x^{2}+1}\, x^{2}-3 i x +\sqrt {-x^{2}+1}\right )}{72}+\frac {\left (\arccos \relax (x )+i\right ) \left (i x -\sqrt {-x^{2}+1}\right )}{8}-\frac {\left (\arccos \relax (x )-i\right ) \left (i x +\sqrt {-x^{2}+1}\right )}{8}+\frac {\left (-i+3 \arccos \relax (x )\right ) \left (4 i x^{3}+4 \sqrt {-x^{2}+1}\, x^{2}-3 i x -\sqrt {-x^{2}+1}\right )}{72}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(I+3*arccos(x))*(4*I*x^3-4*(-x^2+1)^(1/2)*x^2-3*I*x+(-x^2+1)^(1/2))+1/8*(arccos(x)+I)*(I*x-(-x^2+1)^(1/2
))-1/8*(arccos(x)-I)*(I*x+(-x^2+1)^(1/2))+1/72*(-I+3*arccos(x))*(4*I*x^3+4*(-x^2+1)^(1/2)*x^2-3*I*x-(-x^2+1)^(
1/2))

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maxima [A]  time = 0.96, size = 22, normalized size = 0.73 \[ \frac {1}{9} \, x^{3} - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \relax (x) - \frac {1}{3} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*acos(x)*(1 - x^2)^(1/2),x)

[Out]

int(x*acos(x)*(1 - x^2)^(1/2), x)

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sympy [A]  time = 1.06, size = 37, normalized size = 1.23 \[ \frac {x^{3}}{9} + \frac {x^{2} \sqrt {1 - x^{2}} \operatorname {acos}{\relax (x )}}{3} - \frac {x}{3} - \frac {\sqrt {1 - x^{2}} \operatorname {acos}{\relax (x )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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