Integrand size = 14, antiderivative size = 103 \[ \int \frac {1}{x^2 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=-\frac {c \sqrt {1-\frac {1}{c^2 x^2}}}{2 b \left (a+b \sec ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 x \left (a+b \sec ^{-1}(c x)\right )}+\frac {c \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{2 b^3}-\frac {c \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{2 b^3} \]
-1/2/b^2/x/(a+b*arcsec(c*x))-1/2*c*cos(a/b)*Si(a/b+arcsec(c*x))/b^3+1/2*c* Ci(a/b+arcsec(c*x))*sin(a/b)/b^3-1/2*c*(1-1/c^2/x^2)^(1/2)/b/(a+b*arcsec(c *x))^2
Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=-\frac {\frac {b \left (a+b c \sqrt {1-\frac {1}{c^2 x^2}} x+b \sec ^{-1}(c x)\right )}{x \left (a+b \sec ^{-1}(c x)\right )^2}-c \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )+c \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{2 b^3} \]
-1/2*((b*(a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x + b*ArcSec[c*x]))/(x*(a + b*ArcS ec[c*x])^2) - c*CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b] + c*Cos[a/b]*SinIn tegral[a/b + ArcSec[c*x]])/b^3
Time = 0.66 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5745, 3042, 3778, 3042, 3778, 25, 3042, 3784, 3042, 3780, 3783}
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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 5745 |
| \(\displaystyle c \int \frac {\sqrt {1-\frac {1}{c^2 x^2}}}{\left (a+b \sec ^{-1}(c x)\right )^3}d\sec ^{-1}(c x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \int \frac {\sin \left (\sec ^{-1}(c x)\right )}{\left (a+b \sec ^{-1}(c x)\right )^3}d\sec ^{-1}(c x)\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle c \left (\frac {\int \frac {1}{c x \left (a+b \sec ^{-1}(c x)\right )^2}d\sec ^{-1}(c x)}{2 b}-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 b \left (a+b \sec ^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (\frac {\int \frac {\sin \left (\sec ^{-1}(c x)+\frac {\pi }{2}\right )}{\left (a+b \sec ^{-1}(c x)\right )^2}d\sec ^{-1}(c x)}{2 b}-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 b \left (a+b \sec ^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle c \left (\frac {\frac {\int -\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{a+b \sec ^{-1}(c x)}d\sec ^{-1}(c x)}{b}-\frac {1}{b c x \left (a+b \sec ^{-1}(c x)\right )}}{2 b}-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 b \left (a+b \sec ^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {-\frac {\int \frac {\sqrt {1-\frac {1}{c^2 x^2}}}{a+b \sec ^{-1}(c x)}d\sec ^{-1}(c x)}{b}-\frac {1}{b c x \left (a+b \sec ^{-1}(c x)\right )}}{2 b}-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 b \left (a+b \sec ^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (\frac {-\frac {\int \frac {\sin \left (\sec ^{-1}(c x)\right )}{a+b \sec ^{-1}(c x)}d\sec ^{-1}(c x)}{b}-\frac {1}{b c x \left (a+b \sec ^{-1}(c x)\right )}}{2 b}-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 b \left (a+b \sec ^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle c \left (\frac {-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{a+b \sec ^{-1}(c x)}d\sec ^{-1}(c x)-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{a+b \sec ^{-1}(c x)}d\sec ^{-1}(c x)}{b}-\frac {1}{b c x \left (a+b \sec ^{-1}(c x)\right )}}{2 b}-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 b \left (a+b \sec ^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (\frac {-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{a+b \sec ^{-1}(c x)}d\sec ^{-1}(c x)-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a}{b}+\sec ^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \sec ^{-1}(c x)}d\sec ^{-1}(c x)}{b}-\frac {1}{b c x \left (a+b \sec ^{-1}(c x)\right )}}{2 b}-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 b \left (a+b \sec ^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle c \left (\frac {-\frac {\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b}-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a}{b}+\sec ^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \sec ^{-1}(c x)}d\sec ^{-1}(c x)}{b}-\frac {1}{b c x \left (a+b \sec ^{-1}(c x)\right )}}{2 b}-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 b \left (a+b \sec ^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle c \left (\frac {-\frac {\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b}-\frac {\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b}}{b}-\frac {1}{b c x \left (a+b \sec ^{-1}(c x)\right )}}{2 b}-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 b \left (a+b \sec ^{-1}(c x)\right )^2}\right )\) |
c*(-1/2*Sqrt[1 - 1/(c^2*x^2)]/(b*(a + b*ArcSec[c*x])^2) + (-(1/(b*c*x*(a + b*ArcSec[c*x]))) - (-((CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b])/b) + (Cos [a/b]*SinIntegral[a/b + ArcSec[c*x]])/b)/b)/(2*b))
3.1.48.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 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 = 0.47 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.50
| method | result | size |
| derivativedivides | \(c \left (-\frac {\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{2 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )^{2} b}-\frac {\operatorname {arcsec}\left (c x \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) b c x -\operatorname {arcsec}\left (c x \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) b c x +\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) a c x -\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) a c x +b}{2 c x \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b^{3}}\right )\) | \(154\) |
| default | \(c \left (-\frac {\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{2 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )^{2} b}-\frac {\operatorname {arcsec}\left (c x \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) b c x -\operatorname {arcsec}\left (c x \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) b c x +\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) a c x -\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) a c x +b}{2 c x \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b^{3}}\right )\) | \(154\) |
c*(-1/2*((c^2*x^2-1)/c^2/x^2)^(1/2)/(a+b*arcsec(c*x))^2/b-1/2*(arcsec(c*x) *cos(a/b)*Si(a/b+arcsec(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*arcsec(c*x))/b^3)
\[ \int \frac {1}{x^2 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3} x^{2}} \,d x } \]
integral(1/(b^3*x^2*arcsec(c*x)^3 + 3*a*b^2*x^2*arcsec(c*x)^2 + 3*a^2*b*x^ 2*arcsec(c*x) + a^3*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^{2} \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}}\, dx \]
\[ \int \frac {1}{x^2 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3} x^{2}} \,d x } \]
-(8*b^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^3 + 24*a*b^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 2*a*b^2*log(c^2*x^2)^2 + 8*a*b^2*log(c)^2 + 16*a*b^ 2*log(c)*log(x) + 8*a*b^2*log(x)^2 + 8*a^3 + 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*(b^3*log(c^2*x^2)^2 + 4*b^3*log(c)^2 + 8*b^3*log(c)*log(x) + 4*b^3*lo g(x)^2 + 12*a^2*b - 4*(b^3*log(c) + b^3*log(x))*log(c^2*x^2))*arctan(sqrt( c*x + 1)*sqrt(c*x - 1)) + (16*b^6*x*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^4 + b^6*x*log(c^2*x^2)^4 + 64*b^6*x*log(c)*log(x)^3 + 16*b^6*x*log(x)^4 + 64 *a*b^5*x*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 8*(b^6*x*log(c) + b^6*x*l og(x))*log(c^2*x^2)^3 + 32*(3*b^6*log(c)^2 + a^2*b^4)*x*log(x)^2 + 8*(b^6* x*log(c^2*x^2)^2 + 8*b^6*x*log(c)*log(x) + 4*b^6*x*log(x)^2 + 4*(b^6*log(c )^2 + 3*a^2*b^4)*x - 4*(b^6*x*log(c) + b^6*x*log(x))*log(c^2*x^2))*arctan( sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 8*(6*b^6*x*log(c)*log(x) + 3*b^6*x*log(x) ^2 + (3*b^6*log(c)^2 + a^2*b^4)*x)*log(c^2*x^2)^2 + 64*(b^6*log(c)^3 + a^2 *b^4*log(c))*x*log(x) + 16*(b^6*log(c)^4 + 2*a^2*b^4*log(c)^2 + a^4*b^2)*x + 16*(a*b^5*x*log(c^2*x^2)^2 + 8*a*b^5*x*log(c)*log(x) + 4*a*b^5*x*log(x) ^2 + 4*(a*b^5*log(c)^2 + a^3*b^3)*x - 4*(a*b^5*x*log(c) + a*b^5*x*log(x))* log(c^2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - 32*(3*b^6*x*log(c)*...
Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (93) = 186\).
Time = 0.31 (sec) , antiderivative size = 580, normalized size of antiderivative = 5.63 \[ \int \frac {1}{x^2 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\frac {1}{2} \, {\left (\frac {b^{2} \arccos \left (\frac {1}{c x}\right )^{2} \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{5} \arccos \left (\frac {1}{c x}\right )^{2} + 2 \, a b^{4} \arccos \left (\frac {1}{c x}\right ) + a^{2} b^{3}} - \frac {b^{2} \arccos \left (\frac {1}{c x}\right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b^{5} \arccos \left (\frac {1}{c x}\right )^{2} + 2 \, a b^{4} \arccos \left (\frac {1}{c x}\right ) + a^{2} b^{3}} + \frac {2 \, a b \arccos \left (\frac {1}{c x}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{5} \arccos \left (\frac {1}{c x}\right )^{2} + 2 \, a b^{4} \arccos \left (\frac {1}{c x}\right ) + a^{2} b^{3}} - \frac {2 \, a b \arccos \left (\frac {1}{c x}\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b^{5} \arccos \left (\frac {1}{c x}\right )^{2} + 2 \, a b^{4} \arccos \left (\frac {1}{c x}\right ) + a^{2} b^{3}} + \frac {a^{2} \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{5} \arccos \left (\frac {1}{c x}\right )^{2} + 2 \, a b^{4} \arccos \left (\frac {1}{c x}\right ) + a^{2} b^{3}} - \frac {a^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b^{5} \arccos \left (\frac {1}{c x}\right )^{2} + 2 \, a b^{4} \arccos \left (\frac {1}{c x}\right ) + a^{2} b^{3}} - \frac {b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{b^{5} \arccos \left (\frac {1}{c x}\right )^{2} + 2 \, a b^{4} \arccos \left (\frac {1}{c x}\right ) + a^{2} b^{3}} - \frac {b^{2} \arccos \left (\frac {1}{c x}\right )}{{\left (b^{5} \arccos \left (\frac {1}{c x}\right )^{2} + 2 \, a b^{4} \arccos \left (\frac {1}{c x}\right ) + a^{2} b^{3}\right )} c x} - \frac {a b}{{\left (b^{5} \arccos \left (\frac {1}{c x}\right )^{2} + 2 \, a b^{4} \arccos \left (\frac {1}{c x}\right ) + a^{2} b^{3}\right )} c x}\right )} c \]
1/2*(b^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) - b^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) + 2*a*b*arccos(1/(c*x))*cos_integra l(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) - 2*a*b*arccos(1/(c*x))*cos(a/b)*sin_integral(a/b + ar ccos(1/(c*x)))/(b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3) + a^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) - a^2*cos(a/b)*sin_integral(a/b + ar ccos(1/(c*x)))/(b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3) - b^2*sqrt(-1/(c^2*x^2) + 1)/(b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c *x)) + a^2*b^3) - b^2*arccos(1/(c*x))/((b^5*arccos(1/(c*x))^2 + 2*a*b^4*ar ccos(1/(c*x)) + a^2*b^3)*c*x) - a*b/((b^5*arccos(1/(c*x))^2 + 2*a*b^4*arcc os(1/(c*x)) + a^2*b^3)*c*x))*c
Timed out. \[ \int \frac {1}{x^2 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3} \,d x \]