3.7.93 \(\int \frac {(-1+x^2)^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx\) [693]

3.7.93.1 Optimal result

Integrand size = 17, antiderivative size = 133 \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\frac {\sqrt {-1+x^2} \left (-2+17 x^2\right )}{64 x^4}-\frac {3 \sec ^{-1}(x)}{8 x \sqrt {x^2}}+\frac {9 x \sec ^{-1}(x)}{64 \sqrt {x^2}}+\frac {\left (-1+x^2\right )^2 \sec ^{-1}(x)}{8 x^3 \sqrt {x^2}}-\frac {3 \sqrt {-1+x^2} \sec ^{-1}(x)^2}{8 x^2}-\frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{4 x^4}+\frac {x \sec ^{-1}(x)^3}{8 \sqrt {x^2}} \]

output

-1/4*(x^2-1)^(3/2)*arcsec(x)^2/x^4-3/8*arcsec(x)/x/(x^2)^(1/2)+9/64*x*arcs 
ec(x)/(x^2)^(1/2)+1/8*(x^2-1)^2*arcsec(x)/x^3/(x^2)^(1/2)+1/8*x*arcsec(x)^ 
3/(x^2)^(1/2)+1/64*(17*x^2-2)*(x^2-1)^(1/2)/x^4-3/8*arcsec(x)^2*(x^2-1)^(1 
/2)/x^2
 

3.7.93.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.63 \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\frac {\sqrt {-1+x^2} \left (32 \sec ^{-1}(x)^3+4 \sec ^{-1}(x) \left (-16 \cos \left (2 \sec ^{-1}(x)\right )+\cos \left (4 \sec ^{-1}(x)\right )\right )+32 \sin \left (2 \sec ^{-1}(x)\right )-\sin \left (4 \sec ^{-1}(x)\right )+8 \sec ^{-1}(x)^2 \left (-8 \sin \left (2 \sec ^{-1}(x)\right )+\sin \left (4 \sec ^{-1}(x)\right )\right )\right )}{256 \sqrt {1-\frac {1}{x^2}} x} \]

input

Integrate[((-1 + x^2)^(3/2)*ArcSec[x]^2)/x^5,x]
 

output

(Sqrt[-1 + x^2]*(32*ArcSec[x]^3 + 4*ArcSec[x]*(-16*Cos[2*ArcSec[x]] + Cos[ 
4*ArcSec[x]]) + 32*Sin[2*ArcSec[x]] - Sin[4*ArcSec[x]] + 8*ArcSec[x]^2*(-8 
*Sin[2*ArcSec[x]] + Sin[4*ArcSec[x]])))/(256*Sqrt[1 - x^(-2)]*x)
 

3.7.93.3 Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.46, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {5765, 5159, 5157, 5139, 262, 223, 5153, 5183, 211, 211, 223}

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.

\(\displaystyle \int \frac {\left (x^2-1\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx\)

\(\Big \downarrow \) 5765

\(\displaystyle -\frac {\sqrt {x^2} \int \left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2d\frac {1}{x}}{x}\)

\(\Big \downarrow \) 5159

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {1}{2} \int \frac {\left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )}{x}d\frac {1}{x}+\frac {3}{4} \int \sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^2d\frac {1}{x}+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2}{4 x}\right )}{x}\)

\(\Big \downarrow \) 5157

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {1}{2} \int \frac {\left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )}{x}d\frac {1}{x}+\frac {3}{4} \left (\frac {1}{2} \int \frac {\arccos \left (\frac {1}{x}\right )^2}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}+\int \frac {\arccos \left (\frac {1}{x}\right )}{x}d\frac {1}{x}+\frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^2}{2 x}\right )+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2}{4 x}\right )}{x}\)

\(\Big \downarrow \) 5139

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {1}{2} \int \frac {\left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )}{x}d\frac {1}{x}+\frac {3}{4} \left (\frac {1}{2} \int \frac {\arccos \left (\frac {1}{x}\right )^2}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}+\frac {1}{2} \int \frac {1}{\sqrt {1-\frac {1}{x^2}} x^2}d\frac {1}{x}+\frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^2}{2 x}+\frac {\arccos \left (\frac {1}{x}\right )}{2 x^2}\right )+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2}{4 x}\right )}{x}\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {1}{2} \int \frac {\left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )}{x}d\frac {1}{x}+\frac {3}{4} \left (\frac {1}{2} \int \frac {\arccos \left (\frac {1}{x}\right )^2}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}-\frac {\sqrt {1-\frac {1}{x^2}}}{2 x}\right )+\frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^2}{2 x}+\frac {\arccos \left (\frac {1}{x}\right )}{2 x^2}\right )+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2}{4 x}\right )}{x}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\arccos \left (\frac {1}{x}\right )^2}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}+\frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^2}{2 x}+\frac {\arccos \left (\frac {1}{x}\right )}{2 x^2}+\frac {1}{2} \left (\frac {1}{2} \arcsin \left (\frac {1}{x}\right )-\frac {\sqrt {1-\frac {1}{x^2}}}{2 x}\right )\right )+\frac {1}{2} \int \frac {\left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )}{x}d\frac {1}{x}+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2}{4 x}\right )}{x}\)

\(\Big \downarrow \) 5153

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {1}{2} \int \frac {\left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )}{x}d\frac {1}{x}+\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^2}{2 x}+\frac {\arccos \left (\frac {1}{x}\right )}{2 x^2}-\frac {1}{6} \arccos \left (\frac {1}{x}\right )^3+\frac {1}{2} \left (\frac {1}{2} \arcsin \left (\frac {1}{x}\right )-\frac {\sqrt {1-\frac {1}{x^2}}}{2 x}\right )\right )+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2}{4 x}\right )}{x}\)

\(\Big \downarrow \) 5183

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {1}{2} \left (-\frac {1}{4} \int \left (1-\frac {1}{x^2}\right )^{3/2}d\frac {1}{x}-\frac {1}{4} \left (1-\frac {1}{x^2}\right )^2 \arccos \left (\frac {1}{x}\right )\right )+\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^2}{2 x}+\frac {\arccos \left (\frac {1}{x}\right )}{2 x^2}-\frac {1}{6} \arccos \left (\frac {1}{x}\right )^3+\frac {1}{2} \left (\frac {1}{2} \arcsin \left (\frac {1}{x}\right )-\frac {\sqrt {1-\frac {1}{x^2}}}{2 x}\right )\right )+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2}{4 x}\right )}{x}\)

\(\Big \downarrow \) 211

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {3}{4} \int \sqrt {1-\frac {1}{x^2}}d\frac {1}{x}-\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{4 x}\right )-\frac {1}{4} \left (1-\frac {1}{x^2}\right )^2 \arccos \left (\frac {1}{x}\right )\right )+\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^2}{2 x}+\frac {\arccos \left (\frac {1}{x}\right )}{2 x^2}-\frac {1}{6} \arccos \left (\frac {1}{x}\right )^3+\frac {1}{2} \left (\frac {1}{2} \arcsin \left (\frac {1}{x}\right )-\frac {\sqrt {1-\frac {1}{x^2}}}{2 x}\right )\right )+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2}{4 x}\right )}{x}\)

\(\Big \downarrow \) 211

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}+\frac {\sqrt {1-\frac {1}{x^2}}}{2 x}\right )-\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{4 x}\right )-\frac {1}{4} \left (1-\frac {1}{x^2}\right )^2 \arccos \left (\frac {1}{x}\right )\right )+\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^2}{2 x}+\frac {\arccos \left (\frac {1}{x}\right )}{2 x^2}-\frac {1}{6} \arccos \left (\frac {1}{x}\right )^3+\frac {1}{2} \left (\frac {1}{2} \arcsin \left (\frac {1}{x}\right )-\frac {\sqrt {1-\frac {1}{x^2}}}{2 x}\right )\right )+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2}{4 x}\right )}{x}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {3}{4} \left (\frac {1}{2} \arcsin \left (\frac {1}{x}\right )+\frac {\sqrt {1-\frac {1}{x^2}}}{2 x}\right )-\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{4 x}\right )-\frac {1}{4} \left (1-\frac {1}{x^2}\right )^2 \arccos \left (\frac {1}{x}\right )\right )+\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^2}{2 x}+\frac {\arccos \left (\frac {1}{x}\right )}{2 x^2}-\frac {1}{6} \arccos \left (\frac {1}{x}\right )^3+\frac {1}{2} \left (\frac {1}{2} \arcsin \left (\frac {1}{x}\right )-\frac {\sqrt {1-\frac {1}{x^2}}}{2 x}\right )\right )+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2}{4 x}\right )}{x}\)

input

Int[((-1 + x^2)^(3/2)*ArcSec[x]^2)/x^5,x]
 

output

-((Sqrt[x^2]*(((1 - x^(-2))^(3/2)*ArcCos[x^(-1)]^2)/(4*x) + (-1/4*((1 - x^ 
(-2))^2*ArcCos[x^(-1)]) + (-1/4*(1 - x^(-2))^(3/2)/x - (3*(Sqrt[1 - x^(-2) 
]/(2*x) + ArcSin[x^(-1)]/2))/4)/4)/2 + (3*(ArcCos[x^(-1)]/(2*x^2) + (Sqrt[ 
1 - x^(-2)]*ArcCos[x^(-1)]^2)/(2*x) - ArcCos[x^(-1)]^3/6 + (-1/2*Sqrt[1 - 
x^(-2)]/x + ArcSin[x^(-1)]/2)/2))/4))/x)
 

3.7.93.3.1 Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 5139

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

rule 5153

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] 
]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 
2*d + e, 0] && NeQ[n, -1]
 

rule 5157

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCos[c*x])^n/2), x] + (Simp[(1/2 
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[(a + b*ArcCos[c*x])^n/Sqrt[ 
1 - c^2*x^2], x], x] + Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 
]]   Int[x*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
 

rule 5159

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCos[c*x])^n/(2*p + 1)), x] + (S 
imp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], 
x] + Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[x*(1 
- c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c 
, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
 

rule 5183

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 
1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   I 
nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
 

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]
 

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

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

Time = 0.53 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (8 \operatorname {arcsec}\left (x \right )^{3} x^{4}-40 \operatorname {arcsec}\left (x \right )^{2} \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+17 \,\operatorname {arcsec}\left (x \right ) x^{4}+16 \operatorname {arcsec}\left (x \right )^{2} \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +17 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-40 \,\operatorname {arcsec}\left (x \right ) x^{2}-2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +8 \,\operatorname {arcsec}\left (x \right )\right )}{64 x^{4}}\)	\(114\)

input

int((x^2-1)^(3/2)*arcsec(x)^2/x^5,x,method=_RETURNVERBOSE)
 

output

1/64*csgn(x*(1-1/x^2)^(1/2))*(8*arcsec(x)^3*x^4-40*arcsec(x)^2*((x^2-1)/x^ 
2)^(1/2)*x^3+17*arcsec(x)*x^4+16*arcsec(x)^2*((x^2-1)/x^2)^(1/2)*x+17*((x^ 
2-1)/x^2)^(1/2)*x^3-40*arcsec(x)*x^2-2*((x^2-1)/x^2)^(1/2)*x+8*arcsec(x))/ 
x^4
 

3.7.93.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.44 \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\frac {8 \, x^{4} \operatorname {arcsec}\left (x\right )^{3} + {\left (17 \, x^{4} - 40 \, x^{2} + 8\right )} \operatorname {arcsec}\left (x\right ) - {\left (8 \, {\left (5 \, x^{2} - 2\right )} \operatorname {arcsec}\left (x\right )^{2} - 17 \, x^{2} + 2\right )} \sqrt {x^{2} - 1}}{64 \, x^{4}} \]

input

integrate((x^2-1)^(3/2)*arcsec(x)^2/x^5,x, algorithm="fricas")
 

output

1/64*(8*x^4*arcsec(x)^3 + (17*x^4 - 40*x^2 + 8)*arcsec(x) - (8*(5*x^2 - 2) 
*arcsec(x)^2 - 17*x^2 + 2)*sqrt(x^2 - 1))/x^4
 

3.7.93.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\text {Timed out} \]

input

integrate((x**2-1)**(3/2)*asec(x)**2/x**5,x)
 

output

Timed out
 

3.7.93.7 Maxima [F]

\[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\int { \frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\left (x\right )^{2}}{x^{5}} \,d x } \]

input

integrate((x^2-1)^(3/2)*arcsec(x)^2/x^5,x, algorithm="maxima")
 

output

integrate((x^2 - 1)^(3/2)*arcsec(x)^2/x^5, x)
 

3.7.93.8 Giac [F]

\[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\int { \frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\left (x\right )^{2}}{x^{5}} \,d x } \]

input

integrate((x^2-1)^(3/2)*arcsec(x)^2/x^5,x, algorithm="giac")
 

output

integrate((x^2 - 1)^(3/2)*arcsec(x)^2/x^5, x)
 

3.7.93.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\int \frac {{\mathrm {acos}\left (\frac {1}{x}\right )}^2\,{\left (x^2-1\right )}^{3/2}}{x^5} \,d x \]

input

int((acos(1/x)^2*(x^2 - 1)^(3/2))/x^5,x)
 

output

int((acos(1/x)^2*(x^2 - 1)^(3/2))/x^5, x)
 

3.7.93.10 Reduce [F]

\[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=-\left (\int \frac {\sqrt {x^{2}-1}\, \mathit {asec} \left (x \right )^{2}}{x^{5}}d x \right )+\int \frac {\sqrt {x^{2}-1}\, \mathit {asec} \left (x \right )^{2}}{x^{3}}d x \]

input

int((sqrt(x**2 - 1)*asec(x)**2*(x**2 - 1))/x**5,x)
 

output

 - int((sqrt(x**2 - 1)*asec(x)**2)/x**5,x) + int((sqrt(x**2 - 1)*asec(x)** 
2)/x**3,x)