Integrand size = 14, antiderivative size = 170 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=-\frac {14}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}+\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x}+\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3} \]
2/27*b^3*c^3*(1-1/c^2/x^2)^(3/2)+2/9*b^2*(a+b*arcsec(c*x))/x^3+4/3*b^2*c^2 *(a+b*arcsec(c*x))/x-1/3*(a+b*arcsec(c*x))^3/x^3-14/9*b^3*c^3*(1-1/c^2/x^2 )^(1/2)+2/3*b*c^3*(a+b*arcsec(c*x))^2*(1-1/c^2/x^2)^(1/2)+1/3*b*c*(a+b*arc sec(c*x))^2*(1-1/c^2/x^2)^(1/2)/x^2
Time = 0.21 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=\frac {-9 a^3+9 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )+6 a b^2 \left (1+6 c^2 x^2\right )-2 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+20 c^2 x^2\right )+3 b \left (-9 a^2+6 a b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )+2 b^2 \left (1+6 c^2 x^2\right )\right ) \sec ^{-1}(c x)+9 b^2 \left (-3 a+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )\right ) \sec ^{-1}(c x)^2-9 b^3 \sec ^{-1}(c x)^3}{27 x^3} \]
(-9*a^3 + 9*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) + 6*a*b^2*(1 + 6*c^2*x^2) - 2*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 20*c^2*x^2) + 3*b*(-9*a ^2 + 6*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) + 2*b^2*(1 + 6*c^2*x^ 2))*ArcSec[c*x] + 9*b^2*(-3*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2 ))*ArcSec[c*x]^2 - 9*b^3*ArcSec[c*x]^3)/(27*x^3)
Time = 0.74 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {5745, 4905, 3042, 3792, 3042, 3113, 2009, 3777, 25, 3042, 3777, 3042, 3117}
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.
| \(\displaystyle \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx\) |
| \(\Big \downarrow \) 5745 |
| \(\displaystyle c^3 \int \frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^3}{c^2 x^2}d\sec ^{-1}(c x)\) |
| \(\Big \downarrow \) 4905 |
| \(\displaystyle c^3 \left (b \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{c^3 x^3}d\sec ^{-1}(c x)-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^3 \left (b \int \left (a+b \sec ^{-1}(c x)\right )^2 \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^3d\sec ^{-1}(c x)-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle c^3 \left (b \left (\frac {2}{3} \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{c x}d\sec ^{-1}(c x)-\frac {2}{9} b^2 \int \frac {1}{c^3 x^3}d\sec ^{-1}(c x)+\frac {2 b \left (a+b \sec ^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 c^2 x^2}\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^3 \left (b \left (\frac {2}{3} \int \left (a+b \sec ^{-1}(c x)\right )^2 \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )d\sec ^{-1}(c x)-\frac {2}{9} b^2 \int \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^3d\sec ^{-1}(c x)+\frac {2 b \left (a+b \sec ^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 c^2 x^2}\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle c^3 \left (b \left (\frac {2}{3} \int \left (a+b \sec ^{-1}(c x)\right )^2 \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )d\sec ^{-1}(c x)+\frac {2}{9} b^2 \int \frac {1}{c^2 x^2}d\left (-\sqrt {1-\frac {1}{c^2 x^2}}\right )+\frac {2 b \left (a+b \sec ^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 c^2 x^2}\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c^3 \left (b \left (\frac {2}{3} \int \left (a+b \sec ^{-1}(c x)\right )^2 \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )d\sec ^{-1}(c x)+\frac {2 b \left (a+b \sec ^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle c^3 \left (b \left (\frac {2}{3} \left (2 b \int -\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )d\sec ^{-1}(c x)+\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2\right )+\frac {2 b \left (a+b \sec ^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c^3 \left (b \left (\frac {2}{3} \left (\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2-2 b \int \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )d\sec ^{-1}(c x)\right )+\frac {2 b \left (a+b \sec ^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^3 \left (b \left (\frac {2}{3} \left (\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2-2 b \int \left (a+b \sec ^{-1}(c x)\right ) \sin \left (\sec ^{-1}(c x)\right )d\sec ^{-1}(c x)\right )+\frac {2 b \left (a+b \sec ^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle c^3 \left (b \left (\frac {2}{3} \left (\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2-2 b \left (b \int \frac {1}{c x}d\sec ^{-1}(c x)-\frac {a+b \sec ^{-1}(c x)}{c x}\right )\right )+\frac {2 b \left (a+b \sec ^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^3 \left (b \left (\frac {2}{3} \left (\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2-2 b \left (b \int \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )d\sec ^{-1}(c x)-\frac {a+b \sec ^{-1}(c x)}{c x}\right )\right )+\frac {2 b \left (a+b \sec ^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle c^3 \left (b \left (\frac {2 b \left (a+b \sec ^{-1}(c x)\right )}{9 c^3 x^3}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{3} \left (\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2-2 b \left (b \sqrt {1-\frac {1}{c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{c x}\right )\right )+\frac {2}{9} b^2 \left (\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 c^3 x^3}\right )\) |
c^3*(-1/3*(a + b*ArcSec[c*x])^3/(c^3*x^3) + b*((2*b^2*(-Sqrt[1 - 1/(c^2*x^ 2)] + (1 - 1/(c^2*x^2))^(3/2)/3))/9 + (2*b*(a + b*ArcSec[c*x]))/(9*c^3*x^3 ) + (Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x])^2)/(3*c^2*x^2) + (2*(Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x])^2 - 2*b*(b*Sqrt[1 - 1/(c^2*x^2)] - (a + b*ArcSec[c*x])/(c*x))))/3))
3.1.31.3.1 Defintions of rubi rules used
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[a + b*x]^(n + 1)/(b*(n + 1 ))), x] + Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Cos[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[1 /c^(m + 1) Subst[Int[(a + b*x)^n*Sec[x]^(m + 1)*Tan[x], x], x, ArcSec[c*x ]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] | | LtQ[m, -1])
Leaf count of result is larger than twice the leaf count of optimal. \(298\) vs. \(2(148)=296\).
Time = 1.16 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.76
| method | result | size |
| derivativedivides | \(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {\operatorname {arcsec}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}-\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\operatorname {arcsec}\left (c x \right )}{3 c x}+\frac {2 \,\operatorname {arcsec}\left (c x \right )}{9 c^{3} x^{3}}-\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsec}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\) | \(299\) |
| default | \(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {\operatorname {arcsec}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}-\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\operatorname {arcsec}\left (c x \right )}{3 c x}+\frac {2 \,\operatorname {arcsec}\left (c x \right )}{9 c^{3} x^{3}}-\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsec}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\) | \(299\) |
| parts | \(-\frac {a^{3}}{3 x^{3}}+b^{3} c^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {\operatorname {arcsec}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}-\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\operatorname {arcsec}\left (c x \right )}{3 c x}+\frac {2 \,\operatorname {arcsec}\left (c x \right )}{9 c^{3} x^{3}}-\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} c^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsec}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \,c^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\) | \(301\) |
c^3*(-1/3*a^3/c^3/x^3+b^3*(-1/3/c^3/x^3*arcsec(c*x)^3+1/3*arcsec(c*x)^2*(2 *c^2*x^2+1)/c^2/x^2*((c^2*x^2-1)/c^2/x^2)^(1/2)-4/3*((c^2*x^2-1)/c^2/x^2)^ (1/2)+4/3/c/x*arcsec(c*x)+2/9/c^3/x^3*arcsec(c*x)-2/27*(2*c^2*x^2+1)/c^2/x ^2*((c^2*x^2-1)/c^2/x^2)^(1/2))+3*a*b^2*(-1/3/c^3/x^3*arcsec(c*x)^2+2/9*ar csec(c*x)*(2*c^2*x^2+1)/c^2/x^2*((c^2*x^2-1)/c^2/x^2)^(1/2)+2/27/c^3/x^3+4 /9/c/x)+3*a^2*b*(-1/3/c^3/x^3*arcsec(c*x)+1/9*(c^2*x^2-1)*(2*c^2*x^2+1)/(( c^2*x^2-1)/c^2/x^2)^(1/2)/c^4/x^4))
Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=\frac {36 \, a b^{2} c^{2} x^{2} - 9 \, b^{3} \operatorname {arcsec}\left (c x\right )^{3} - 27 \, a b^{2} \operatorname {arcsec}\left (c x\right )^{2} - 9 \, a^{3} + 6 \, a b^{2} + 3 \, {\left (12 \, b^{3} c^{2} x^{2} - 9 \, a^{2} b + 2 \, b^{3}\right )} \operatorname {arcsec}\left (c x\right ) + {\left (2 \, {\left (9 \, a^{2} b - 20 \, b^{3}\right )} c^{2} x^{2} + 9 \, a^{2} b - 2 \, b^{3} + 9 \, {\left (2 \, b^{3} c^{2} x^{2} + b^{3}\right )} \operatorname {arcsec}\left (c x\right )^{2} + 18 \, {\left (2 \, a b^{2} c^{2} x^{2} + a b^{2}\right )} \operatorname {arcsec}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{27 \, x^{3}} \]
1/27*(36*a*b^2*c^2*x^2 - 9*b^3*arcsec(c*x)^3 - 27*a*b^2*arcsec(c*x)^2 - 9* a^3 + 6*a*b^2 + 3*(12*b^3*c^2*x^2 - 9*a^2*b + 2*b^3)*arcsec(c*x) + (2*(9*a ^2*b - 20*b^3)*c^2*x^2 + 9*a^2*b - 2*b^3 + 9*(2*b^3*c^2*x^2 + b^3)*arcsec( c*x)^2 + 18*(2*a*b^2*c^2*x^2 + a*b^2)*arcsec(c*x))*sqrt(c^2*x^2 - 1))/x^3
\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=\int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (148) = 296\).
Time = 0.70 (sec) , antiderivative size = 575, normalized size of antiderivative = 3.38 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=-\frac {1}{216} \, {\left (\frac {72 \, {\left (c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )} \operatorname {arcsec}\left (c x\right )^{2}}{c} + \frac {72 \, c^{4} {\left (\frac {c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {2 \, \sqrt {c^{2} x^{2} - 1} c}{x} - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}}{c} - \frac {c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {2 \, \sqrt {c^{2} x^{2} - 1} c}{x} - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}}{c} - \frac {4 \, \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\right )} + c^{2} {\left (\frac {9 \, c^{4} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {16 \, \sqrt {c^{2} x^{2} - 1} c^{3}}{x} - \frac {9 \, \sqrt {c^{2} x^{2} - 1} c^{2}}{x^{2}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} c}{x^{3}} - \frac {6 \, \sqrt {c^{2} x^{2} - 1}}{x^{4}}}{c} - \frac {9 \, c^{4} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {16 \, \sqrt {c^{2} x^{2} - 1} c^{3}}{x} - \frac {9 \, \sqrt {c^{2} x^{2} - 1} c^{2}}{x^{2}} - \frac {8 \, \sqrt {c^{2} x^{2} - 1} c}{x^{3}} - \frac {6 \, \sqrt {c^{2} x^{2} - 1}}{x^{4}}}{c} - \frac {48 \, \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )}{x^{3}}\right )}}{c^{2}}\right )} b^{3} - \frac {1}{3} \, a^{2} b {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {3 \, \operatorname {arcsec}\left (c x\right )}{x^{3}}\right )} - \frac {b^{3} \operatorname {arcsec}\left (c x\right )^{3}}{3 \, x^{3}} - \frac {a b^{2} \operatorname {arcsec}\left (c x\right )^{2}}{x^{3}} - \frac {a^{3}}{3 \, x^{3}} + \frac {2 \, {\left ({\left (6 \, c^{3} x^{2} + c\right )} \sqrt {c x + 1} \sqrt {c x - 1} + 3 \, {\left (2 \, c^{5} x^{4} - c^{3} x^{2} - c\right )} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} a b^{2}}{9 \, \sqrt {c x + 1} \sqrt {c x - 1} c x^{3}} \]
-1/216*(72*(c^4*(-1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(-1/(c^2*x^2) + 1))*a rcsec(c*x)^2/c + (72*c^4*((c^2*arcsin(1/(c*abs(x))) + 2*sqrt(c^2*x^2 - 1)* c/x - sqrt(c^2*x^2 - 1)/x^2)/c - (c^2*arcsin(1/(c*abs(x))) - 2*sqrt(c^2*x^ 2 - 1)*c/x - sqrt(c^2*x^2 - 1)/x^2)/c - 4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/x) + c^2*((9*c^4*arcsin(1/(c*abs(x))) + 16*sqrt(c^2*x^2 - 1)*c^3/x - 9 *sqrt(c^2*x^2 - 1)*c^2/x^2 + 8*sqrt(c^2*x^2 - 1)*c/x^3 - 6*sqrt(c^2*x^2 - 1)/x^4)/c - (9*c^4*arcsin(1/(c*abs(x))) - 16*sqrt(c^2*x^2 - 1)*c^3/x - 9*s qrt(c^2*x^2 - 1)*c^2/x^2 - 8*sqrt(c^2*x^2 - 1)*c/x^3 - 6*sqrt(c^2*x^2 - 1) /x^4)/c - 48*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/x^3))/c^2)*b^3 - 1/3*a^2* b*((c^4*(-1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(-1/(c^2*x^2) + 1))/c + 3*arc sec(c*x)/x^3) - 1/3*b^3*arcsec(c*x)^3/x^3 - a*b^2*arcsec(c*x)^2/x^3 - 1/3* a^3/x^3 + 2/9*((6*c^3*x^2 + c)*sqrt(c*x + 1)*sqrt(c*x - 1) + 3*(2*c^5*x^4 - c^3*x^2 - c)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)))*a*b^2/(sqrt(c*x + 1)*s qrt(c*x - 1)*c*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (148) = 296\).
Time = 0.32 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=\frac {1}{27} \, {\left (18 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )^{2} + 36 \, a b^{2} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right ) + 18 \, a^{2} b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 40 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {36 \, b^{3} c \arccos \left (\frac {1}{c x}\right )}{x} + \frac {9 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )^{2}}{x^{2}} + \frac {36 \, a b^{2} c}{x} + \frac {18 \, a b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )}{x^{2}} - \frac {9 \, b^{3} \arccos \left (\frac {1}{c x}\right )^{3}}{c x^{3}} + \frac {9 \, a^{2} b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{2}} - \frac {2 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{2}} - \frac {27 \, a b^{2} \arccos \left (\frac {1}{c x}\right )^{2}}{c x^{3}} - \frac {27 \, a^{2} b \arccos \left (\frac {1}{c x}\right )}{c x^{3}} + \frac {6 \, b^{3} \arccos \left (\frac {1}{c x}\right )}{c x^{3}} - \frac {9 \, a^{3}}{c x^{3}} + \frac {6 \, a b^{2}}{c x^{3}}\right )} c \]
1/27*(18*b^3*c^2*sqrt(-1/(c^2*x^2) + 1)*arccos(1/(c*x))^2 + 36*a*b^2*c^2*s qrt(-1/(c^2*x^2) + 1)*arccos(1/(c*x)) + 18*a^2*b*c^2*sqrt(-1/(c^2*x^2) + 1 ) - 40*b^3*c^2*sqrt(-1/(c^2*x^2) + 1) + 36*b^3*c*arccos(1/(c*x))/x + 9*b^3 *sqrt(-1/(c^2*x^2) + 1)*arccos(1/(c*x))^2/x^2 + 36*a*b^2*c/x + 18*a*b^2*sq rt(-1/(c^2*x^2) + 1)*arccos(1/(c*x))/x^2 - 9*b^3*arccos(1/(c*x))^3/(c*x^3) + 9*a^2*b*sqrt(-1/(c^2*x^2) + 1)/x^2 - 2*b^3*sqrt(-1/(c^2*x^2) + 1)/x^2 - 27*a*b^2*arccos(1/(c*x))^2/(c*x^3) - 27*a^2*b*arccos(1/(c*x))/(c*x^3) + 6 *b^3*arccos(1/(c*x))/(c*x^3) - 9*a^3/(c*x^3) + 6*a*b^2/(c*x^3))*c
Timed out. \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3}{x^4} \,d x \]