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]

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

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.46, size = 22, normalized size = 0.81 \[ \frac {\sqrt {x} \sqrt {-x + 1} - \arccos \left (\sqrt {x}\right )}{x} \]

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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giac [A]  time = 2.99, size = 40, normalized size = 1.48 \[ \frac {\sqrt {-x + 1} - 1}{2 \, \sqrt {x}} - \frac {\arccos \left (\sqrt {x}\right )}{x} - \frac {\sqrt {x}}{2 \, {\left (\sqrt {-x + 1} - 1\right )}} \]

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)

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maple [A]  time = 0.00, size = 22, normalized size = 0.81 \[ -\frac {\arccos \left (\sqrt {x}\right )}{x}+\frac {\sqrt {1-x}}{\sqrt {x}} \]

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 = 0.42, size = 21, normalized size = 0.78 \[ \frac {\sqrt {-x + 1}}{\sqrt {x}} - \frac {\arccos \left (\sqrt {x}\right )}{x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {acos}\left (\sqrt {x}\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acos(x^(1/2))/x^2, x)

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sympy [C]  time = 3.11, size = 44, normalized size = 1.63 \[ - \frac {\begin {cases} - \frac {2 i \sqrt {x - 1}}{\sqrt {x}} & \text {for}\: \left |{x}\right | > 1 \\- \frac {2 \sqrt {1 - x}}{\sqrt {x}} & \text {otherwise} \end {cases}}{2} - \frac {\operatorname {acos}{\left (\sqrt {x} \right )}}{x} \]

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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