Integrand size = 14, antiderivative size = 208 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^5} \, dx=-\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{128 x^3}-\frac {45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{256 x}-\frac {45}{256} b^3 c^4 \sec ^{-1}(c x)+\frac {3 b^2 \left (a+b \sec ^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \sec ^{-1}(c x)\right )}{32 x^2}+\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{16 x^3}+\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{32 x}+\frac {3}{32} c^4 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 x^4} \]
-45/256*b^3*c^4*arcsec(c*x)+3/32*b^2*(a+b*arcsec(c*x))/x^4+9/32*b^2*c^2*(a +b*arcsec(c*x))/x^2+3/32*c^4*(a+b*arcsec(c*x))^3-1/4*(a+b*arcsec(c*x))^3/x ^4-3/128*b^3*c*(1-1/c^2/x^2)^(1/2)/x^3-45/256*b^3*c^3*(1-1/c^2/x^2)^(1/2)/ x+3/16*b*c*(a+b*arcsec(c*x))^2*(1-1/c^2/x^2)^(1/2)/x^3+9/32*b*c^3*(a+b*arc sec(c*x))^2*(1-1/c^2/x^2)^(1/2)/x
Time = 0.23 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^5} \, dx=\frac {-64 a^3+24 a b^2+48 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x-6 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x+72 a b^2 c^2 x^2+72 a^2 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3-45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3+24 b \left (-8 a^2+b^2 \left (1+3 c^2 x^2\right )+2 a b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (2+3 c^2 x^2\right )\right ) \sec ^{-1}(c x)+24 b^2 \left (b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (2+3 c^2 x^2\right )+a \left (-8+3 c^4 x^4\right )\right ) \sec ^{-1}(c x)^2+8 b^3 \left (-8+3 c^4 x^4\right ) \sec ^{-1}(c x)^3+9 b \left (-8 a^2+5 b^2\right ) c^4 x^4 \arcsin \left (\frac {1}{c x}\right )}{256 x^4} \]
(-64*a^3 + 24*a*b^2 + 48*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x - 6*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x + 72*a*b^2*c^2*x^2 + 72*a^2*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*x ^3 - 45*b^3*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 + 24*b*(-8*a^2 + b^2*(1 + 3*c^2* x^2) + 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + 3*c^2*x^2))*ArcSec[c*x] + 24*b ^2*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + 3*c^2*x^2) + a*(-8 + 3*c^4*x^4))*ArcS ec[c*x]^2 + 8*b^3*(-8 + 3*c^4*x^4)*ArcSec[c*x]^3 + 9*b*(-8*a^2 + 5*b^2)*c^ 4*x^4*ArcSin[1/(c*x)])/(256*x^4)
Time = 0.77 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.29, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5745, 4905, 3042, 3792, 3042, 3115, 3042, 3115, 24, 3792, 17, 3042, 3115, 24}
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.
| \(\displaystyle \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^5} \, dx\) |
| \(\Big \downarrow \) 5745 |
| \(\displaystyle c^4 \int \frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^3}{c^3 x^3}d\sec ^{-1}(c x)\) |
| \(\Big \downarrow \) 4905 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{c^4 x^4}d\sec ^{-1}(c x)-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \int \left (a+b \sec ^{-1}(c x)\right )^2 \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^4d\sec ^{-1}(c x)-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \left (\frac {3}{4} \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{c^2 x^2}d\sec ^{-1}(c x)-\frac {1}{8} b^2 \int \frac {1}{c^4 x^4}d\sec ^{-1}(c x)+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c^3 x^3}\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \sec ^{-1}(c x)\right )^2 \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^2d\sec ^{-1}(c x)-\frac {1}{8} b^2 \int \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^4d\sec ^{-1}(c x)+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c^3 x^3}\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \sec ^{-1}(c x)\right )^2 \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^2d\sec ^{-1}(c x)-\frac {1}{8} b^2 \left (\frac {3}{4} \int \frac {1}{c^2 x^2}d\sec ^{-1}(c x)+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c^3 x^3}\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \sec ^{-1}(c x)\right )^2 \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^2d\sec ^{-1}(c x)-\frac {1}{8} b^2 \left (\frac {3}{4} \int \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^2d\sec ^{-1}(c x)+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c^3 x^3}\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \sec ^{-1}(c x)\right )^2 \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^2d\sec ^{-1}(c x)-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \int 1d\sec ^{-1}(c x)+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c^3 x^3}\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \sec ^{-1}(c x)\right )^2 \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^2d\sec ^{-1}(c x)+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}+\frac {1}{2} \sec ^{-1}(c x)\right )+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \int \left (a+b \sec ^{-1}(c x)\right )^2d\sec ^{-1}(c x)-\frac {1}{2} b^2 \int \frac {1}{c^2 x^2}d\sec ^{-1}(c x)+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{2 c x}\right )+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}+\frac {1}{2} \sec ^{-1}(c x)\right )+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \left (\frac {3}{4} \left (-\frac {1}{2} b^2 \int \frac {1}{c^2 x^2}d\sec ^{-1}(c x)+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{6 b}\right )+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}+\frac {1}{2} \sec ^{-1}(c x)\right )+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \left (\frac {3}{4} \left (-\frac {1}{2} b^2 \int \sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )^2d\sec ^{-1}(c x)+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{6 b}\right )+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}+\frac {1}{2} \sec ^{-1}(c x)\right )+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \left (\frac {3}{4} \left (-\frac {1}{2} b^2 \left (\frac {1}{2} \int 1d\sec ^{-1}(c x)+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{6 b}\right )+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}+\frac {1}{2} \sec ^{-1}(c x)\right )+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle c^4 \left (\frac {3}{4} b \left (\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{6 b}-\frac {1}{2} b^2 \left (\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}+\frac {1}{2} \sec ^{-1}(c x)\right )\right )+\frac {b \left (a+b \sec ^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}+\frac {1}{2} \sec ^{-1}(c x)\right )+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{4 c^4 x^4}\right )\) |
c^4*(-1/4*(a + b*ArcSec[c*x])^3/(c^4*x^4) + (3*b*(-1/8*(b^2*(Sqrt[1 - 1/(c ^2*x^2)]/(4*c^3*x^3) + (3*(Sqrt[1 - 1/(c^2*x^2)]/(2*c*x) + ArcSec[c*x]/2)) /4)) + (b*(a + b*ArcSec[c*x]))/(8*c^4*x^4) + (Sqrt[1 - 1/(c^2*x^2)]*(a + b *ArcSec[c*x])^2)/(4*c^3*x^3) + (3*(-1/2*(b^2*(Sqrt[1 - 1/(c^2*x^2)]/(2*c*x ) + ArcSec[c*x]/2)) + (b*(a + b*ArcSec[c*x]))/(2*c^2*x^2) + (Sqrt[1 - 1/(c ^2*x^2)]*(a + b*ArcSec[c*x])^2)/(2*c*x) + (a + b*ArcSec[c*x])^3/(6*b)))/4) )/4)
3.1.32.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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. \(469\) vs. \(2(182)=364\).
Time = 1.10 (sec) , antiderivative size = 470, normalized size of antiderivative = 2.26
| method | result | size |
| parts | \(-\frac {a^{3}}{4 x^{4}}+b^{3} c^{4} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {3 \operatorname {arcsec}\left (c x \right )^{2} \left (3 c^{3} x^{3} \operatorname {arcsec}\left (c x \right )+3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{32 c^{3} x^{3}}+\frac {3 \,\operatorname {arcsec}\left (c x \right )}{32 c^{4} x^{4}}-\frac {3 \left (3 c^{2} x^{2}+2\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{256 c^{3} x^{3}}-\frac {45 \,\operatorname {arcsec}\left (c x \right )}{256}+\frac {9 \,\operatorname {arcsec}\left (c x \right )}{32 c^{2} x^{2}}-\frac {9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{64 c x}-\frac {3 \operatorname {arcsec}\left (c x \right )^{3}}{16}\right )+3 a \,b^{2} c^{4} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arcsec}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arcsec}\left (c x \right )+3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arcsec}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {3 a^{2} b \,\operatorname {arcsec}\left (c x \right )}{4 x^{4}}-\frac {9 a^{2} b \,c^{3} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {9 a^{2} b c \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}}+\frac {3 a^{2} b \left (c^{2} x^{2}-1\right )}{16 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{5}}\) | \(470\) |
| derivativedivides | \(c^{4} \left (-\frac {a^{3}}{4 c^{4} x^{4}}+b^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {3 \operatorname {arcsec}\left (c x \right )^{2} \left (3 c^{3} x^{3} \operatorname {arcsec}\left (c x \right )+3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{32 c^{3} x^{3}}+\frac {3 \,\operatorname {arcsec}\left (c x \right )}{32 c^{4} x^{4}}-\frac {3 \left (3 c^{2} x^{2}+2\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{256 c^{3} x^{3}}-\frac {45 \,\operatorname {arcsec}\left (c x \right )}{256}+\frac {9 \,\operatorname {arcsec}\left (c x \right )}{32 c^{2} x^{2}}-\frac {9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{64 c x}-\frac {3 \operatorname {arcsec}\left (c x \right )^{3}}{16}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arcsec}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arcsec}\left (c x \right )+3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arcsec}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {3 a^{2} b \,\operatorname {arcsec}\left (c x \right )}{4 c^{4} x^{4}}-\frac {9 a^{2} b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {9 a^{2} b \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {3 a^{2} b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(476\) |
| default | \(c^{4} \left (-\frac {a^{3}}{4 c^{4} x^{4}}+b^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {3 \operatorname {arcsec}\left (c x \right )^{2} \left (3 c^{3} x^{3} \operatorname {arcsec}\left (c x \right )+3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{32 c^{3} x^{3}}+\frac {3 \,\operatorname {arcsec}\left (c x \right )}{32 c^{4} x^{4}}-\frac {3 \left (3 c^{2} x^{2}+2\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{256 c^{3} x^{3}}-\frac {45 \,\operatorname {arcsec}\left (c x \right )}{256}+\frac {9 \,\operatorname {arcsec}\left (c x \right )}{32 c^{2} x^{2}}-\frac {9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{64 c x}-\frac {3 \operatorname {arcsec}\left (c x \right )^{3}}{16}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arcsec}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arcsec}\left (c x \right )+3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arcsec}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {3 a^{2} b \,\operatorname {arcsec}\left (c x \right )}{4 c^{4} x^{4}}-\frac {9 a^{2} b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {9 a^{2} b \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {3 a^{2} b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(476\) |
-1/4*a^3/x^4+b^3*c^4*(-1/4/c^4/x^4*arcsec(c*x)^3+3/32*arcsec(c*x)^2*(3*c^3 *x^3*arcsec(c*x)+3*c^2*x^2*((c^2*x^2-1)/c^2/x^2)^(1/2)+2*((c^2*x^2-1)/c^2/ x^2)^(1/2))/c^3/x^3+3/32*arcsec(c*x)/c^4/x^4-3/256*(3*c^2*x^2+2)/c^3/x^3*( (c^2*x^2-1)/c^2/x^2)^(1/2)-45/256*arcsec(c*x)+9/32/c^2/x^2*arcsec(c*x)-9/6 4*((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x-3/16*arcsec(c*x)^3)+3*a*b^2*c^4*(-1/4/c^ 4/x^4*arcsec(c*x)^2+1/16*arcsec(c*x)*(3*c^3*x^3*arcsec(c*x)+3*c^2*x^2*((c^ 2*x^2-1)/c^2/x^2)^(1/2)+2*((c^2*x^2-1)/c^2/x^2)^(1/2))/c^3/x^3-3/32*arcsec (c*x)^2+1/128*(3*c^2*x^2+2)^2/c^4/x^4)-3/4*a^2*b/x^4*arcsec(c*x)-9/32*a^2* b*c^3*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*arctan(1/(c^2*x^2-1) ^(1/2))+9/32*a^2*b*c*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^3+3/16*a^2* b/c*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^5
Time = 0.28 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^5} \, dx=\frac {72 \, a b^{2} c^{2} x^{2} + 8 \, {\left (3 \, b^{3} c^{4} x^{4} - 8 \, b^{3}\right )} \operatorname {arcsec}\left (c x\right )^{3} - 64 \, a^{3} + 24 \, a b^{2} + 24 \, {\left (3 \, a b^{2} c^{4} x^{4} - 8 \, a b^{2}\right )} \operatorname {arcsec}\left (c x\right )^{2} + 3 \, {\left (3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c^{4} x^{4} + 24 \, b^{3} c^{2} x^{2} - 64 \, a^{2} b + 8 \, b^{3}\right )} \operatorname {arcsec}\left (c x\right ) + 3 \, {\left (3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c^{2} x^{2} + 16 \, a^{2} b - 2 \, b^{3} + 8 \, {\left (3 \, b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \operatorname {arcsec}\left (c x\right )^{2} + 16 \, {\left (3 \, a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \operatorname {arcsec}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{256 \, x^{4}} \]
1/256*(72*a*b^2*c^2*x^2 + 8*(3*b^3*c^4*x^4 - 8*b^3)*arcsec(c*x)^3 - 64*a^3 + 24*a*b^2 + 24*(3*a*b^2*c^4*x^4 - 8*a*b^2)*arcsec(c*x)^2 + 3*(3*(8*a^2*b - 5*b^3)*c^4*x^4 + 24*b^3*c^2*x^2 - 64*a^2*b + 8*b^3)*arcsec(c*x) + 3*(3* (8*a^2*b - 5*b^3)*c^2*x^2 + 16*a^2*b - 2*b^3 + 8*(3*b^3*c^2*x^2 + 2*b^3)*a rcsec(c*x)^2 + 16*(3*a*b^2*c^2*x^2 + 2*a*b^2)*arcsec(c*x))*sqrt(c^2*x^2 - 1))/x^4
\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^5} \, dx=\int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}}{x^{5}}\, dx \]
\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3}}{x^{5}} \,d x } \]
3/32*a^2*b*((3*c^5*arctan(c*x*sqrt(-1/(c^2*x^2) + 1)) + (3*c^8*x^3*(-1/(c^ 2*x^2) + 1)^(3/2) + 5*c^6*x*sqrt(-1/(c^2*x^2) + 1))/(c^4*x^4*(1/(c^2*x^2) - 1)^2 - 2*c^2*x^2*(1/(c^2*x^2) - 1) + 1))/c - 8*arcsec(c*x)/x^4) - 1/4*a^ 3/x^4 - 1/16*(4*b^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 3*b^3*arctan(s qrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2 + 12*(2*(c^2*log(c*x + 1) + c^2 *log(c*x - 1) - 2*c^2*log(x) + 1/x^2)*a*b^2*c^2*log(c)^2 + 64*b^3*c^2*inte grate(1/16*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/(c^2*x^7 - x^5), x)*log (c)^2 - 64*b^3*c^2*integrate(1/16*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))* log(c^2*x^2)/(c^2*x^7 - x^5), x)*log(c) + 128*b^3*c^2*integrate(1/16*x^2*a rctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(x)/(c^2*x^7 - x^5), x)*log(c) - 64* a*b^2*c^2*integrate(1/16*x^2*log(c^2*x^2)/(c^2*x^7 - x^5), x)*log(c) + 128 *a*b^2*c^2*integrate(1/16*x^2*log(x)/(c^2*x^7 - x^5), x)*log(c) - 64*b^3*c ^2*integrate(1/16*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)*log (x)/(c^2*x^7 - x^5), x) + 64*b^3*c^2*integrate(1/16*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(x)^2/(c^2*x^7 - x^5), x) - 64*a*b^2*c^2*integrate(1/ 16*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2/(c^2*x^7 - x^5), x) + 16*b^3* c^2*integrate(1/16*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)/(c ^2*x^7 - x^5), x) + 16*a*b^2*c^2*integrate(1/16*x^2*log(c^2*x^2)^2/(c^2*x^ 7 - x^5), x) - 64*a*b^2*c^2*integrate(1/16*x^2*log(c^2*x^2)*log(x)/(c^2*x^ 7 - x^5), x) + 64*a*b^2*c^2*integrate(1/16*x^2*log(x)^2/(c^2*x^7 - x^5)...
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (182) = 364\).
Time = 0.31 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^5} \, dx=\frac {1}{256} \, {\left (24 \, b^{3} c^{3} \arccos \left (\frac {1}{c x}\right )^{3} + 72 \, a b^{2} c^{3} \arccos \left (\frac {1}{c x}\right )^{2} + 72 \, a^{2} b c^{3} \arccos \left (\frac {1}{c x}\right ) - 45 \, b^{3} c^{3} \arccos \left (\frac {1}{c x}\right ) + \frac {72 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )^{2}}{x} - 45 \, a b^{2} c^{3} + \frac {144 \, a b^{2} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )}{x} + \frac {72 \, a^{2} b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} - \frac {45 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} + \frac {72 \, b^{3} c \arccos \left (\frac {1}{c x}\right )}{x^{2}} + \frac {48 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )^{2}}{x^{3}} + \frac {72 \, a b^{2} c}{x^{2}} + \frac {96 \, a b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )}{x^{3}} - \frac {64 \, b^{3} \arccos \left (\frac {1}{c x}\right )^{3}}{c x^{4}} + \frac {48 \, a^{2} b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{3}} - \frac {6 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{3}} - \frac {192 \, a b^{2} \arccos \left (\frac {1}{c x}\right )^{2}}{c x^{4}} - \frac {192 \, a^{2} b \arccos \left (\frac {1}{c x}\right )}{c x^{4}} + \frac {24 \, b^{3} \arccos \left (\frac {1}{c x}\right )}{c x^{4}} - \frac {64 \, a^{3}}{c x^{4}} + \frac {24 \, a b^{2}}{c x^{4}}\right )} c \]
1/256*(24*b^3*c^3*arccos(1/(c*x))^3 + 72*a*b^2*c^3*arccos(1/(c*x))^2 + 72* a^2*b*c^3*arccos(1/(c*x)) - 45*b^3*c^3*arccos(1/(c*x)) + 72*b^3*c^2*sqrt(- 1/(c^2*x^2) + 1)*arccos(1/(c*x))^2/x - 45*a*b^2*c^3 + 144*a*b^2*c^2*sqrt(- 1/(c^2*x^2) + 1)*arccos(1/(c*x))/x + 72*a^2*b*c^2*sqrt(-1/(c^2*x^2) + 1)/x - 45*b^3*c^2*sqrt(-1/(c^2*x^2) + 1)/x + 72*b^3*c*arccos(1/(c*x))/x^2 + 48 *b^3*sqrt(-1/(c^2*x^2) + 1)*arccos(1/(c*x))^2/x^3 + 72*a*b^2*c/x^2 + 96*a* b^2*sqrt(-1/(c^2*x^2) + 1)*arccos(1/(c*x))/x^3 - 64*b^3*arccos(1/(c*x))^3/ (c*x^4) + 48*a^2*b*sqrt(-1/(c^2*x^2) + 1)/x^3 - 6*b^3*sqrt(-1/(c^2*x^2) + 1)/x^3 - 192*a*b^2*arccos(1/(c*x))^2/(c*x^4) - 192*a^2*b*arccos(1/(c*x))/( c*x^4) + 24*b^3*arccos(1/(c*x))/(c*x^4) - 64*a^3/(c*x^4) + 24*a*b^2/(c*x^4 ))*c
Timed out. \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3}{x^5} \,d x \]