3.57 \(\int \frac{\cos ^{-1}(\frac{a}{x})}{x^2} \, dx\)

Optimal. Leaf size=30 \[ \frac{\sqrt{1-\frac{a^2}{x^2}}}{a}-\frac{\sec ^{-1}\left (\frac{x}{a}\right )}{x} \]

[Out]

Sqrt[1 - a^2/x^2]/a - ArcSec[x/a]/x

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Rubi [A]  time = 0.0218937, antiderivative size = 30, 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 = {4833, 5220, 261} \[ \frac{\sqrt{1-\frac{a^2}{x^2}}}{a}-\frac{\sec ^{-1}\left (\frac{x}{a}\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - a^2/x^2]/a - ArcSec[x/a]/x

Rule 4833

Int[ArcCos[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[u*ArcSec[a/c + (b*x^n)/c]^m, x] /;
FreeQ[{a, b, c, n, m}, x]

Rule 5220

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

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.0185694, size = 30, normalized size = 1. \[ \frac{\sqrt{1-\frac{a^2}{x^2}}}{a}-\frac{\cos ^{-1}\left (\frac{a}{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - a^2/x^2]/a - ArcCos[a/x]/x

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Maple [A]  time = 0.001, size = 32, normalized size = 1.1 \begin{align*} -{\frac{1}{a} \left ({\frac{a}{x}\arccos \left ({\frac{a}{x}} \right ) }-\sqrt{1-{\frac{{a}^{2}}{{x}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a/x)/x^2,x)

[Out]

-1/a*(1/x*a*arccos(a/x)-(1-a^2/x^2)^(1/2))

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Maxima [A]  time = 1.43795, size = 42, normalized size = 1.4 \begin{align*} -\frac{\frac{a \arccos \left (\frac{a}{x}\right )}{x} - \sqrt{-\frac{a^{2}}{x^{2}} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.39408, size = 69, normalized size = 2.3 \begin{align*} -\frac{a \arccos \left (\frac{a}{x}\right ) - x \sqrt{-\frac{a^{2} - x^{2}}{x^{2}}}}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*arccos(a/x) - x*sqrt(-(a^2 - x^2)/x^2))/(a*x)

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Sympy [A]  time = 2.34016, size = 26, normalized size = 0.87 \begin{align*} \begin{cases} - \frac{\operatorname{acos}{\left (\frac{a}{x} \right )}}{x} + \frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{a} & \text{for}\: a \neq 0 \\- \frac{\pi }{2 x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a/x)/x**2,x)

[Out]

Piecewise((-acos(a/x)/x + sqrt(-a**2/x**2 + 1)/a, Ne(a, 0)), (-pi/(2*x), True))

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Giac [A]  time = 1.2283, size = 42, normalized size = 1.4 \begin{align*} -\frac{\frac{a \arccos \left (\frac{a}{x}\right )}{x} - \sqrt{-\frac{a^{2}}{x^{2}} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a/x)/x^2,x, algorithm="giac")

[Out]

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