3.7.83 \(\int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^2} \, dx\) [683]

3.7.83.1 Optimal result

Integrand size = 15, antiderivative size = 107 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^2} \, dx=-\frac {1}{\sqrt {x^2}}-\frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x}-\frac {2 i \sqrt {x^2} \sec ^{-1}(x) \arctan \left (e^{i \sec ^{-1}(x)}\right )}{x}+\frac {i \sqrt {x^2} \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(x)}\right )}{x}-\frac {i \sqrt {x^2} \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(x)}\right )}{x} \]

output

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

3.7.83.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^2} \, dx=-\frac {\sqrt {1-\frac {1}{x^2}} \left (1+\sqrt {1-\frac {1}{x^2}} x \sec ^{-1}(x)-x \sec ^{-1}(x) \log \left (1-i e^{i \sec ^{-1}(x)}\right )+x \sec ^{-1}(x) \log \left (1+i e^{i \sec ^{-1}(x)}\right )-i x \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(x)}\right )+i x \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(x)}\right )\right )}{\sqrt {-1+x^2}} \]

input

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

output

-((Sqrt[1 - x^(-2)]*(1 + Sqrt[1 - x^(-2)]*x*ArcSec[x] - x*ArcSec[x]*Log[1 
- I*E^(I*ArcSec[x])] + x*ArcSec[x]*Log[1 + I*E^(I*ArcSec[x])] - I*x*PolyLo 
g[2, (-I)*E^(I*ArcSec[x])] + I*x*PolyLog[2, I*E^(I*ArcSec[x])]))/Sqrt[-1 + 
 x^2])
 

3.7.83.3 Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {5765, 5199, 24, 5219, 3042, 4669, 2715, 2838}

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.

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

\(\Big \downarrow \) 5765

\(\displaystyle -\frac {\sqrt {x^2} \int \sqrt {1-\frac {1}{x^2}} x \arccos \left (\frac {1}{x}\right )d\frac {1}{x}}{x}\)

\(\Big \downarrow \) 5199

\(\displaystyle -\frac {\sqrt {x^2} \left (\int \frac {x \arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}+\int 1d\frac {1}{x}+\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )\right )}{x}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {\sqrt {x^2} \left (\int \frac {x \arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}+\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )+\frac {1}{x}\right )}{x}\)

\(\Big \downarrow \) 5219

\(\displaystyle -\frac {\sqrt {x^2} \left (-\int x \arccos \left (\frac {1}{x}\right )d\arccos \left (\frac {1}{x}\right )+\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )+\frac {1}{x}\right )}{x}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\sqrt {x^2} \left (-\int \arccos \left (\frac {1}{x}\right ) \csc \left (\arccos \left (\frac {1}{x}\right )+\frac {\pi }{2}\right )d\arccos \left (\frac {1}{x}\right )+\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )+\frac {1}{x}\right )}{x}\)

\(\Big \downarrow \) 4669

\(\displaystyle -\frac {\sqrt {x^2} \left (\int \log \left (1-i e^{i \arccos \left (\frac {1}{x}\right )}\right )d\arccos \left (\frac {1}{x}\right )-\int \log \left (1+i e^{i \arccos \left (\frac {1}{x}\right )}\right )d\arccos \left (\frac {1}{x}\right )+2 i \arccos \left (\frac {1}{x}\right ) \arctan \left (e^{i \arccos \left (\frac {1}{x}\right )}\right )+\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )+\frac {1}{x}\right )}{x}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {\sqrt {x^2} \left (-i \int x \log \left (1-i e^{i \arccos \left (\frac {1}{x}\right )}\right )de^{i \arccos \left (\frac {1}{x}\right )}+i \int x \log \left (1+i e^{i \arccos \left (\frac {1}{x}\right )}\right )de^{i \arccos \left (\frac {1}{x}\right )}+2 i \arccos \left (\frac {1}{x}\right ) \arctan \left (e^{i \arccos \left (\frac {1}{x}\right )}\right )+\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )+\frac {1}{x}\right )}{x}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {\sqrt {x^2} \left (2 i \arccos \left (\frac {1}{x}\right ) \arctan \left (e^{i \arccos \left (\frac {1}{x}\right )}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos \left (\frac {1}{x}\right )}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos \left (\frac {1}{x}\right )}\right )+\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )+\frac {1}{x}\right )}{x}\)

input

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

output

-((Sqrt[x^2]*(x^(-1) + Sqrt[1 - x^(-2)]*ArcCos[x^(-1)] + (2*I)*ArcCos[x^(- 
1)]*ArcTan[E^(I*ArcCos[x^(-1)])] - I*PolyLog[2, (-I)*E^(I*ArcCos[x^(-1)])] 
 + I*PolyLog[2, I*E^(I*ArcCos[x^(-1)])]))/x)
 

3.7.83.3.1 Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5199

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
(e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcC 
os[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 
 - c^2*x^2]]   Int[(f*x)^m*((a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2]), x], x 
] + Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[ 
(f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, 
 f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 5219

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* 
(x_)^2], x_Symbol] :> Simp[(-(c^(m + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[ 
d + e*x^2]]   Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]], x] /; Fr 
eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
 

rule 5765

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[-Sqrt[x^2]/x   Subst[Int[(e + d*x^2)^p*((a + b* 
ArcCos[x/c])^n/x^(m + 2*(p + 1))), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, 
 n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1 
/2] && GtQ[e, 0] && LtQ[d, 0]
 

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

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.66 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.81

method	result	size
default	\(-\frac {\left (\sqrt {\frac {x^{2}-1}{x^{2}}}\, x -i\right ) \operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (\operatorname {arcsec}\left (x \right )+i\right )}{2 x}-\frac {\left (\sqrt {\frac {x^{2}-1}{x^{2}}}\, x +i\right ) \operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (\operatorname {arcsec}\left (x \right )-i\right )}{2 x}-\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (\operatorname {arcsec}\left (x \right ) \ln \left (1+i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )-\operatorname {arcsec}\left (x \right ) \ln \left (1-i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )-i \operatorname {dilog}\left (1+i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )+i \operatorname {dilog}\left (1-i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )\right )\)	\(194\)

input

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

output

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

3.7.83.5 Fricas [F]

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

input

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

output

integral(sqrt(x^2 - 1)*arcsec(x)/x^2, x)
 

3.7.83.6 Sympy [F]

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

input

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

output

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

3.7.83.7 Maxima [F]

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

input

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

output

integrate(sqrt(x^2 - 1)*arcsec(x)/x^2, x)
 

3.7.83.8 Giac [F]

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

input

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

output

integrate(sqrt(x^2 - 1)*arcsec(x)/x^2, x)
 

3.7.83.9 Mupad [F(-1)]

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

input

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

output

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

3.7.83.10 Reduce [F]

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

input

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

output

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