\(\int \frac {\arcsin (\sqrt {1-x^2})}{\sqrt {1-x^2}} \, dx\) [699]

Optimal result

Integrand size = 24, antiderivative size = 28 \[ \int \frac {\arcsin \left (\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \, dx=-\frac {\sqrt {x^2} \arcsin \left (\sqrt {1-x^2}\right )^2}{2 x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4918, 4737} \[ \int \frac {\arcsin \left (\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \, dx=-\frac {\sqrt {x^2} \arcsin \left (\sqrt {1-x^2}\right )^2}{2 x} \]

[In]

Int[ArcSin[Sqrt[1 - x^2]]/Sqrt[1 - x^2],x]

[Out]

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

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 4918

Int[ArcSin[Sqrt[1 + (b_.)*(x_)^2]]^(n_.)/Sqrt[1 + (b_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[(-b)*x^2]/(b*x), Subst
[Int[ArcSin[x]^n/Sqrt[1 - x^2], x], x, Sqrt[1 + b*x^2]], x] /; FreeQ[{b, n}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {\arcsin (x)}{\sqrt {1-x^2}} \, dx,x,\sqrt {1-x^2}\right )}{x} \\ & = -\frac {\sqrt {x^2} \arcsin \left (\sqrt {1-x^2}\right )^2}{2 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin \left (\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \, dx=-\frac {\sqrt {x^2} \arcsin \left (\sqrt {1-x^2}\right )^2}{2 x} \]

[In]

Integrate[ArcSin[Sqrt[1 - x^2]]/Sqrt[1 - x^2],x]

[Out]

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

Maple [F]

\[\int \frac {\arcsin \left (\sqrt {-x^{2}+1}\right )}{\sqrt {-x^{2}+1}}d x\]

[In]

int(arcsin((-x^2+1)^(1/2))/(-x^2+1)^(1/2),x)

[Out]

int(arcsin((-x^2+1)^(1/2))/(-x^2+1)^(1/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.50 \[ \int \frac {\arcsin \left (\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \, dx=-\frac {1}{2} \, \arcsin \left (\sqrt {-x^{2} + 1}\right )^{2} \]

[In]

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

[Out]

-1/2*arcsin(sqrt(-x^2 + 1))^2

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\arcsin \left (\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \, dx=\frac {x \operatorname {asin}^{2}{\left (x \right )}}{2 \sqrt {x^{2}}} + \operatorname {asin}{\left (x \right )} \operatorname {asin}{\left (\sqrt {1 - x^{2}} \right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\arcsin \left (\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \, dx=\int { \frac {\arcsin \left (\sqrt {-x^{2} + 1}\right )}{\sqrt {-x^{2} + 1}} \,d x } \]

[In]

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

[Out]

integrate(arcsin(sqrt(-x^2 + 1))/sqrt(-x^2 + 1), x)

Giac [F]

\[ \int \frac {\arcsin \left (\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \, dx=\int { \frac {\arcsin \left (\sqrt {-x^{2} + 1}\right )}{\sqrt {-x^{2} + 1}} \,d x } \]

[In]

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

[Out]

integrate(arcsin(sqrt(-x^2 + 1))/sqrt(-x^2 + 1), x)

Mupad [F(-1)]

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

[In]

int(asin((1 - x^2)^(1/2))/(1 - x^2)^(1/2),x)

[Out]

int(asin((1 - x^2)^(1/2))/(1 - x^2)^(1/2), x)