\(\int x^5 (d+e x^2) (a+b \sec ^{-1}(c x)) \, dx\) [76]

Optimal result

Integrand size = 19, antiderivative size = 196 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx=-\frac {b \left (4 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}-\frac {b \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}-\frac {b \left (4 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}-\frac {b e x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right ) \]

[Out]

1/6*d*x^6*(a+b*arcsec(c*x))+1/8*e*x^8*(a+b*arcsec(c*x))-1/72*b*(8*c^2*d+9*e)*x*(c^2*x^2-1)^(3/2)/c^7/(c^2*x^2)
^(1/2)-1/120*b*(4*c^2*d+9*e)*x*(c^2*x^2-1)^(5/2)/c^7/(c^2*x^2)^(1/2)-1/56*b*e*x*(c^2*x^2-1)^(7/2)/c^7/(c^2*x^2
)^(1/2)-1/24*b*(4*c^2*d+3*e)*x*(c^2*x^2-1)^(1/2)/c^7/(c^2*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 5346, 12, 457, 78} \[ \int x^5 \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {b x \left (c^2 x^2-1\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^7 \sqrt {c^2 x^2}}-\frac {b x \left (c^2 x^2-1\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^7 \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (4 c^2 d+3 e\right )}{24 c^7 \sqrt {c^2 x^2}}-\frac {b e x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}} \]

[In]

Int[x^5*(d + e*x^2)*(a + b*ArcSec[c*x]),x]

[Out]

-1/24*(b*(4*c^2*d + 3*e)*x*Sqrt[-1 + c^2*x^2])/(c^7*Sqrt[c^2*x^2]) - (b*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(3/2)
)/(72*c^7*Sqrt[c^2*x^2]) - (b*(4*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7*Sqrt[c^2*x^2]) - (b*e*x*(-1 + c
^2*x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (d*x^6*(a + b*ArcSec[c*x]))/6 + (e*x^8*(a + b*ArcSec[c*x]))/8

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5346

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSec[c*x], u, x] - Dist[b*c*(x/Sqrt[c^2*x^2]), Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{24 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{24 \sqrt {c^2 x^2}} \\ & = \frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \text {Subst}\left (\int \frac {x^2 (4 d+3 e x)}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}} \\ & = \frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \text {Subst}\left (\int \left (\frac {4 c^2 d+3 e}{c^6 \sqrt {-1+c^2 x}}+\frac {\left (8 c^2 d+9 e\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {\left (4 c^2 d+9 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {3 e \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}} \\ & = -\frac {b \left (4 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}-\frac {b \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}-\frac {b \left (4 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}-\frac {b e x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.60 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{24} a x^6 \left (4 d+3 e x^2\right )-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (144 e+8 c^2 \left (28 d+9 e x^2\right )+2 c^4 \left (56 d x^2+27 e x^4\right )+c^6 \left (84 d x^4+45 e x^6\right )\right )}{2520 c^7}+\frac {1}{24} b x^6 \left (4 d+3 e x^2\right ) \sec ^{-1}(c x) \]

[In]

Integrate[x^5*(d + e*x^2)*(a + b*ArcSec[c*x]),x]

[Out]

(a*x^6*(4*d + 3*e*x^2))/24 - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(144*e + 8*c^2*(28*d + 9*e*x^2) + 2*c^4*(56*d*x^2 + 27
*e*x^4) + c^6*(84*d*x^4 + 45*e*x^6)))/(2520*c^7) + (b*x^6*(4*d + 3*e*x^2)*ArcSec[c*x])/24

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.71

method	result	size
parts	\(a \left (\frac {1}{8} e \,x^{8}+\frac {1}{6} d \,x^{6}\right )+\frac {b \left (\frac {c^{6} \operatorname {arcsec}\left (c x \right ) e \,x^{8}}{8}+\frac {\operatorname {arcsec}\left (c x \right ) d \,x^{6} c^{6}}{6}-\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )}{c^{6}}\)	\(139\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arcsec}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\operatorname {arcsec}\left (c x \right ) e \,c^{8} x^{8}}{8}-\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}}\)	\(152\)
default	\(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arcsec}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\operatorname {arcsec}\left (c x \right ) e \,c^{8} x^{8}}{8}-\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}}\)	\(152\)

[In]

int(x^5*(e*x^2+d)*(a+b*arcsec(c*x)),x,method=_RETURNVERBOSE)

[Out]

a*(1/8*e*x^8+1/6*d*x^6)+b/c^6*(1/8*c^6*arcsec(c*x)*e*x^8+1/6*arcsec(c*x)*d*x^6*c^6-1/2520/c^3*(c^2*x^2-1)*(45*
c^6*e*x^6+84*c^6*d*x^4+54*c^4*e*x^4+112*c^4*d*x^2+72*c^2*e*x^2+224*c^2*d+144*e)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.65 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {315 \, a c^{8} e x^{8} + 420 \, a c^{8} d x^{6} + 105 \, {\left (3 \, b c^{8} e x^{8} + 4 \, b c^{8} d x^{6}\right )} \operatorname {arcsec}\left (c x\right ) - {\left (45 \, b c^{6} e x^{6} + 6 \, {\left (14 \, b c^{6} d + 9 \, b c^{4} e\right )} x^{4} + 224 \, b c^{2} d + 8 \, {\left (14 \, b c^{4} d + 9 \, b c^{2} e\right )} x^{2} + 144 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{2520 \, c^{8}} \]

[In]

integrate(x^5*(e*x^2+d)*(a+b*arcsec(c*x)),x, algorithm="fricas")

[Out]

1/2520*(315*a*c^8*e*x^8 + 420*a*c^8*d*x^6 + 105*(3*b*c^8*e*x^8 + 4*b*c^8*d*x^6)*arcsec(c*x) - (45*b*c^6*e*x^6
+ 6*(14*b*c^6*d + 9*b*c^4*e)*x^4 + 224*b*c^2*d + 8*(14*b*c^4*d + 9*b*c^2*e)*x^2 + 144*b*e)*sqrt(c^2*x^2 - 1))/
c^8

Sympy [A] (verification not implemented)

Time = 4.04 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.86 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {a d x^{6}}{6} + \frac {a e x^{8}}{8} + \frac {b d x^{6} \operatorname {asec}{\left (c x \right )}}{6} + \frac {b e x^{8} \operatorname {asec}{\left (c x \right )}}{8} - \frac {b d \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} - \frac {b e \left (\begin {cases} \frac {x^{6} \sqrt {c^{2} x^{2} - 1}}{7 c} + \frac {6 x^{4} \sqrt {c^{2} x^{2} - 1}}{35 c^{3}} + \frac {8 x^{2} \sqrt {c^{2} x^{2} - 1}}{35 c^{5}} + \frac {16 \sqrt {c^{2} x^{2} - 1}}{35 c^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{6} \sqrt {- c^{2} x^{2} + 1}}{7 c} + \frac {6 i x^{4} \sqrt {- c^{2} x^{2} + 1}}{35 c^{3}} + \frac {8 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{5}} + \frac {16 i \sqrt {- c^{2} x^{2} + 1}}{35 c^{7}} & \text {otherwise} \end {cases}\right )}{8 c} \]

[In]

integrate(x**5*(e*x**2+d)*(a+b*asec(c*x)),x)

[Out]

a*d*x**6/6 + a*e*x**8/8 + b*d*x**6*asec(c*x)/6 + b*e*x**8*asec(c*x)/8 - b*d*Piecewise((x**4*sqrt(c**2*x**2 - 1
)/(5*c) + 4*x**2*sqrt(c**2*x**2 - 1)/(15*c**3) + 8*sqrt(c**2*x**2 - 1)/(15*c**5), Abs(c**2*x**2) > 1), (I*x**4
*sqrt(-c**2*x**2 + 1)/(5*c) + 4*I*x**2*sqrt(-c**2*x**2 + 1)/(15*c**3) + 8*I*sqrt(-c**2*x**2 + 1)/(15*c**5), Tr
ue))/(6*c) - b*e*Piecewise((x**6*sqrt(c**2*x**2 - 1)/(7*c) + 6*x**4*sqrt(c**2*x**2 - 1)/(35*c**3) + 8*x**2*sqr
t(c**2*x**2 - 1)/(35*c**5) + 16*sqrt(c**2*x**2 - 1)/(35*c**7), Abs(c**2*x**2) > 1), (I*x**6*sqrt(-c**2*x**2 +
1)/(7*c) + 6*I*x**4*sqrt(-c**2*x**2 + 1)/(35*c**3) + 8*I*x**2*sqrt(-c**2*x**2 + 1)/(35*c**5) + 16*I*sqrt(-c**2
*x**2 + 1)/(35*c**7), True))/(8*c)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.94 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{8} \, a e x^{8} + \frac {1}{6} \, a d x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arcsec}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arcsec}\left (c x\right ) - \frac {5 \, c^{6} x^{7} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} + 21 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 35 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e \]

[In]

integrate(x^5*(e*x^2+d)*(a+b*arcsec(c*x)),x, algorithm="maxima")

[Out]

1/8*a*e*x^8 + 1/6*a*d*x^6 + 1/90*(15*x^6*arcsec(c*x) - (3*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 10*c^2*x^3*(-1/(c
^2*x^2) + 1)^(3/2) + 15*x*sqrt(-1/(c^2*x^2) + 1))/c^5)*b*d + 1/280*(35*x^8*arcsec(c*x) - (5*c^6*x^7*(-1/(c^2*x
^2) + 1)^(7/2) + 21*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 35*c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 35*x*sqrt(-1/(c^2
*x^2) + 1))/c^7)*b*e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 13018 vs. \(2 (168) = 336\).

Time = 0.47 (sec) , antiderivative size = 13018, normalized size of antiderivative = 66.42 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]

[In]

integrate(x^5*(e*x^2+d)*(a+b*arcsec(c*x)),x, algorithm="giac")

[Out]

1/2520*(420*b*c^2*d*arccos(1/(c*x))/(c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^
2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 +
56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2)
- 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16) + 420*a*c^2*d/(c^9 + 8*c^9*(1/(c^2*x^2) -
1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6
+ 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^
2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1
)^16) - 1680*b*c^2*d*(1/(c^2*x^2) - 1)*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^
9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^
4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12
+ 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^2) - 84
0*b*c^2*d*sqrt(-1/(c^2*x^2) + 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/
(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56
*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) -
1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)) - 1680*a*c^2*d*(1/(c^2*x^2)
- 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/
(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/
(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9
*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^2) + 315*b*e*arccos(1/(c*x))/(c^9 + 8*c^9*(1/(c^2*x^2) -
1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 +
70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2
) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)
^16) + 1680*b*c^2*d*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c
^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)
^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12
+ 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^4) + 3
640*b*c^2*d*(-1/(c^2*x^2) + 1)^(3/2)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1
)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8
+ 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^
2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^3) + 315*a*e/(c^9 + 8*c^9
*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(
1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28
*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)
^8/(1/(c*x) + 1)^16) + 1680*a*c^2*d*(1/(c^2*x^2) - 1)^2/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c
^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)
^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12
+ 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^4) - 2
520*b*e*(1/(c^2*x^2) - 1)*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2
) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) +
1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c
^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^2) + 1680*b*c^2*d*(1
/(c^2*x^2) - 1)^3*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2
/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 5
6*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2)
- 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^6) - 630*b*e*sqrt(-1/(c^2*x^
2) + 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*
(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/
(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 +
c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)) - 9128*b*c^2*d*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2)
+ 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/
(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/
(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9
*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^5) - 2520*a*e*(1/(c^2*x^2) - 1)/((c^9 + 8*c^9*(1/(c^2*x^2
) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)
^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2
*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x)
+ 1)^16)*(1/(c*x) + 1)^2) + 1680*a*c^2*d*(1/(c^2*x^2) - 1)^3/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 +
28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2)
- 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) +
1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^6
) + 8820*b*e*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(
c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(
c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^
9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^4) - 4200*b*c
^2*d*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2)
- 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1
)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^
2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^8) - 15064*b*c^2*d*(1
/(c^2*x^2) - 1)^3*sqrt(-1/(c^2*x^2) + 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2)
- 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1
)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^
2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^7) + 1890*b*e*(-1/(c^
2*x^2) + 1)^(3/2)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4
+ 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^
2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x)
+ 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^3) + 8820*a*e*(1/(c^2*x^2) - 1)^2/((c^9 + 8*
c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^
3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 +
28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) -
1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^4) - 4200*a*c^2*d*(1/(c^2*x^2) - 1)^4/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1
/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c
^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1
)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*
(1/(c*x) + 1)^8) - 17640*b*e*(1/(c^2*x^2) - 1)^3*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)
^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*
x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x
) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) +
1)^6) + 1680*b*c^2*d*(1/(c^2*x^2) - 1)^5*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*
c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1
)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^1
2 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^10) -
6678*b*e*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*
(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/
(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 +
8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^5) - 1506
4*b*c^2*d*(1/(c^2*x^2) - 1)^4*sqrt(-1/(c^2*x^2) + 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*
(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/
(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 +
8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^9) - 1764
0*a*e*(1/(c^2*x^2) - 1)^3/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x
) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1
/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(
1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^6) + 1680*a*c^2*d*(1/(c^2*x^2) - 1)^
5/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^
2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*
x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1
/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^10) + 22050*b*e*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))/((c^9 +
8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1
)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10
+ 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2)
- 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^8) + 1680*b*c^2*d*(1/(c^2*x^2) - 1)^6*arccos(1/(c*x))/((c^9 + 8*c^9*(1
/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(
c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^
9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/
(1/(c*x) + 1)^16)*(1/(c*x) + 1)^12) - 9234*b*e*(1/(c^2*x^2) - 1)^3*sqrt(-1/(c^2*x^2) + 1)/((c^9 + 8*c^9*(1/(c^
2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x)
+ 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1
/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(
c*x) + 1)^16)*(1/(c*x) + 1)^7) - 9128*b*c^2*d*(1/(c^2*x^2) - 1)^5*sqrt(-1/(c^2*x^2) + 1)/((c^9 + 8*c^9*(1/(c^2
*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x)
+ 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/
(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c
*x) + 1)^16)*(1/(c*x) + 1)^11) + 22050*a*e*(1/(c^2*x^2) - 1)^4/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2
+ 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^
2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x)
+ 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)
^8) + 1680*a*c^2*d*(1/(c^2*x^2) - 1)^6/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) -
1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^
8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*
x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^12) - 17640*b*e*(1/(c^2
*x^2) - 1)^5*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(
c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9
*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^
7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^10) - 1680*b*c^2*d*(1/(c^2*x^2) -
1)^7*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) +
1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^
2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c
*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^14) - 9234*b*e*(1/(c^2*x^2) - 1)^4*sqrt(
-1/(c^2*x^2) + 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4
+ 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^
2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x)
+ 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^9) - 3640*b*c^2*d*(1/(c^2*x^2) - 1)^6*sqrt(-
1/(c^2*x^2) + 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4
+ 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2
) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) +
1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^13) - 17640*a*e*(1/(c^2*x^2) - 1)^5/((c^9 + 8
*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)
^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10
+ 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2)
- 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^10) - 1680*a*c^2*d*(1/(c^2*x^2) - 1)^7/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/
(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70
*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) -
1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16
)*(1/(c*x) + 1)^14) + 8820*b*e*(1/(c^2*x^2) - 1)^6*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) +
1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^
2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c
*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x)
+ 1)^12) + 420*b*c^2*d*(1/(c^2*x^2) - 1)^8*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2
8*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) -
1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)
^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^16)
- 6678*b*e*(1/(c^2*x^2) - 1)^5*sqrt(-1/(c^2*x^2) + 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^
9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^
4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12
+ 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^11) - 8
40*b*c^2*d*(1/(c^2*x^2) - 1)^7*sqrt(-1/(c^2*x^2) + 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9
*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4
/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 +
8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^15) + 88
20*a*e*(1/(c^2*x^2) - 1)^6/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*
x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(
1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/
(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^12) + 420*a*c^2*d*(1/(c^2*x^2) - 1)
^8/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c
^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c
*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(
1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^16) - 2520*b*e*(1/(c^2*x^2) - 1)^7*arccos(1/(c*x))/((c^9 +
8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1
)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10
+ 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2)
- 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^14) - 1890*b*e*(1/(c^2*x^2) - 1)^6*sqrt(-1/(c^2*x^2) + 1)/((c^9 + 8*c^
9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/
(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 2
8*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1
)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^13) - 2520*a*e*(1/(c^2*x^2) - 1)^7/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x
) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1
/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(
1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c
*x) + 1)^14) + 315*b*e*(1/(c^2*x^2) - 1)^8*arccos(1/(c*x))/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2
8*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) -
1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)
^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^16)
- 630*b*e*(1/(c^2*x^2) - 1)^7*sqrt(-1/(c^2*x^2) + 1)/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9
*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4
/(1/(c*x) + 1)^8 + 56*c^9*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 +
8*c^9*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^15) + 31
5*a*e*(1/(c^2*x^2) - 1)^8/((c^9 + 8*c^9*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 28*c^9*(1/(c^2*x^2) - 1)^2/(1/(c*x
) + 1)^4 + 56*c^9*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 70*c^9*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 56*c^9*(1
/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 28*c^9*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + 8*c^9*(1/(c^2*x^2) - 1)^7/(
1/(c*x) + 1)^14 + c^9*(1/(c^2*x^2) - 1)^8/(1/(c*x) + 1)^16)*(1/(c*x) + 1)^16))*c

Mupad [F(-1)]

Timed out. \[ \int x^5 \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int x^5\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

[In]

int(x^5*(d + e*x^2)*(a + b*acos(1/(c*x))),x)

[Out]

int(x^5*(d + e*x^2)*(a + b*acos(1/(c*x))), x)