\(\int x \sec ^{-1}(\frac {a}{x}) \, dx\) [11]

Optimal result

Integrand size = 8, antiderivative size = 47 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=-\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{2} x^2 \arccos \left (\frac {x}{a}\right )+\frac {1}{4} a^2 \arcsin \left (\frac {x}{a}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5372, 4724, 327, 222} \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {1}{4} a^2 \arcsin \left (\frac {x}{a}\right )-\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{2} x^2 \arccos \left (\frac {x}{a}\right ) \]

[In]

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

[Out]

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

Rule 222

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]

Rule 327

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 4724

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 5372

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]

Rubi steps \begin{align*} \text {integral}& = \int x \arccos \left (\frac {x}{a}\right ) \, dx \\ & = \frac {1}{2} x^2 \arccos \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 \arccos \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 \arccos \left (\frac {x}{a}\right )+\frac {1}{4} a^2 \arcsin \left (\frac {x}{a}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {1}{4} \left (-a x \sqrt {1-\frac {x^2}{a^2}}+2 x^2 \sec ^{-1}\left (\frac {a}{x}\right )+a^2 \arcsin \left (\frac {x}{a}\right )\right ) \]

[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

Maple [A] (verified)

Time = 1.37 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.43

method	result	size
parts	\(\frac {x^{2} \operatorname {arcsec}\left (\frac {a}{x}\right )}{2}+\frac {-\frac {x \,a^{2} \sqrt {1-\frac {x^{2}}{a^{2}}}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {\frac {1}{a^{2}}}\, x}{\sqrt {1-\frac {x^{2}}{a^{2}}}}\right )}{2 \sqrt {\frac {1}{a^{2}}}}}{2 a}\)	\(67\)
derivativedivides	\(-a^{2} \left (-\frac {x^{2} \operatorname {arcsec}\left (\frac {a}{x}\right )}{2 a^{2}}-\frac {\sqrt {\frac {a^{2}}{x^{2}}-1}\, \left (\frac {\arctan \left (\frac {1}{\sqrt {\frac {a^{2}}{x^{2}}-1}}\right ) a^{2}}{x^{2}}-\sqrt {\frac {a^{2}}{x^{2}}-1}\right ) x^{3}}{4 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{3}}\right )\)	\(91\)
default	\(-a^{2} \left (-\frac {x^{2} \operatorname {arcsec}\left (\frac {a}{x}\right )}{2 a^{2}}-\frac {\sqrt {\frac {a^{2}}{x^{2}}-1}\, \left (\frac {\arctan \left (\frac {1}{\sqrt {\frac {a^{2}}{x^{2}}-1}}\right ) a^{2}}{x^{2}}-\sqrt {\frac {a^{2}}{x^{2}}-1}\right ) x^{3}}{4 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{3}}\right )\)	\(91\)

[In]

int(x*arcsec(a/x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=-\frac {1}{4} \, x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} - \frac {1}{4} \, {\left (a^{2} - 2 \, x^{2}\right )} \operatorname {arcsec}\left (\frac {a}{x}\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=\begin {cases} - \frac {a^{2} \operatorname {asec}{\left (\frac {a}{x} \right )}}{4} - \frac {a x \sqrt {1 - \frac {x^{2}}{a^{2}}}}{4} + \frac {x^{2} \operatorname {asec}{\left (\frac {a}{x} \right )}}{2} & \text {for}\: a \neq 0 \\\tilde {\infty } x^{2} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {arcsec}\left (\frac {a}{x}\right ) + \frac {a^{3} \arcsin \left (\frac {x}{a}\right ) - a^{2} x \sqrt {-\frac {x^{2}}{a^{2}} + 1}}{4 \, a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=-\frac {1}{4} \, a^{2} \arccos \left (\frac {x}{a}\right ) + \frac {1}{2} \, x^{2} \arccos \left (\frac {x}{a}\right ) - \frac {1}{4} \, a x \sqrt {-\frac {x^{2}}{a^{2}} + 1} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {a^2\,\mathrm {acos}\left (\frac {x}{a}\right )\,\left (\frac {2\,x^2}{a^2}-1\right )}{4}-\frac {a\,x\,\sqrt {1-\frac {x^2}{a^2}}}{4} \]

[In]

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

[Out]

(a^2*acos(x/a)*((2*x^2)/a^2 - 1))/4 - (a*x*(1 - x^2/a^2)^(1/2))/4