3.12 \(\int \sec ^{-1}(\frac{a}{x}) \, dx\)

Optimal. Leaf size=26 \[ x \cos ^{-1}\left (\frac{x}{a}\right )-a \sqrt{1-\frac{x^2}{a^2}} \]

[Out]

-(a*Sqrt[1 - x^2/a^2]) + x*ArcCos[x/a]

________________________________________________________________________________________

Rubi [A]  time = 0.0092175, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5264, 4620, 261} \[ x \cos ^{-1}\left (\frac{x}{a}\right )-a \sqrt{1-\frac{x^2}{a^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSec[a/x],x]

[Out]

-(a*Sqrt[1 - x^2/a^2]) + x*ArcCos[x/a]

Rule 5264

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

Rule 4620

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[
(x*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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

Mathematica [A]  time = 0.0125269, size = 26, normalized size = 1. \[ x \sec ^{-1}\left (\frac{a}{x}\right )-a \sqrt{1-\frac{x^2}{a^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSec[a/x],x]

[Out]

-(a*Sqrt[1 - x^2/a^2]) + x*ArcSec[a/x]

________________________________________________________________________________________

Maple [B]  time = 0.151, size = 51, normalized size = 2. \begin{align*} -a \left ( -{\frac{x}{a}{\rm arcsec} \left ({\frac{a}{x}}\right )}+{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ){\frac{1}{\sqrt{{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) }}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsec(a/x),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.964025, size = 32, normalized size = 1.23 \begin{align*} x \operatorname{arcsec}\left (\frac{a}{x}\right ) - a \sqrt{-\frac{x^{2}}{a^{2}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(a/x),x, algorithm="maxima")

[Out]

x*arcsec(a/x) - a*sqrt(-x^2/a^2 + 1)

________________________________________________________________________________________

Fricas [A]  time = 2.60646, size = 55, normalized size = 2.12 \begin{align*} x \operatorname{arcsec}\left (\frac{a}{x}\right ) - x \sqrt{\frac{a^{2} - x^{2}}{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(a/x),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.164664, size = 22, normalized size = 0.85 \begin{align*} \begin{cases} - a \sqrt{1 - \frac{x^{2}}{a^{2}}} + x \operatorname{asec}{\left (\frac{a}{x} \right )} & \text{for}\: a \neq 0 \\\tilde{\infty } x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asec(a/x),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.08103, size = 38, normalized size = 1.46 \begin{align*} a{\left (\frac{x \arccos \left (\frac{x}{a}\right )}{a} - \sqrt{-\frac{x^{2}}{a^{2}} + 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(a/x),x, algorithm="giac")

[Out]

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