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

Optimal. Leaf size=34 \[ \frac {1}{2} x^2 \sec ^{-1}\left (\frac {x}{a}\right )-\frac {1}{2} a x \sqrt {1-\frac {a^2}{x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4833, 5220, 191} \[ \frac {1}{2} x^2 \sec ^{-1}\left (\frac {x}{a}\right )-\frac {1}{2} a x \sqrt {1-\frac {a^2}{x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 33, normalized size = 0.97 \[ \frac {1}{2} \left (x^2 \cos ^{-1}\left (\frac {a}{x}\right )-a x \sqrt {1-\frac {a^2}{x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 32, normalized size = 0.94 \[ \frac {1}{2} \, x^{2} \arccos \left (\frac {a}{x}\right ) - \frac {1}{2} \, a x \sqrt {-\frac {a^{2} - x^{2}}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.21, size = 64, normalized size = 1.88 \[ -\frac {a^{3} {\left (\frac {x {\left (\sqrt {-\frac {a^{2}}{x^{2}} + 1} - 1\right )}}{a} - \frac {a}{x {\left (\sqrt {-\frac {a^{2}}{x^{2}} + 1} - 1\right )}}\right )} - 2 \, a x^{2} \arccos \left (\frac {a}{x}\right )}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 39, normalized size = 1.15 \[ -a^{2} \left (-\frac {x^{2} \arccos \left (\frac {a}{x}\right )}{2 a^{2}}+\frac {\sqrt {1-\frac {a^{2}}{x^{2}}}\, x}{2 a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 28, normalized size = 0.82 \[ \frac {1}{2} \, x^{2} \arccos \left (\frac {a}{x}\right ) - \frac {1}{2} \, a x \sqrt {-\frac {a^{2}}{x^{2}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.24, size = 28, normalized size = 0.82 \[ \frac {x^2\,\mathrm {acos}\left (\frac {a}{x}\right )}{2}-\frac {a\,x\,\sqrt {1-\frac {a^2}{x^2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.54, size = 48, normalized size = 1.41 \[ - \frac {a \left (\begin {cases} a \sqrt {-1 + \frac {x^{2}}{a^{2}}} & \text {for}\: \left |{\frac {x^{2}}{a^{2}}}\right | > 1 \\i a \sqrt {1 - \frac {x^{2}}{a^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x^{2} \operatorname {acos}{\left (\frac {a}{x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*Piecewise((a*sqrt(-1 + x**2/a**2), Abs(x**2/a**2) > 1), (I*a*sqrt(1 - x**2/a**2), True))/2 + x**2*acos(a/x)
/2

________________________________________________________________________________________