Integrand size = 17, antiderivative size = 110 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx=\frac {2 \left (1-21 x^2\right )}{27 \left (x^2\right )^{3/2}}-\frac {4 \sqrt {-1+x^2} \sec ^{-1}(x)}{3 x}-\frac {2 \left (-1+x^2\right )^{3/2} \sec ^{-1}(x)}{9 x^3}+\frac {2 \sec ^{-1}(x)^2}{3 \sqrt {x^2}}+\frac {\left (-1+x^2\right ) \sec ^{-1}(x)^2}{3 \left (x^2\right )^{3/2}}+\frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^3}{3 x^3} \]
-2/9*(x^2-1)^(3/2)*arcsec(x)/x^3+1/3*(x^2-1)^(3/2)*arcsec(x)^3/x^3+2/27*(- 21*x^2+1)/x^2/(x^2)^(1/2)+2/3*arcsec(x)^2/(x^2)^(1/2)+1/3*(x^2-1)*arcsec(x )^2/x^2/(x^2)^(1/2)-4/3*arcsec(x)*(x^2-1)^(1/2)/x
Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx=\frac {2 \sqrt {1-\frac {1}{x^2}} x \left (1-21 x^2\right )-6 \left (1-8 x^2+7 x^4\right ) \sec ^{-1}(x)+9 \sqrt {1-\frac {1}{x^2}} x \left (-1+3 x^2\right ) \sec ^{-1}(x)^2+9 \left (-1+x^2\right )^2 \sec ^{-1}(x)^3}{27 x^3 \sqrt {-1+x^2}} \]
(2*Sqrt[1 - x^(-2)]*x*(1 - 21*x^2) - 6*(1 - 8*x^2 + 7*x^4)*ArcSec[x] + 9*S qrt[1 - x^(-2)]*x*(-1 + 3*x^2)*ArcSec[x]^2 + 9*(-1 + x^2)^2*ArcSec[x]^3)/( 27*x^3*Sqrt[-1 + x^2])
Time = 0.59 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5765, 5183, 5159, 5131, 5183, 24, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {x^2-1} \sec ^{-1}(x)^3}{x^4} \, dx\) |
| \(\Big \downarrow \) 5765 |
| \(\displaystyle -\frac {\sqrt {x^2} \int \frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )^3}{x}d\frac {1}{x}}{x}\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle -\frac {\sqrt {x^2} \left (-\int \left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )^2d\frac {1}{x}-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^3\right )}{x}\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {2}{3} \int \frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )}{x}d\frac {1}{x}-\frac {2}{3} \int \arccos \left (\frac {1}{x}\right )^2d\frac {1}{x}-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^3-\frac {\left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )^2}{3 x}\right )}{x}\) |
| \(\Big \downarrow \) 5131 |
| \(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {2}{3} \left (2 \int \frac {\arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}} x}d\frac {1}{x}+\frac {\arccos \left (\frac {1}{x}\right )^2}{x}\right )-\frac {2}{3} \int \frac {\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )}{x}d\frac {1}{x}-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^3-\frac {\left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )^2}{3 x}\right )}{x}\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {2}{3} \left (2 \left (-\int 1d\frac {1}{x}-\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )\right )+\frac {\arccos \left (\frac {1}{x}\right )^2}{x}\right )-\frac {2}{3} \left (-\frac {1}{3} \int \left (1-\frac {1}{x^2}\right )d\frac {1}{x}-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )\right )-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^3-\frac {\left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )^2}{3 x}\right )}{x}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {2}{3} \left (-\frac {1}{3} \int \left (1-\frac {1}{x^2}\right )d\frac {1}{x}-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )\right )-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^3-\frac {\left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )^2}{3 x}-\frac {2}{3} \left (2 \left (-\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )-\frac {1}{x}\right )+\frac {\arccos \left (\frac {1}{x}\right )^2}{x}\right )\right )}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )^3-\frac {\left (1-\frac {1}{x^2}\right ) \arccos \left (\frac {1}{x}\right )^2}{3 x}-\frac {2}{3} \left (2 \left (-\sqrt {1-\frac {1}{x^2}} \arccos \left (\frac {1}{x}\right )-\frac {1}{x}\right )+\frac {\arccos \left (\frac {1}{x}\right )^2}{x}\right )-\frac {2}{3} \left (\frac {1}{3} \left (\frac {1}{3 x^3}-\frac {1}{x}\right )-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \arccos \left (\frac {1}{x}\right )\right )\right )}{x}\) |
-((Sqrt[x^2]*(-1/3*((1 - x^(-2))*ArcCos[x^(-1)]^2)/x - ((1 - x^(-2))^(3/2) *ArcCos[x^(-1)]^3)/3 - (2*((1/(3*x^3) - x^(-1))/3 - ((1 - x^(-2))^(3/2)*Ar cCos[x^(-1)])/3))/3 - (2*(ArcCos[x^(-1)]^2/x + 2*(-x^(-1) - Sqrt[1 - x^(-2 )]*ArcCos[x^(-1)])))/3))/x)
3.7.94.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cCos[c*x])^n, x] + Simp[b*c*n Int[x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && 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 = 0.54 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.34
| method | result | size |
| default | \(\frac {\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \sqrt {\frac {x^{2}-1}{x^{2}}}\, \left (9 \operatorname {arcsec}\left (x \right )^{3} x^{4}+27 \operatorname {arcsec}\left (x \right )^{2} \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-18 x^{2} \operatorname {arcsec}\left (x \right )^{3}-42 \,\operatorname {arcsec}\left (x \right ) x^{4}-9 \operatorname {arcsec}\left (x \right )^{2} \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -42 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+9 \operatorname {arcsec}\left (x \right )^{3}+48 \,\operatorname {arcsec}\left (x \right ) x^{2}+2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -6 \,\operatorname {arcsec}\left (x \right )\right )}{27 \left (x^{2}-1\right ) x^{2}}\) | \(147\) |
1/27*csgn(x*(1-1/x^2)^(1/2))*((x^2-1)/x^2)^(1/2)/(x^2-1)/x^2*(9*arcsec(x)^ 3*x^4+27*arcsec(x)^2*((x^2-1)/x^2)^(1/2)*x^3-18*x^2*arcsec(x)^3-42*arcsec( x)*x^4-9*arcsec(x)^2*((x^2-1)/x^2)^(1/2)*x-42*((x^2-1)/x^2)^(1/2)*x^3+9*ar csec(x)^3+48*arcsec(x)*x^2+2*((x^2-1)/x^2)^(1/2)*x-6*arcsec(x))
Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx=\frac {9 \, {\left (3 \, x^{2} - 1\right )} \operatorname {arcsec}\left (x\right )^{2} - 42 \, x^{2} + 3 \, {\left (3 \, {\left (x^{2} - 1\right )} \operatorname {arcsec}\left (x\right )^{3} - 2 \, {\left (7 \, x^{2} - 1\right )} \operatorname {arcsec}\left (x\right )\right )} \sqrt {x^{2} - 1} + 2}{27 \, x^{3}} \]
1/27*(9*(3*x^2 - 1)*arcsec(x)^2 - 42*x^2 + 3*(3*(x^2 - 1)*arcsec(x)^3 - 2* (7*x^2 - 1)*arcsec(x))*sqrt(x^2 - 1) + 2)/x^3
\[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx=\int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right )} \operatorname {asec}^{3}{\left (x \right )}}{x^{4}}\, dx \]
Time = 0.53 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx=\frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\left (x\right )^{3}}{3 \, x^{3}} + \frac {{\left (3 \, x^{2} - 1\right )} \operatorname {arcsec}\left (x\right )^{2}}{3 \, x^{3}} - \frac {2 \, {\left ({\left (21 \, x^{2} - 1\right )} \sqrt {x + 1} \sqrt {x - 1} + 3 \, {\left (7 \, x^{4} - 8 \, x^{2} + 1\right )} \arctan \left (\sqrt {x + 1} \sqrt {x - 1}\right )\right )}}{27 \, \sqrt {x + 1} \sqrt {x - 1} x^{3}} \]
1/3*(x^2 - 1)^(3/2)*arcsec(x)^3/x^3 + 1/3*(3*x^2 - 1)*arcsec(x)^2/x^3 - 2/ 27*((21*x^2 - 1)*sqrt(x + 1)*sqrt(x - 1) + 3*(7*x^4 - 8*x^2 + 1)*arctan(sq rt(x + 1)*sqrt(x - 1)))/(sqrt(x + 1)*sqrt(x - 1)*x^3)
\[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx=\int { \frac {\sqrt {x^{2} - 1} \operatorname {arcsec}\left (x\right )^{3}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx=\int \frac {{\mathrm {acos}\left (\frac {1}{x}\right )}^3\,\sqrt {x^2-1}}{x^4} \,d x \]
\[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx=\int \frac {\sqrt {x^{2}-1}\, \mathit {asec} \left (x \right )^{3}}{x^{4}}d x \]