3.7.53 \(\int x \sqrt {1-x^2} \cos ^{-1}(x) \, dx\) [653]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {4768} \begin {gather*} -\frac {1}{3} \left (1-x^2\right )^{3/2} \text {ArcCos}(x)+\frac {x^3}{9}-\frac {x}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

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

Rule 4768

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 = 0.02, size = 26, normalized size = 0.87 \begin {gather*} \frac {1}{9} \left (-3 x+x^3-3 \left (1-x^2\right )^{3/2} \cos ^{-1}(x)\right ) \end {gather*}

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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Maple [C] Result contains complex when optimal does not.
time = 0.20, size = 134, normalized size = 4.47

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

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*(-x^2+1)^(1/2)-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*(-x^2+1)^(1/2)-3*I*x-(-x^2+1)^(
1/2))

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Maxima [A]
time = 2.07, size = 22, normalized size = 0.73 \begin {gather*} \frac {1}{9} \, x^{3} - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right ) - \frac {1}{3} \, x \end {gather*}

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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Fricas [A]
time = 0.49, size = 27, normalized size = 0.90 \begin {gather*} \frac {1}{9} \, x^{3} + \frac {1}{3} \, {\left (x^{2} - 1\right )} \sqrt {-x^{2} + 1} \arccos \left (x\right ) - \frac {1}{3} \, x \end {gather*}

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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Sympy [A]
time = 0.18, size = 37, normalized size = 1.23 \begin {gather*} \frac {x^{3}}{9} + \frac {x^{2} \sqrt {1 - x^{2}} \operatorname {acos}{\left (x \right )}}{3} - \frac {x}{3} - \frac {\sqrt {1 - x^{2}} \operatorname {acos}{\left (x \right )}}{3} \end {gather*}

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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Giac [A]
time = 1.70, size = 22, normalized size = 0.73 \begin {gather*} \frac {1}{9} \, x^{3} - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right ) - \frac {1}{3} \, x \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x\,\mathrm {acos}\left (x\right )\,\sqrt {1-x^2} \,d x \end {gather*}

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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