\(\int \sqrt {1-x^2} \arcsin (x) \, dx\) [651]

Optimal result

Integrand size = 14, antiderivative size = 34 \[ \int \sqrt {1-x^2} \arcsin (x) \, dx=-\frac {x^2}{4}+\frac {1}{2} x \sqrt {1-x^2} \arcsin (x)+\frac {\arcsin (x)^2}{4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4741, 4737, 30} \[ \int \sqrt {1-x^2} \arcsin (x) \, dx=\frac {1}{2} \sqrt {1-x^2} x \arcsin (x)+\frac {\arcsin (x)^2}{4}-\frac {x^2}{4} \]

[In]

Int[Sqrt[1 - x^2]*ArcSin[x],x]

[Out]

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

Rule 30

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

Rule 4737

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

Rule 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*((
a + b*ArcSin[c*x])^n/2), x] + (Dist[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[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*ArcSin[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*} \text {integral}& = \frac {1}{2} x \sqrt {1-x^2} \arcsin (x)-\frac {\int x \, dx}{2}+\frac {1}{2} \int \frac {\arcsin (x)}{\sqrt {1-x^2}} \, dx \\ & = -\frac {x^2}{4}+\frac {1}{2} x \sqrt {1-x^2} \arcsin (x)+\frac {\arcsin (x)^2}{4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \sqrt {1-x^2} \arcsin (x) \, dx=\frac {1}{4} \left (-x^2+2 x \sqrt {1-x^2} \arcsin (x)+\arcsin (x)^2\right ) \]

[In]

Integrate[Sqrt[1 - x^2]*ArcSin[x],x]

[Out]

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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {\arcsin \left (x \right ) \left (x \sqrt {-x^{2}+1}+\arcsin \left (x \right )\right )}{2}-\frac {\arcsin \left (x \right )^{2}}{4}-\frac {x^{2}}{4}\)	\(31\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \sqrt {1-x^2} \arcsin (x) \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right ) - \frac {1}{4} \, x^{2} + \frac {1}{4} \, \arcsin \left (x\right )^{2} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \sqrt {1-x^2} \arcsin (x) \, dx=- \frac {x^{2}}{4} + \left (\frac {x \sqrt {1 - x^{2}}}{2} + \frac {\operatorname {asin}{\left (x \right )}}{2}\right ) \operatorname {asin}{\left (x \right )} - \frac {\operatorname {asin}^{2}{\left (x \right )}}{4} \]

[In]

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

[Out]

-x**2/4 + (x*sqrt(1 - x**2)/2 + asin(x)/2)*asin(x) - asin(x)**2/4

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \sqrt {1-x^2} \arcsin (x) \, dx=-\frac {1}{4} \, x^{2} + \frac {1}{2} \, {\left (\sqrt {-x^{2} + 1} x + \arcsin \left (x\right )\right )} \arcsin \left (x\right ) - \frac {1}{4} \, \arcsin \left (x\right )^{2} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \sqrt {1-x^2} \arcsin (x) \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right ) - \frac {1}{4} \, x^{2} + \frac {1}{4} \, \arcsin \left (x\right )^{2} + \frac {1}{8} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {1-x^2} \arcsin (x) \, dx=\int \mathrm {asin}\left (x\right )\,\sqrt {1-x^2} \,d x \]

[In]

int(asin(x)*(1 - x^2)^(1/2),x)

[Out]

int(asin(x)*(1 - x^2)^(1/2), x)