3.7.52 \(\int \sqrt {1-x^2} \cos ^{-1}(x) \, dx\) [652]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, 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 = {4742, 4738, 30} \begin {gather*} \frac {1}{2} \sqrt {1-x^2} x \text {ArcCos}(x)-\frac {\text {ArcCos}(x)^2}{4}+\frac {x^2}{4} \end {gather*}

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 4738

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

Rule 4742

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[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcCos[c*x])^n/S
qrt[1 - c^2*x^2], x], x] + Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 30, normalized size = 0.88 \begin {gather*} \frac {1}{4} \left (x^2+2 x \sqrt {1-x^2} \cos ^{-1}(x)-\cos ^{-1}(x)^2\right ) \end {gather*}

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

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 33, normalized size = 0.97

method	result	size
default	\(-\frac {\arccos \left (x \right ) \left (-x \sqrt {-x^{2}+1}+\arccos \left (x \right )\right )}{2}+\frac {\arccos \left (x \right )^{2}}{4}+\frac {x^{2}}{4}-\frac {1}{4}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 1.73, size = 30, normalized size = 0.88 \begin {gather*} \frac {1}{4} \, x^{2} + \frac {1}{2} \, {\left (\sqrt {-x^{2} + 1} x + \arcsin \left (x\right )\right )} \arccos \left (x\right ) + \frac {1}{4} \, \arcsin \left (x\right )^{2} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.51, size = 26, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x \arccos \left (x\right ) + \frac {1}{4} \, x^{2} - \frac {1}{4} \, \arccos \left (x\right )^{2} \end {gather*}

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

________________________________________________________________________________________

Sympy [A]
time = 1.07, size = 39, normalized size = 1.15 \begin {gather*} \left (\begin {cases} \frac {x \sqrt {1 - x^{2}}}{2} + \frac {\operatorname {asin}{\left (x \right )}}{2} & \text {for}\: x > -1 \wedge x < 1 \end {cases}\right ) \operatorname {acos}{\left (x \right )} + \begin {cases} \text {NaN} & \text {for}\: x < -1 \\\frac {x^{2}}{4} + \frac {\operatorname {asin}^{2}{\left (x \right )}}{4} & \text {for}\: x < 1 \\\text {NaN} & \text {otherwise} \end {cases} \end {gather*}

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, x < 1), (nan, True))

________________________________________________________________________________________

Giac [A]
time = 1.40, size = 27, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x \arccos \left (x\right ) + \frac {1}{4} \, x^{2} - \frac {1}{4} \, \arccos \left (x\right )^{2} - \frac {1}{8} \end {gather*}

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \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(acos(x)*(1 - x^2)^(1/2),x)

[Out]

int(acos(x)*(1 - x^2)^(1/2), x)

________________________________________________________________________________________