3.365 \(\int \frac{\sin ^{-1}(\sqrt{x})}{x^2} \, dx\)

Optimal. Leaf size=28 \[ -\frac{\sqrt{1-x}}{\sqrt{x}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{x} \]

[Out]

-(Sqrt[1 - x]/Sqrt[x]) - ArcSin[Sqrt[x]]/x

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Rubi [A]  time = 0.0132777, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4842, 12, 37} \[ -\frac{\sqrt{1-x}}{\sqrt{x}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - x]/Sqrt[x]) - ArcSin[Sqrt[x]]/x

Rule 4842

Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 - u^2], x]
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(
m + 1), 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 37

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

Rubi steps

\begin{align*} \int \frac{\sin ^{-1}\left (\sqrt{x}\right )}{x^2} \, dx &=-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{x}+\int \frac{1}{2 \sqrt{1-x} x^{3/2}} \, dx\\ &=-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{x}+\frac{1}{2} \int \frac{1}{\sqrt{1-x} x^{3/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{\sqrt{x}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{x}\\ \end{align*}

Mathematica [A]  time = 0.0129373, size = 23, normalized size = 0.82 \[ -\frac{\sqrt{x-x^2}+\sin ^{-1}\left (\sqrt{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 23, normalized size = 0.8 \begin{align*} -{\frac{1}{x}\arcsin \left ( \sqrt{x} \right ) }-{\sqrt{1-x}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.41121, size = 30, normalized size = 1.07 \begin{align*} -\frac{\sqrt{-x + 1}}{\sqrt{x}} - \frac{\arcsin \left (\sqrt{x}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.3704, size = 61, normalized size = 2.18 \begin{align*} -\frac{\sqrt{x} \sqrt{-x + 1} + \arcsin \left (\sqrt{x}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 4.39882, size = 42, normalized size = 1.5 \begin{align*} \frac{\begin{cases} - \frac{2 i \sqrt{x - 1}}{\sqrt{x}} & \text{for}\: \left |{x}\right | > 1 \\- \frac{2 \sqrt{1 - x}}{\sqrt{x}} & \text{otherwise} \end{cases}}{2} - \frac{\operatorname{asin}{\left (\sqrt{x} \right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*I*sqrt(x - 1)/sqrt(x), Abs(x) > 1), (-2*sqrt(1 - x)/sqrt(x), True))/2 - asin(sqrt(x))/x

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Giac [A]  time = 1.20766, size = 54, normalized size = 1.93 \begin{align*} -\frac{\sqrt{-x + 1} - 1}{2 \, \sqrt{x}} - \frac{\arcsin \left (\sqrt{x}\right )}{x} + \frac{\sqrt{x}}{2 \,{\left (\sqrt{-x + 1} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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