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

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Rubi [A]  time = 0.0478767, 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, 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 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]))

Rule 30

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

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*}

Mathematica [A]  time = 0.0344072, 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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Maple [C]  time = 0.223, size = 186, normalized size = 8.1 \begin{align*} -{\frac{1}{4\,x} \left ( \sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-3\,i{x}^{2}-4\,\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+4\,i \right ){\frac{1}{\sqrt{{x}^{2}-1}}} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+1 \right ) ^{-1}}+{\frac{{\rm arcsec} \left (x\right )}{4\,x} \left ({x}^{2}-2-2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{{\rm arcsec} \left (x\right )x}{2}{\frac{1}{\sqrt{{x}^{2}-1}}}}-{{\frac{i}{4}}{x}^{3}{\frac{1}{\sqrt{{x}^{2}-1}}} \left ({x}^{2}-2-2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x \right ) ^{-1}}+{\frac{{\rm arcsec} \left (x\right )}{4\,x} \left ({x}^{2}-2+2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsec(x)/x^2/(x^2-1)^(1/2),x)

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

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Maxima [A]  time = 1.47155, size = 23, normalized size = 1. \begin{align*} \frac{\sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right )}{x} + \frac{1}{x} \end{align*}

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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Fricas [A]  time = 2.57726, size = 45, normalized size = 1.96 \begin{align*} \frac{\sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right ) + 1}{x} \end{align*}

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

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

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Giac [B]  time = 1.10502, size = 68, normalized size = 2.96 \begin{align*} \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}\left (x\right )} + \frac{1}{x \mathrm{sgn}\left (x\right )} \end{align*}

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))