3.64 \(\int \frac{\cos ^{-1}(\sqrt{x})}{x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{\sqrt{1-x}}{\sqrt{x}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{x} \]

[Out]

Sqrt[1 - x]/Sqrt[x] - ArcCos[Sqrt[x]]/x

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Rubi [A]  time = 0.0119337, antiderivative size = 27, 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 = {4843, 12, 37} \[ \frac{\sqrt{1-x}}{\sqrt{x}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - x]/Sqrt[x] - ArcCos[Sqrt[x]]/x

Rule 4843

Int[((a_.) + ArcCos[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcCos[
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{\cos ^{-1}\left (\sqrt{x}\right )}{x^2} \, dx &=-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{x}-\int \frac{1}{2 \sqrt{1-x} x^{3/2}} \, dx\\ &=-\frac{\cos ^{-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{\cos ^{-1}\left (\sqrt{x}\right )}{x}\\ \end{align*}

Mathematica [A]  time = 0.0156122, size = 24, normalized size = 0.89 \[ \frac{\sqrt{x-x^2}-\cos ^{-1}\left (\sqrt{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x - x^2] - ArcCos[Sqrt[x]])/x

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Maple [A]  time = 0.001, size = 22, normalized size = 0.8 \begin{align*} -{\frac{1}{x}\arccos \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(arccos(x^(1/2))/x^2,x)

[Out]

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

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Maxima [A]  time = 1.49926, size = 28, normalized size = 1.04 \begin{align*} \frac{\sqrt{-x + 1}}{\sqrt{x}} - \frac{\arccos \left (\sqrt{x}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.56133, size = 59, normalized size = 2.19 \begin{align*} \frac{\sqrt{x} \sqrt{-x + 1} - \arccos \left (\sqrt{x}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 4.22227, size = 44, normalized size = 1.63 \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{acos}{\left (\sqrt{x} \right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(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 - acos(sqrt(x))/x

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Giac [A]  time = 1.17879, size = 54, normalized size = 2. \begin{align*} \frac{\sqrt{-x + 1} - 1}{2 \, \sqrt{x}} - \frac{\arccos \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(arccos(x^(1/2))/x^2,x, algorithm="giac")

[Out]

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