Integrand size = 17, antiderivative size = 133 \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\frac {\sqrt {-1+x^2} \left (-2+17 x^2\right )}{64 x^4}-\frac {3 \sec ^{-1}(x)}{8 x \sqrt {x^2}}+\frac {9 x \sec ^{-1}(x)}{64 \sqrt {x^2}}+\frac {\left (-1+x^2\right )^2 \sec ^{-1}(x)}{8 x^3 \sqrt {x^2}}-\frac {3 \sqrt {-1+x^2} \sec ^{-1}(x)^2}{8 x^2}-\frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{4 x^4}+\frac {x \sec ^{-1}(x)^3}{8 \sqrt {x^2}} \]
-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
Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.63 \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\frac {\sqrt {-1+x^2} \left (32 \sec ^{-1}(x)^3+4 \sec ^{-1}(x) \left (-16 \cos \left (2 \sec ^{-1}(x)\right )+\cos \left (4 \sec ^{-1}(x)\right )\right )+32 \sin \left (2 \sec ^{-1}(x)\right )-\sin \left (4 \sec ^{-1}(x)\right )+8 \sec ^{-1}(x)^2 \left (-8 \sin \left (2 \sec ^{-1}(x)\right )+\sin \left (4 \sec ^{-1}(x)\right )\right )\right )}{256 \sqrt {1-\frac {1}{x^2}} x} \]
(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)
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 |
\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {3}{4} \left (\frac {1}{2} \arcsin \left (\frac {1}{x}\right )+\frac {\sqrt {1-\frac {1}{x^2}}}{2 x}\right )-\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{4 x}\right )-\frac {1}{4} \left (1-\frac {1}{x^2}\right )^2 \arccos \left (\frac {1}{x}\right )\right )+\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^2}{2 x}+\frac {\arccos \left (\frac {1}{x}\right )}{2 x^2}-\frac {1}{6} \arccos \left (\frac {1}{x}\right )^3+\frac {1}{2} \left (\frac {1}{2} \arcsin \left (\frac {1}{x}\right )-\frac {\sqrt {1-\frac {1}{x^2}}}{2 x}\right )\right )+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^2}{4 x}\right )}{x}\) |
-((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
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])
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]
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]
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]
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]
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]
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]
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]
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]
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 | \(\frac {\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (8 \operatorname {arcsec}\left (x \right )^{3} x^{4}-40 \operatorname {arcsec}\left (x \right )^{2} \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+17 \,\operatorname {arcsec}\left (x \right ) x^{4}+16 \operatorname {arcsec}\left (x \right )^{2} \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +17 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-40 \,\operatorname {arcsec}\left (x \right ) x^{2}-2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +8 \,\operatorname {arcsec}\left (x \right )\right )}{64 x^{4}}\) | \(114\) |
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
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.44 \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\frac {8 \, x^{4} \operatorname {arcsec}\left (x\right )^{3} + {\left (17 \, x^{4} - 40 \, x^{2} + 8\right )} \operatorname {arcsec}\left (x\right ) - {\left (8 \, {\left (5 \, x^{2} - 2\right )} \operatorname {arcsec}\left (x\right )^{2} - 17 \, x^{2} + 2\right )} \sqrt {x^{2} - 1}}{64 \, x^{4}} \]
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
Timed out. \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\int { \frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\left (x\right )^{2}}{x^{5}} \,d x } \]
\[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\int { \frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\left (x\right )^{2}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\int \frac {{\mathrm {acos}\left (\frac {1}{x}\right )}^2\,{\left (x^2-1\right )}^{3/2}}{x^5} \,d x \]