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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0818331, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{\sqrt{x^2}}+\frac{\sqrt{x^2-1} \sec ^{-1}(x)}{x} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.97558, size = 20, normalized size = 0.87 \[ \frac{1}{\sqrt{x^{2}}} + \frac{\sqrt{x^{2} - 1} \operatorname{asec}{\left (x \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asec(x)/x**2/(x**2-1)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0328063, 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])

_______________________________________________________________________________________

Maple [C]  time = 0.223, size = 76, normalized size = 3.3 \[{\frac{{\rm arcsec} \left (x\right )+i}{2\,x} \left ( -i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+{x}^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{{\rm arcsec} \left (x\right )-i}{2\,x} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+{x}^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsec(x)/x^2/(x^2-1)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.52973, size = 23, normalized size = 1. \[ \frac{\sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right )}{x} + \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsec(x)/(sqrt(x^2 - 1)*x^2),x, algorithm="maxima")

[Out]

sqrt(x^2 - 1)*arcsec(x)/x + 1/x

_______________________________________________________________________________________

Fricas [A]  time = 0.221303, size = 22, normalized size = 0.96 \[ \frac{\sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right ) + 1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsec(x)/(sqrt(x^2 - 1)*x^2),x, algorithm="fricas")

[Out]

(sqrt(x^2 - 1)*arcsec(x) + 1)/x

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(asec(x)/x**2/(x**2-1)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220471, size = 68, normalized size = 2.96 \[ \frac{2 \, \arccos \left (\frac{1}{x}\right )}{{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 1} - \frac{2 \, \arctan \left (-x + \sqrt{x^{2} - 1}\right )}{{\rm sign}\left (x\right )} + \frac{1}{x{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsec(x)/(sqrt(x^2 - 1)*x^2),x, algorithm="giac")

[Out]

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