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

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4648, 4642, 30} \[ \frac {x^2}{4}+\frac {1}{2} \sqrt {1-x^2} x \cos ^{-1}(x)-\frac {1}{4} \cos ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

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

Rule 30

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

Rule 4642

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> -Simp[(a + b*ArcCos[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 4648

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

Rubi steps

\begin {align*} \int \sqrt {1-x^2} \cos ^{-1}(x) \, dx &=\frac {1}{2} x \sqrt {1-x^2} \cos ^{-1}(x)+\frac {\int x \, dx}{2}+\frac {1}{2} \int \frac {\cos ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=\frac {x^2}{4}+\frac {1}{2} x \sqrt {1-x^2} \cos ^{-1}(x)-\frac {1}{4} \cos ^{-1}(x)^2\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 0.88 \[ \frac {1}{4} \left (x^2+2 \sqrt {1-x^2} x \cos ^{-1}(x)-\cos ^{-1}(x)^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

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

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fricas [A]  time = 0.46, size = 26, normalized size = 0.76 \[ \frac {1}{2} \, \sqrt {-x^{2} + 1} x \arccos \relax (x) + \frac {1}{4} \, x^{2} - \frac {1}{4} \, \arccos \relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-x^2 + 1)*x*arccos(x) + 1/4*x^2 - 1/4*arccos(x)^2

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giac [A]  time = 0.85, size = 27, normalized size = 0.79 \[ \frac {1}{2} \, \sqrt {-x^{2} + 1} x \arccos \relax (x) + \frac {1}{4} \, x^{2} - \frac {1}{4} \, \arccos \relax (x)^{2} - \frac {1}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-x^2 + 1)*x*arccos(x) + 1/4*x^2 - 1/4*arccos(x)^2 - 1/8

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maple [A]  time = 0.21, size = 33, normalized size = 0.97 \[ -\frac {\arccos \relax (x ) \left (-x \sqrt {-x^{2}+1}+\arccos \relax (x )\right )}{2}+\frac {\arccos \relax (x )^{2}}{4}+\frac {x^{2}}{4}-\frac {1}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.41, size = 30, normalized size = 0.88 \[ \frac {1}{4} \, x^{2} + \frac {1}{2} \, {\left (\sqrt {-x^{2} + 1} x + \arcsin \relax (x)\right )} \arccos \relax (x) + \frac {1}{4} \, \arcsin \relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(x)*(1 - x^2)^(1/2),x)

[Out]

int(acos(x)*(1 - x^2)^(1/2), x)

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sympy [A]  time = 20.18, size = 48, normalized size = 1.41 \[ \left (\begin {cases} \frac {x \sqrt {1 - x^{2}}}{2} + \frac {\operatorname {asin}{\relax (x )}}{2} & \text {for}\: x > -1 \wedge x < 1 \end {cases}\right ) \operatorname {acos}{\relax (x )} + \begin {cases} \text {NaN} & \text {for}\: x < -1 \\\frac {x^{2}}{4} + \frac {\operatorname {asin}^{2}{\relax (x )}}{4} - \frac {\pi ^{2}}{16} - \frac {1}{4} & \text {for}\: x < 1 \\\text {NaN} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*sqrt(1 - x**2)/2 + asin(x)/2, (x > -1) & (x < 1)))*acos(x) + Piecewise((nan, x < -1), (x**2/4 + a
sin(x)**2/4 - pi**2/16 - 1/4, x < 1), (nan, True))

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