3.690 \(\int \frac {\sec ^{-1}(x)}{x^2 \sqrt {-1+x^2}} \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{\sqrt {x^2}}+\frac {\sqrt {x^2-1} \sec ^{-1}(x)}{x} \]

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Rubi [A]  time = 0.05, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {264, 5238, 30} \[ \frac {1}{\sqrt {x^2}}+\frac {\sqrt {x^2-1} \sec ^{-1}(x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcSec[x]/(x^2*Sqrt[-1 + x^2]),x]

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 5238

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSec[c*x], u, x] - Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 35, normalized size = 1.52 \[ \frac {\sqrt {1-\frac {1}{x^2}} x+\left (x^2-1\right ) \sec ^{-1}(x)}{x \sqrt {x^2-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{-1}(x)}{x^2 \sqrt {-1+x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 1.17, size = 16, normalized size = 0.70 \[ \frac {\sqrt {x^{2} - 1} \operatorname {arcsec}\relax (x) + 1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(x)/x^2/(x^2-1)^(1/2),x, algorithm="fricas")

[Out]

(sqrt(x^2 - 1)*arcsec(x) + 1)/x

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giac [B]  time = 1.04, size = 50, normalized size = 2.17 \[ \frac {2 \, \arccos \left (\frac {1}{x}\right )}{{\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1} - \frac {2 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right )}{\mathrm {sgn}\relax (x)} + \frac {1}{x \mathrm {sgn}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(x)/x^2/(x^2-1)^(1/2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.51, size = 178, normalized size = 7.74

method	result	size
default	\(-\frac {\sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-3 i x^{2}-4 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +4 i}{4 \sqrt {x^{2}-1}\, \left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +1\right ) x}+\frac {\left (x^{2}-2-2 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \right ) \mathrm {arcsec}\relax (x )}{4 \sqrt {x^{2}-1}\, x}-\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -1\right ) \left (\mathrm {arcsec}\relax (x )+i\right )}{4 \sqrt {x^{2}-1}\, x}+\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) \left (3 \,\mathrm {arcsec}\relax (x )-i\right )}{4 \sqrt {x^{2}-1}\, x}\)	\(178\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsec(x)/x^2/(x^2-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 1.03, size = 17, normalized size = 0.74 \[ \frac {\sqrt {x^{2} - 1} \operatorname {arcsec}\relax (x)}{x} + \frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(x)/x^2/(x^2-1)^(1/2),x, algorithm="maxima")

[Out]

sqrt(x^2 - 1)*arcsec(x)/x + 1/x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {acos}\left (\frac {1}{x}\right )}{x^2\,\sqrt {x^2-1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(1/x)/(x^2*(x^2 - 1)^(1/2)),x)

[Out]

int(acos(1/x)/(x^2*(x^2 - 1)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asec}{\relax (x )}}{x^{2} \sqrt {\left (x - 1\right ) \left (x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asec(x)/x**2/(x**2-1)**(1/2),x)

[Out]

Integral(asec(x)/(x**2*sqrt((x - 1)*(x + 1))), x)

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