3.7.90 \(\int \frac {\sec ^{-1}(x)}{x^2 \sqrt {-1+x^2}} \, dx\) [690]

3.7.90.1 Optimal result

Integrand size = 15, antiderivative size = 23 \[ \int \frac {\sec ^{-1}(x)}{x^2 \sqrt {-1+x^2}} \, dx=\frac {1}{\sqrt {x^2}}+\frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x} \]

output

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

3.7.90.2 Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52 \[ \int \frac {\sec ^{-1}(x)}{x^2 \sqrt {-1+x^2}} \, dx=\frac {\sqrt {1-\frac {1}{x^2}} x+\left (-1+x^2\right ) \sec ^{-1}(x)}{x \sqrt {-1+x^2}} \]

input

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

output

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

3.7.90.3 Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5761, 15}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5761

\(\displaystyle \frac {\sqrt {x^2-1} \sec ^{-1}(x)}{x}-\frac {x \int \frac {1}{x^2}dx}{\sqrt {x^2}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{\sqrt {x^2}}+\frac {\sqrt {x^2-1} \sec ^{-1}(x)}{x}\)

input

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

output

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

3.7.90.3.1 Defintions of rubi rules used

rule 15

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

rule 5761

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]}, Sim 
p[(a + b*ArcSec[c*x])   u, x] - Simp[b*c*(x/Sqrt[c^2*x^2])   Int[SimplifyIn 
tegrand[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])) || (ILtQ[(m 
 + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))
 

3.7.90.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.35 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43

method	result	size
default	\(\frac {\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \sqrt {\frac {x^{2}-1}{x^{2}}}\, \left (\operatorname {arcsec}\left (x \right ) x^{2}-\operatorname {arcsec}\left (x \right )+\sqrt {\frac {x^{2}-1}{x^{2}}}\, x \right )}{x^{2}-1}\)	\(56\)

input

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

output

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

3.7.90.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {\sec ^{-1}(x)}{x^2 \sqrt {-1+x^2}} \, dx=\frac {\sqrt {x^{2} - 1} \operatorname {arcsec}\left (x\right ) + 1}{x} \]

input

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

output

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

3.7.90.6 Sympy [F]

\[ \int \frac {\sec ^{-1}(x)}{x^2 \sqrt {-1+x^2}} \, dx=\int \frac {\operatorname {asec}{\left (x \right )}}{x^{2} \sqrt {\left (x - 1\right ) \left (x + 1\right )}}\, dx \]

input

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

output

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

3.7.90.7 Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {\sec ^{-1}(x)}{x^2 \sqrt {-1+x^2}} \, dx=\frac {\sqrt {x^{2} - 1} \operatorname {arcsec}\left (x\right )}{x} + \frac {1}{x} \]

input

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

output

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

3.7.90.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (19) = 38\).

Time = 0.32 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.17 \[ \int \frac {\sec ^{-1}(x)}{x^2 \sqrt {-1+x^2}} \, dx=\frac {2 \, \arccos \left (\frac {1}{x}\right )}{{\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1} - \frac {2 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right )}{\mathrm {sgn}\left (x\right )} + \frac {1}{x \mathrm {sgn}\left (x\right )} \]

input

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

output

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

3.7.90.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^{-1}(x)}{x^2 \sqrt {-1+x^2}} \, dx=\int \frac {\mathrm {acos}\left (\frac {1}{x}\right )}{x^2\,\sqrt {x^2-1}} \,d x \]

input

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

output

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

3.7.90.10 Reduce [F]

\[ \int \frac {\sec ^{-1}(x)}{x^2 \sqrt {-1+x^2}} \, dx=\int \frac {\mathit {asec} \left (x \right )}{\sqrt {x^{2}-1}\, x^{2}}d x \]

input

int(asec(x)/(sqrt(x**2 - 1)*x**2),x)
 

output

int(asec(x)/(sqrt(x**2 - 1)*x**2),x)