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

Optimal. Leaf size=47 \[ -\frac{1}{4} a x \sqrt{1-\frac{x^2}{a^2}}+\frac{1}{4} a^2 \sin ^{-1}\left (\frac{x}{a}\right )+\frac{1}{2} x^2 \cos ^{-1}\left (\frac{x}{a}\right ) \]

[Out]

-(a*x*Sqrt[1 - x^2/a^2])/4 + (x^2*ArcCos[x/a])/2 + (a^2*ArcSin[x/a])/4

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Rubi [A]  time = 0.0201988, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5264, 4628, 321, 216} \[ -\frac{1}{4} a x \sqrt{1-\frac{x^2}{a^2}}+\frac{1}{4} a^2 \sin ^{-1}\left (\frac{x}{a}\right )+\frac{1}{2} x^2 \cos ^{-1}\left (\frac{x}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*ArcSec[a/x],x]

[Out]

-(a*x*Sqrt[1 - x^2/a^2])/4 + (x^2*ArcCos[x/a])/2 + (a^2*ArcSin[x/a])/4

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 4628

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int x \sec ^{-1}\left (\frac{a}{x}\right ) \, dx &=\int x \cos ^{-1}\left (\frac{x}{a}\right ) \, dx\\ &=\frac{1}{2} x^2 \cos ^{-1}\left (\frac{x}{a}\right )+\frac{\int \frac{x^2}{\sqrt{1-\frac{x^2}{a^2}}} \, dx}{2 a}\\ &=-\frac{1}{4} a x \sqrt{1-\frac{x^2}{a^2}}+\frac{1}{2} x^2 \cos ^{-1}\left (\frac{x}{a}\right )+\frac{1}{4} a \int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx\\ &=-\frac{1}{4} a x \sqrt{1-\frac{x^2}{a^2}}+\frac{1}{2} x^2 \cos ^{-1}\left (\frac{x}{a}\right )+\frac{1}{4} a^2 \sin ^{-1}\left (\frac{x}{a}\right )\\ \end{align*}

Mathematica [A]  time = 0.0268036, size = 44, normalized size = 0.94 \[ \frac{1}{4} \left (-a x \sqrt{1-\frac{x^2}{a^2}}+a^2 \sin ^{-1}\left (\frac{x}{a}\right )+2 x^2 \sec ^{-1}\left (\frac{a}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*ArcSec[a/x],x]

[Out]

(-(a*x*Sqrt[1 - x^2/a^2]) + 2*x^2*ArcSec[a/x] + a^2*ArcSin[x/a])/4

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Maple [B]  time = 0.152, size = 93, normalized size = 2. \begin{align*}{\frac{{x}^{2}}{2}{\rm arcsec} \left ({\frac{a}{x}}\right )}+{\frac{ax}{4}\sqrt{-1+{\frac{{a}^{2}}{{x}^{2}}}}\arctan \left ({\frac{1}{\sqrt{-1+{\frac{{a}^{2}}{{x}^{2}}}}}} \right ){\frac{1}{\sqrt{{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) }}}}}-{\frac{{x}^{3}}{4\,a} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ){\frac{1}{\sqrt{{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) }}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arcsec(a/x),x)

[Out]

1/2*x^2*arcsec(a/x)+1/4*a*(-1+a^2/x^2)^(1/2)/((-1+a^2/x^2)/a^2*x^2)^(1/2)*x*arctan(1/(-1+a^2/x^2)^(1/2))-1/4/a
*(-1+a^2/x^2)/((-1+a^2/x^2)/a^2*x^2)^(1/2)*x^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.54974, size = 86, normalized size = 1.83 \begin{align*} -\frac{1}{4} \, x^{2} \sqrt{\frac{a^{2} - x^{2}}{x^{2}}} - \frac{1}{4} \,{\left (a^{2} - 2 \, x^{2}\right )} \operatorname{arcsec}\left (\frac{a}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^2*sqrt((a^2 - x^2)/x^2) - 1/4*(a^2 - 2*x^2)*arcsec(a/x)

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Sympy [A]  time = 0.266126, size = 41, normalized size = 0.87 \begin{align*} \begin{cases} - \frac{a^{2} \operatorname{asec}{\left (\frac{a}{x} \right )}}{4} - \frac{a x \sqrt{1 - \frac{x^{2}}{a^{2}}}}{4} + \frac{x^{2} \operatorname{asec}{\left (\frac{a}{x} \right )}}{2} & \text{for}\: a \neq 0 \\\tilde{\infty } x^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asec(a/x),x)

[Out]

Piecewise((-a**2*asec(a/x)/4 - a*x*sqrt(1 - x**2/a**2)/4 + x**2*asec(a/x)/2, Ne(a, 0)), (zoo*x**2, True))

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Giac [A]  time = 1.09669, size = 53, normalized size = 1.13 \begin{align*} -\frac{1}{4} \, a^{2} \arccos \left (\frac{x}{a}\right ) + \frac{1}{2} \, x^{2} \arccos \left (\frac{x}{a}\right ) - \frac{1}{4} \, a x \sqrt{-\frac{x^{2}}{a^{2}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^2*arccos(x/a) + 1/2*x^2*arccos(x/a) - 1/4*a*x*sqrt(-x^2/a^2 + 1)