3.15 \(\int \frac {\sec ^{-1}(\frac {a}{x})}{x^3} \, dx\)

Optimal. Leaf size=38 \[ \frac {\sqrt {1-\frac {x^2}{a^2}}}{2 a x}-\frac {\cos ^{-1}\left (\frac {x}{a}\right )}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5264, 4628, 264} \[ \frac {\sqrt {1-\frac {x^2}{a^2}}}{2 a x}-\frac {\cos ^{-1}\left (\frac {x}{a}\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcSec[a/x]/x^3,x]

[Out]

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

Rule 264

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

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 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]

Rubi steps

\begin {align*} \int \frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx &=\int \frac {\cos ^{-1}\left (\frac {x}{a}\right )}{x^3} \, dx\\ &=-\frac {\cos ^{-1}\left (\frac {x}{a}\right )}{2 x^2}-\frac {\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx}{2 a}\\ &=\frac {\sqrt {1-\frac {x^2}{a^2}}}{2 a x}-\frac {\cos ^{-1}\left (\frac {x}{a}\right )}{2 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 36, normalized size = 0.95 \[ \frac {x \sqrt {1-\frac {x^2}{a^2}}-a \sec ^{-1}\left (\frac {a}{x}\right )}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSec[a/x]/x^3,x]

[Out]

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

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fricas [A]  time = 0.65, size = 39, normalized size = 1.03 \[ -\frac {a^{2} \operatorname {arcsec}\left (\frac {a}{x}\right ) - x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}}}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(a/x)/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 61, normalized size = 1.61 \[ \frac {a {\left (\frac {a + \sqrt {a^{2} - x^{2}}}{a^{2} x} - \frac {x}{{\left (a + \sqrt {a^{2} - x^{2}}\right )} a^{2}}\right )}}{4 \, {\left | a \right |}} - \frac {\arccos \left (\frac {x}{a}\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(a/x)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 54, normalized size = 1.42 \[ -\frac {\frac {a^{2} \mathrm {arcsec}\left (\frac {a}{x}\right )}{2 x^{2}}-\frac {x \left (-1+\frac {a^{2}}{x^{2}}\right )}{2 \sqrt {\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) x^{2}}{a^{2}}}\, a}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsec(a/x)/x^3,x)

[Out]

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

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maxima [A]  time = 0.42, size = 32, normalized size = 0.84 \[ -\frac {\operatorname {arcsec}\left (\frac {a}{x}\right )}{2 \, x^{2}} + \frac {\sqrt {-\frac {x^{2}}{a^{2}} + 1}}{2 \, a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(a/x)/x^3,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acos}\left (\frac {x}{a}\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(x/a)/x^3,x)

[Out]

int(acos(x/a)/x^3, x)

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sympy [C]  time = 1.25, size = 53, normalized size = 1.39 \[ - \frac {\operatorname {asec}{\left (\frac {a}{x} \right )}}{2 x^{2}} - \frac {\begin {cases} - \frac {\sqrt {\frac {a^{2}}{x^{2}} - 1}}{a} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\- \frac {i \sqrt {- \frac {a^{2}}{x^{2}} + 1}}{a} & \text {otherwise} \end {cases}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asec(a/x)/x**3,x)

[Out]

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

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