\(\int \frac {\arccos (\frac {a}{x})}{x^2} \, dx\) [57]

Optimal result

Integrand size = 10, antiderivative size = 30 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {1-\frac {a^2}{x^2}}}{a}-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{x} \]

[Out]

-arcsec(x/a)/x+(1-a^2/x^2)^(1/2)/a

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4917, 5328, 267} \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {1-\frac {a^2}{x^2}}}{a}-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{x} \]

[In]

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

[Out]

Sqrt[1 - a^2/x^2]/a - ArcSec[x/a]/x

Rule 267

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]

Rule 4917

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 5328

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {1-\frac {a^2}{x^2}}}{a}-\frac {\arccos \left (\frac {a}{x}\right )}{x} \]

[In]

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

[Out]

Sqrt[1 - a^2/x^2]/a - ArcCos[a/x]/x

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(-\frac {\frac {a \arccos \left (\frac {a}{x}\right )}{x}-\sqrt {1-\frac {a^{2}}{x^{2}}}}{a}\)	\(32\)
default	\(-\frac {\frac {a \arccos \left (\frac {a}{x}\right )}{x}-\sqrt {1-\frac {a^{2}}{x^{2}}}}{a}\)	\(32\)
parts	\(-\frac {\arccos \left (\frac {a}{x}\right )}{x}-\frac {a^{2}-x^{2}}{a \sqrt {-\frac {a^{2}-x^{2}}{x^{2}}}\, x^{2}}\)	\(46\)

[In]

int(arccos(a/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {a \arccos \left (\frac {a}{x}\right ) - x \sqrt {-\frac {a^{2} - x^{2}}{x^{2}}}}{a x} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=\begin {cases} - \frac {\operatorname {acos}{\left (\frac {a}{x} \right )}}{x} + \frac {\sqrt {- \frac {a^{2}}{x^{2}} + 1}}{a} & \text {for}\: a \neq 0 \\- \frac {\pi }{2 x} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {\frac {a \arccos \left (\frac {a}{x}\right )}{x} - \sqrt {-\frac {a^{2}}{x^{2}} + 1}}{a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {\frac {a \arccos \left (\frac {a}{x}\right )}{x} - \sqrt {-\frac {a^{2}}{x^{2}} + 1}}{a} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {1-\frac {a^2}{x^2}}}{a}-\frac {\mathrm {acos}\left (\frac {a}{x}\right )}{x} \]

[In]

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

[Out]

(1 - a^2/x^2)^(1/2)/a - acos(a/x)/x