Integrand size = 14, antiderivative size = 228 \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=-\frac {c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{8 b \left (a+b \sec ^{-1}(c x)\right )^2}-\frac {c^2}{8 b^2 x \left (a+b \sec ^{-1}(c x)\right )}-\frac {3 c^3 \cos \left (3 \sec ^{-1}(c x)\right )}{8 b^2 \left (a+b \sec ^{-1}(c x)\right )}+\frac {c^3 \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{8 b^3}+\frac {9 c^3 \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right ) \sin \left (\frac {3 a}{b}\right )}{8 b^3}-\frac {c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{8 b \left (a+b \sec ^{-1}(c x)\right )^2}-\frac {c^3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{8 b^3}-\frac {9 c^3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{8 b^3} \]
-1/8*c^2/b^2/x/(a+b*arcsec(c*x))-3/8*c^3*cos(3*arcsec(c*x))/b^2/(a+b*arcse c(c*x))-1/8*c^3*cos(a/b)*Si(a/b+arcsec(c*x))/b^3-9/8*c^3*cos(3*a/b)*Si(3*a /b+3*arcsec(c*x))/b^3+1/8*c^3*Ci(a/b+arcsec(c*x))*sin(a/b)/b^3+9/8*c^3*Ci( 3*a/b+3*arcsec(c*x))*sin(3*a/b)/b^3-1/8*c^3*sin(3*arcsec(c*x))/b/(a+b*arcs ec(c*x))^2-1/8*c^3*(1-1/c^2/x^2)^(1/2)/b/(a+b*arcsec(c*x))^2
Time = 0.37 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\frac {-\frac {4 b^2 c \sqrt {1-\frac {1}{c^2 x^2}}}{x^2 \left (a+b \sec ^{-1}(c x)\right )^2}-\frac {12 b}{x^3 \left (a+b \sec ^{-1}(c x)\right )}+\frac {8 b c^2}{a x+b x \sec ^{-1}(c x)}+c^3 \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )+9 c^3 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-c^3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )-9 c^3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )}{8 b^3} \]
((-4*b^2*c*Sqrt[1 - 1/(c^2*x^2)])/(x^2*(a + b*ArcSec[c*x])^2) - (12*b)/(x^ 3*(a + b*ArcSec[c*x])) + (8*b*c^2)/(a*x + b*x*ArcSec[c*x]) + c^3*CosIntegr al[a/b + ArcSec[c*x]]*Sin[a/b] + 9*c^3*CosIntegral[3*(a/b + ArcSec[c*x])]* Sin[(3*a)/b] - c^3*Cos[a/b]*SinIntegral[a/b + ArcSec[c*x]] - 9*c^3*Cos[(3* a)/b]*SinIntegral[3*(a/b + ArcSec[c*x])])/(8*b^3)
Time = 0.57 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5745, 4906, 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.
| \(\displaystyle \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 5745 |
| \(\displaystyle c^3 \int \frac {\sqrt {1-\frac {1}{c^2 x^2}}}{c^2 x^2 \left (a+b \sec ^{-1}(c x)\right )^3}d\sec ^{-1}(c x)\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle c^3 \int \left (\frac {\sin \left (3 \sec ^{-1}(c x)\right )}{4 \left (a+b \sec ^{-1}(c x)\right )^3}+\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 \left (a+b \sec ^{-1}(c x)\right )^3}\right )d\sec ^{-1}(c x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c^3 \left (\frac {\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{8 b^3}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{8 b^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{8 b^3}-\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{8 b^3}-\frac {1}{8 b^2 c x \left (a+b \sec ^{-1}(c x)\right )}-\frac {3 \cos \left (3 \sec ^{-1}(c x)\right )}{8 b^2 \left (a+b \sec ^{-1}(c x)\right )}-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{8 b \left (a+b \sec ^{-1}(c x)\right )^2}-\frac {\sin \left (3 \sec ^{-1}(c x)\right )}{8 b \left (a+b \sec ^{-1}(c x)\right )^2}\right )\) |
c^3*(-1/8*Sqrt[1 - 1/(c^2*x^2)]/(b*(a + b*ArcSec[c*x])^2) - 1/(8*b^2*c*x*( a + b*ArcSec[c*x])) - (3*Cos[3*ArcSec[c*x]])/(8*b^2*(a + b*ArcSec[c*x])) + (CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b])/(8*b^3) + (9*CosIntegral[(3*a)/ b + 3*ArcSec[c*x]]*Sin[(3*a)/b])/(8*b^3) - Sin[3*ArcSec[c*x]]/(8*b*(a + b* ArcSec[c*x])^2) - (Cos[a/b]*SinIntegral[a/b + ArcSec[c*x]])/(8*b^3) - (9*C os[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSec[c*x]])/(8*b^3))
3.1.50.3.1 Defintions of rubi rules used
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
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])
Time = 0.39 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.35
| method | result | size |
| derivativedivides | \(c^{3} \left (-\frac {\sin \left (3 \,\operatorname {arcsec}\left (c x \right )\right )}{8 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )^{2} b}-\frac {3 \left (3 \,\operatorname {arcsec}\left (c x \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) b -3 \,\operatorname {arcsec}\left (c x \right ) \sin \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) b +3 \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) a -3 \sin \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) a +\cos \left (3 \,\operatorname {arcsec}\left (c x \right )\right ) b \right )}{8 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b^{3}}-\frac {\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )^{2} b}-\frac {\operatorname {arcsec}\left (c x \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) b c x -\operatorname {arcsec}\left (c x \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) b c x +\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) a c x -\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) a c x +b}{8 c x \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b^{3}}\right )\) | \(307\) |
| default | \(c^{3} \left (-\frac {\sin \left (3 \,\operatorname {arcsec}\left (c x \right )\right )}{8 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )^{2} b}-\frac {3 \left (3 \,\operatorname {arcsec}\left (c x \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) b -3 \,\operatorname {arcsec}\left (c x \right ) \sin \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) b +3 \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) a -3 \sin \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) a +\cos \left (3 \,\operatorname {arcsec}\left (c x \right )\right ) b \right )}{8 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b^{3}}-\frac {\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )^{2} b}-\frac {\operatorname {arcsec}\left (c x \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) b c x -\operatorname {arcsec}\left (c x \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) b c x +\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) a c x -\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) a c x +b}{8 c x \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b^{3}}\right )\) | \(307\) |
c^3*(-1/8*sin(3*arcsec(c*x))/(a+b*arcsec(c*x))^2/b-3/8*(3*arcsec(c*x)*cos( 3*a/b)*Si(3*a/b+3*arcsec(c*x))*b-3*arcsec(c*x)*sin(3*a/b)*Ci(3*a/b+3*arcse c(c*x))*b+3*cos(3*a/b)*Si(3*a/b+3*arcsec(c*x))*a-3*sin(3*a/b)*Ci(3*a/b+3*a rcsec(c*x))*a+cos(3*arcsec(c*x))*b)/(a+b*arcsec(c*x))/b^3-1/8*((c^2*x^2-1) /c^2/x^2)^(1/2)/(a+b*arcsec(c*x))^2/b-1/8*(arcsec(c*x)*cos(a/b)*Si(a/b+arc sec(c*x))*b*c*x-arcsec(c*x)*sin(a/b)*Ci(a/b+arcsec(c*x))*b*c*x+cos(a/b)*Si (a/b+arcsec(c*x))*a*c*x-sin(a/b)*Ci(a/b+arcsec(c*x))*a*c*x+b)/c/x/(a+b*arc sec(c*x))/b^3)
\[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3} x^{4}} \,d x } \]
integral(1/(b^3*x^4*arcsec(c*x)^3 + 3*a*b^2*x^4*arcsec(c*x)^2 + 3*a^2*b*x^ 4*arcsec(c*x) + a^3*x^4), x)
\[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^{4} \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}}\, dx \]
\[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3} x^{4}} \,d x } \]
-(24*a*b^2*log(c)^2 - 8*(2*b^3*c^2*x^2 - 3*b^3)*arctan(sqrt(c*x + 1)*sqrt( c*x - 1))^3 + 24*a^3 - 16*(a*b^2*c^2*log(c)^2 + a^3*c^2)*x^2 - 24*(2*a*b^2 *c^2*x^2 - 3*a*b^2)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 - 2*(2*a*b^2*c^2 *x^2 - 3*a*b^2)*log(c^2*x^2)^2 - 8*(2*a*b^2*c^2*x^2 - 3*a*b^2)*log(x)^2 + 2*(4*b^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 - b^3*log(c^2*x^2)^2 - 4*b^ 3*log(c)^2 - 8*b^3*log(c)*log(x) - 4*b^3*log(x)^2 + 8*a*b^2*arctan(sqrt(c* x + 1)*sqrt(c*x - 1)) + 4*a^2*b + 4*(b^3*log(c) + b^3*log(x))*log(c^2*x^2) )*sqrt(c*x + 1)*sqrt(c*x - 1) + 2*(12*b^3*log(c)^2 + 36*a^2*b - 8*(b^3*c^2 *log(c)^2 + 3*a^2*b*c^2)*x^2 - (2*b^3*c^2*x^2 - 3*b^3)*log(c^2*x^2)^2 - 4* (2*b^3*c^2*x^2 - 3*b^3)*log(x)^2 + 4*(2*b^3*c^2*x^2*log(c) - 3*b^3*log(c) + (2*b^3*c^2*x^2 - 3*b^3)*log(x))*log(c^2*x^2) - 8*(2*b^3*c^2*x^2*log(c) - 3*b^3*log(c))*log(x))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - (16*b^6*x^3*a rctan(sqrt(c*x + 1)*sqrt(c*x - 1))^4 + b^6*x^3*log(c^2*x^2)^4 + 64*b^6*x^3 *log(c)*log(x)^3 + 16*b^6*x^3*log(x)^4 + 64*a*b^5*x^3*arctan(sqrt(c*x + 1) *sqrt(c*x - 1))^3 + 32*(3*b^6*log(c)^2 + a^2*b^4)*x^3*log(x)^2 + 64*(b^6*l og(c)^3 + a^2*b^4*log(c))*x^3*log(x) + 16*(b^6*log(c)^4 + 2*a^2*b^4*log(c) ^2 + a^4*b^2)*x^3 - 8*(b^6*x^3*log(c) + b^6*x^3*log(x))*log(c^2*x^2)^3 + 8 *(b^6*x^3*log(c^2*x^2)^2 + 8*b^6*x^3*log(c)*log(x) + 4*b^6*x^3*log(x)^2 + 4*(b^6*log(c)^2 + 3*a^2*b^4)*x^3 - 4*(b^6*x^3*log(c) + b^6*x^3*log(x))*log (c^2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 8*(6*b^6*x^3*log(c)*...
Leaf count of result is larger than twice the leaf count of optimal. 1640 vs. \(2 (210) = 420\).
Time = 0.31 (sec) , antiderivative size = 1640, normalized size of antiderivative = 7.19 \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\text {Too large to display} \]
1/8*(36*b^2*c^2*arccos(1/(c*x))^2*cos(a/b)^2*cos_integral(3*a/b + 3*arccos (1/(c*x)))*sin(a/b)/(b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2 *b^3) - 36*b^2*c^2*arccos(1/(c*x))^2*cos(a/b)^3*sin_integral(3*a/b + 3*arc cos(1/(c*x)))/(b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3) + 72*a*b*c^2*arccos(1/(c*x))*cos(a/b)^2*cos_integral(3*a/b + 3*arccos(1/(c *x)))*sin(a/b)/(b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3) - 72*a*b*c^2*arccos(1/(c*x))*cos(a/b)^3*sin_integral(3*a/b + 3*arccos(1/( c*x)))/(b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3) - 9*b^2 *c^2*arccos(1/(c*x))^2*cos_integral(3*a/b + 3*arccos(1/(c*x)))*sin(a/b)/(b ^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3) + 36*a^2*c^2*cos (a/b)^2*cos_integral(3*a/b + 3*arccos(1/(c*x)))*sin(a/b)/(b^5*arccos(1/(c* x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3) + b^2*c^2*arccos(1/(c*x))^2*cos _integral(a/b + arccos(1/(c*x)))*sin(a/b)/(b^5*arccos(1/(c*x))^2 + 2*a*b^4 *arccos(1/(c*x)) + a^2*b^3) + 27*b^2*c^2*arccos(1/(c*x))^2*cos(a/b)*sin_in tegral(3*a/b + 3*arccos(1/(c*x)))/(b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos( 1/(c*x)) + a^2*b^3) - 36*a^2*c^2*cos(a/b)^3*sin_integral(3*a/b + 3*arccos( 1/(c*x)))/(b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3) - b^ 2*c^2*arccos(1/(c*x))^2*cos(a/b)*sin_integral(a/b + arccos(1/(c*x)))/(b^5* arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3) - 18*a*b*c^2*arccos (1/(c*x))*cos_integral(3*a/b + 3*arccos(1/(c*x)))*sin(a/b)/(b^5*arccos(...
Timed out. \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^4\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3} \,d x \]