\(\int \frac {1}{x^4 (a+b \sec ^{-1}(c x))^3} \, dx\) [50]

Optimal result

Integrand size = 14, antiderivative size = 228 \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=-\frac {c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{8 b \left (a+b \sec ^{-1}(c x)\right )^2}-\frac {c^2}{8 b^2 x \left (a+b \sec ^{-1}(c x)\right )}-\frac {3 c^3 \cos \left (3 \sec ^{-1}(c x)\right )}{8 b^2 \left (a+b \sec ^{-1}(c x)\right )}+\frac {c^3 \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{8 b^3}+\frac {9 c^3 \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right ) \sin \left (\frac {3 a}{b}\right )}{8 b^3}-\frac {c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{8 b \left (a+b \sec ^{-1}(c x)\right )^2}-\frac {c^3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{8 b^3}-\frac {9 c^3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{8 b^3} \]

[Out]

-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*arcsec(c*x))-1/8*c^3*cos(a/b)*Si(a/b+arcs
ec(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*C
i(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*arcsec(c*x))^2-1/8*c^3*(1-1/c^2/x^2)^(
1/2)/b/(a+b*arcsec(c*x))^2

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5330, 4491, 3378, 3384, 3380, 3383} \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\frac {c^3 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{8 b^3}+\frac {9 c^3 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{8 b^3}-\frac {c^3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{8 b^3}-\frac {9 c^3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{8 b^3}-\frac {3 c^3 \cos \left (3 \sec ^{-1}(c x)\right )}{8 b^2 \left (a+b \sec ^{-1}(c x)\right )}-\frac {c^2}{8 b^2 x \left (a+b \sec ^{-1}(c x)\right )}-\frac {c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{8 b \left (a+b \sec ^{-1}(c x)\right )^2}-\frac {c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{8 b \left (a+b \sec ^{-1}(c x)\right )^2} \]

[In]

Int[1/(x^4*(a + b*ArcSec[c*x])^3),x]

[Out]

-1/8*(c^3*Sqrt[1 - 1/(c^2*x^2)])/(b*(a + b*ArcSec[c*x])^2) - c^2/(8*b^2*x*(a + b*ArcSec[c*x])) - (3*c^3*Cos[3*
ArcSec[c*x]])/(8*b^2*(a + b*ArcSec[c*x])) + (c^3*CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b])/(8*b^3) + (9*c^3*Cos
Integral[(3*a)/b + 3*ArcSec[c*x]]*Sin[(3*a)/b])/(8*b^3) - (c^3*Sin[3*ArcSec[c*x]])/(8*b*(a + b*ArcSec[c*x])^2)
- (c^3*Cos[a/b]*SinIntegral[a/b + ArcSec[c*x]])/(8*b^3) - (9*c^3*Cos[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSec[
c*x]])/(8*b^3)

Rule 3378

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] - Dist[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]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[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]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rule 5330

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
ec[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])

Rubi steps \begin{align*} \text {integral}& = c^3 \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{(a+b x)^3} \, dx,x,\sec ^{-1}(c x)\right ) \\ & = c^3 \text {Subst}\left (\int \left (\frac {\sin (x)}{4 (a+b x)^3}+\frac {\sin (3 x)}{4 (a+b x)^3}\right ) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {1}{4} c^3 \text {Subst}\left (\int \frac {\sin (x)}{(a+b x)^3} \, dx,x,\sec ^{-1}(c x)\right )+\frac {1}{4} c^3 \text {Subst}\left (\int \frac {\sin (3 x)}{(a+b x)^3} \, dx,x,\sec ^{-1}(c x)\right ) \\ & = -\frac {c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{8 b \left (a+b \sec ^{-1}(c x)\right )^2}-\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 \text {Subst}\left (\int \frac {\cos (x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )}{8 b}+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )}{8 b} \\ & = -\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 \sin \left (3 \sec ^{-1}(c x)\right )}{8 b \left (a+b \sec ^{-1}(c x)\right )^2}-\frac {c^3 \text {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{8 b^2}-\frac {\left (9 c^3\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{8 b^2} \\ & = -\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 \sin \left (3 \sec ^{-1}(c x)\right )}{8 b \left (a+b \sec ^{-1}(c x)\right )^2}-\frac {\left (c^3 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{8 b^2}-\frac {\left (9 c^3 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{8 b^2}+\frac {\left (c^3 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{8 b^2}+\frac {\left (9 c^3 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{8 b^2} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\frac {-\frac {4 b^2 c \sqrt {1-\frac {1}{c^2 x^2}}}{x^2 \left (a+b \sec ^{-1}(c x)\right )^2}-\frac {12 b}{x^3 \left (a+b \sec ^{-1}(c x)\right )}+\frac {8 b c^2}{a x+b x \sec ^{-1}(c x)}+c^3 \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )+9 c^3 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-c^3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )-9 c^3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )}{8 b^3} \]

[In]

Integrate[1/(x^4*(a + b*ArcSec[c*x])^3),x]

[Out]

((-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*CosIntegral[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)

Maple [A] (verified)

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	\(c^{3} \left (-\frac {\sin \left (3 \,\operatorname {arcsec}\left (c x \right )\right )}{8 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )^{2} b}-\frac {3 \left (3 \,\operatorname {arcsec}\left (c x \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) b -3 \,\operatorname {arcsec}\left (c x \right ) \sin \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) b +3 \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) a -3 \sin \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) a +\cos \left (3 \,\operatorname {arcsec}\left (c x \right )\right ) b \right )}{8 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b^{3}}-\frac {\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )^{2} b}-\frac {\operatorname {arcsec}\left (c x \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) b c x -\operatorname {arcsec}\left (c x \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) b c x +\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) a c x -\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) a c x +b}{8 c x \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b^{3}}\right )\)	\(307\)

[In]

int(1/x^4/(a+b*arcsec(c*x))^3,x,method=_RETURNVERBOSE)

[Out]

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*a
rcsec(c*x)*sin(3*a/b)*Ci(3*a/b+3*arcsec(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
*arcsec(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+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)

Fricas [F]

\[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3} x^{4}} \,d x } \]

[In]

integrate(1/x^4/(a+b*arcsec(c*x))^3,x, algorithm="fricas")

[Out]

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)

Sympy [F]

\[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^{4} \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}}\, dx \]

[In]

integrate(1/x**4/(a+b*asec(c*x))**3,x)

[Out]

Integral(1/(x**4*(a + b*asec(c*x))**3), x)

Maxima [F]

\[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3} x^{4}} \,d x } \]

[In]

integrate(1/x^4/(a+b*arcsec(c*x))^3,x, algorithm="maxima")

[Out]

-(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(sqr
t(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*arctan(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*log(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)*log(x) + 3*b^6*x^3*log(x)^2 + (3*b^6*log(c)^2 + a^2*b^4)*x^3)*l
og(c^2*x^2)^2 + 16*(a*b^5*x^3*log(c^2*x^2)^2 + 8*a*b^5*x^3*log(c)*log(x) + 4*a*b^5*x^3*log(x)^2 + 4*(a*b^5*log
(c)^2 + a^3*b^3)*x^3 - 4*(a*b^5*x^3*log(c) + a*b^5*x^3*log(x))*log(c^2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1
)) - 32*(3*b^6*x^3*log(c)*log(x)^2 + b^6*x^3*log(x)^3 + (3*b^6*log(c)^2 + a^2*b^4)*x^3*log(x) + (b^6*log(c)^3
+ a^2*b^4*log(c))*x^3)*log(c^2*x^2))*integrate(2*(2*a*c^2*x^2 + (2*b*c^2*x^2 - 9*b)*arctan(sqrt(c*x + 1)*sqrt(
c*x - 1)) - 9*a)/(4*b^4*x^4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + b^4*x^4*log(c^2*x^2)^2 + 8*b^4*x^4*log(c)*
log(x) + 4*b^4*x^4*log(x)^2 + 8*a*b^3*x^4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*(b^4*log(c)^2 + a^2*b^2)*x^4
- 4*(b^4*x^4*log(c) + b^4*x^4*log(x))*log(c^2*x^2)), x) + 8*(2*a*b^2*c^2*x^2*log(c) - 3*a*b^2*log(c) + (2*a*b
^2*c^2*x^2 - 3*a*b^2)*log(x))*log(c^2*x^2) - 16*(2*a*b^2*c^2*x^2*log(c) - 3*a*b^2*log(c))*log(x))/(16*b^6*x^3*
arctan(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*log(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*lo
g(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)*log(x) + 3*b^6*x^3*log(x)^2 + (3*b^6*log(c)^2 + a^2*b^4)*x^3)*log(c^2*
x^2)^2 + 16*(a*b^5*x^3*log(c^2*x^2)^2 + 8*a*b^5*x^3*log(c)*log(x) + 4*a*b^5*x^3*log(x)^2 + 4*(a*b^5*log(c)^2 +
a^3*b^3)*x^3 - 4*(a*b^5*x^3*log(c) + a*b^5*x^3*log(x))*log(c^2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - 32
*(3*b^6*x^3*log(c)*log(x)^2 + b^6*x^3*log(x)^3 + (3*b^6*log(c)^2 + a^2*b^4)*x^3*log(x) + (b^6*log(c)^3 + a^2*b
^4*log(c))*x^3)*log(c^2*x^2))

Giac [B] (verification not implemented)

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 \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/x^4/(a+b*arcsec(c*x))^3,x, algorithm="giac")

[Out]

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*a
rccos(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_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) - 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(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3) + 2*a*b*c^2*arc
cos(1/(c*x))*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) + 54*a*b*c^2*arccos(1/(c*x))*cos(a/b)*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) - 2*a*b*c^2*arccos(1/(c*x))*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) - 9*a^2*c^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) + a^2*c^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*a^2*c^2*cos(a/b)*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) - a^2*c^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) + 8*b^2*c*arccos(1
/(c*x))/((b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3)*x) + 8*a*b*c/((b^5*arccos(1/(c*x))^2 + 2*
a*b^4*arccos(1/(c*x)) + a^2*b^3)*x) - 4*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)*x^2) - 12*b^2*arccos(1/(c*x))/((b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3)*c
*x^3) - 12*a*b/((b^5*arccos(1/(c*x))^2 + 2*a*b^4*arccos(1/(c*x)) + a^2*b^3)*c*x^3))*c

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^4\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3} \,d x \]

[In]

int(1/(x^4*(a + b*acos(1/(c*x)))^3),x)

[Out]

int(1/(x^4*(a + b*acos(1/(c*x)))^3), x)