3.363 \(\int \sin ^{-1}(\sqrt{x}) \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{2} \sqrt{1-x} \sqrt{x}+\frac{1}{4} \sin ^{-1}(1-2 x)+x \sin ^{-1}\left (\sqrt{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.011631, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4840, 12, 50, 53, 619, 216} \[ \frac{1}{2} \sqrt{1-x} \sqrt{x}+\frac{1}{4} \sin ^{-1}(1-2 x)+x \sin ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[Sqrt[x]],x]

[Out]

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

Rule 4840

Int[ArcSin[u_], x_Symbol] :> Simp[x*ArcSin[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/Sqrt[1 - u^2], x], x] /;
InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 53

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

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \sin ^{-1}\left (\sqrt{x}\right ) \, dx &=x \sin ^{-1}\left (\sqrt{x}\right )-\int \frac{\sqrt{x}}{2 \sqrt{1-x}} \, dx\\ &=x \sin ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{\sqrt{x}}{\sqrt{1-x}} \, dx\\ &=\frac{1}{2} \sqrt{1-x} \sqrt{x}+x \sin ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{1}{\sqrt{1-x} \sqrt{x}} \, dx\\ &=\frac{1}{2} \sqrt{1-x} \sqrt{x}+x \sin ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{1}{\sqrt{x-x^2}} \, dx\\ &=\frac{1}{2} \sqrt{1-x} \sqrt{x}+x \sin ^{-1}\left (\sqrt{x}\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,1-2 x\right )\\ &=\frac{1}{2} \sqrt{1-x} \sqrt{x}+\frac{1}{4} \sin ^{-1}(1-2 x)+x \sin ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0126864, size = 34, normalized size = 0.92 \[ \frac{1}{2} \left (\sqrt{-(x-1) x}+\sin ^{-1}\left (\sqrt{1-x}\right )\right )+x \sin ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[Sqrt[x]],x]

[Out]

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

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Maple [A]  time = 0.004, size = 26, normalized size = 0.7 \begin{align*} x\arcsin \left ( \sqrt{x} \right ) +{\frac{1}{2}\sqrt{1-x}\sqrt{x}}-{\frac{1}{2}\arcsin \left ( \sqrt{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.43014, size = 34, normalized size = 0.92 \begin{align*} x \arcsin \left (\sqrt{x}\right ) + \frac{1}{2} \, \sqrt{x} \sqrt{-x + 1} - \frac{1}{2} \, \arcsin \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.21324, size = 78, normalized size = 2.11 \begin{align*} \frac{1}{2} \,{\left (2 \, x - 1\right )} \arcsin \left (\sqrt{x}\right ) + \frac{1}{2} \, \sqrt{x} \sqrt{-x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.365178, size = 29, normalized size = 0.78 \begin{align*} \frac{\sqrt{x} \sqrt{1 - x}}{2} + x \operatorname{asin}{\left (\sqrt{x} \right )} - \frac{\operatorname{asin}{\left (\sqrt{x} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1438, size = 36, normalized size = 0.97 \begin{align*}{\left (x - 1\right )} \arcsin \left (\sqrt{x}\right ) + \frac{1}{2} \, \sqrt{x} \sqrt{-x + 1} + \frac{1}{2} \, \arcsin \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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