\(\int x^3 (d+e x^2)^2 (a+b \sec ^{-1}(c x)) \, dx\) [87]

Optimal result

Integrand size = 21, antiderivative size = 242 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=-\frac {b \left (6 c^4 d^2+8 c^2 d e+3 e^2\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}-\frac {b \left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}-\frac {b e \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}-\frac {b e^2 x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right ) \]

[Out]

1/4*d^2*x^4*(a+b*arcsec(c*x))+1/3*d*e*x^6*(a+b*arcsec(c*x))+1/8*e^2*x^8*(a+b*arcsec(c*x))-1/72*b*(6*c^4*d^2+16
*c^2*d*e+9*e^2)*x*(c^2*x^2-1)^(3/2)/c^7/(c^2*x^2)^(1/2)-1/120*b*e*(8*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^2*x*(c^2*x^2-1)^(7/2)/c^7/(c^2*x^2)^(1/2)-1/24*b*(6*c^4*d^2+8*c^2*d*e+3*e^2)*x*(c^2*x^2-1)^
(1/2)/c^7/(c^2*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {272, 45, 5346, 12, 1265, 785} \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {b e x \left (c^2 x^2-1\right )^{5/2} \left (8 c^2 d+9 e\right )}{120 c^7 \sqrt {c^2 x^2}}-\frac {b e^2 x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}-\frac {b x \left (c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2+16 c^2 d e+9 e^2\right )}{72 c^7 \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (6 c^4 d^2+8 c^2 d e+3 e^2\right )}{24 c^7 \sqrt {c^2 x^2}} \]

[In]

Int[x^3*(d + e*x^2)^2*(a + b*ArcSec[c*x]),x]

[Out]

-1/24*(b*(6*c^4*d^2 + 8*c^2*d*e + 3*e^2)*x*Sqrt[-1 + c^2*x^2])/(c^7*Sqrt[c^2*x^2]) - (b*(6*c^4*d^2 + 16*c^2*d*
e + 9*e^2)*x*(-1 + c^2*x^2)^(3/2))/(72*c^7*Sqrt[c^2*x^2]) - (b*e*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*
c^7*Sqrt[c^2*x^2]) - (b*e^2*x*(-1 + c^2*x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (d^2*x^4*(a + b*ArcSec[c*x]))/4 +
(d*e*x^6*(a + b*ArcSec[c*x]))/3 + (e^2*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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

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}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{24 \sqrt {c^2 x^2}} \\ & = \frac {1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \text {Subst}\left (\int \frac {x \left (6 d^2+8 d e x+3 e^2 x^2\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}} \\ & = \frac {1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \text {Subst}\left (\int \left (\frac {6 c^4 d^2+8 c^2 d e+3 e^2}{c^6 \sqrt {-1+c^2 x}}+\frac {\left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {e \left (8 c^2 d+9 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {3 e^2 \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 (6 c^4 d^2+8 c^2 d e+3 e^2\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}-\frac {b \left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}-\frac {b e \left (8 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^2 x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.67 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{24} a x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (144 e^2+8 c^2 e \left (56 d+9 e x^2\right )+c^4 \left (420 d^2+224 d e x^2+54 e^2 x^4\right )+3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )\right )}{2520 c^7}+\frac {1}{24} b x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \sec ^{-1}(c x) \]

[In]

Integrate[x^3*(d + e*x^2)^2*(a + b*ArcSec[c*x]),x]

[Out]

(a*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4))/24 - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(144*e^2 + 8*c^2*e*(56*d + 9*e*x^2) +
c^4*(420*d^2 + 224*d*e*x^2 + 54*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/(2520*c^7) + (b*x^4*
(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcSec[c*x])/24

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.82

method	result	size
parts	\(a \left (\frac {1}{8} e^{2} x^{8}+\frac {1}{3} d e \,x^{6}+\frac {1}{4} x^{4} d^{2}\right )+\frac {b \left (\frac {c^{4} \operatorname {arcsec}\left (c x \right ) e^{2} x^{8}}{8}+\frac {c^{4} \operatorname {arcsec}\left (c x \right ) d e \,x^{6}}{3}+\frac {\operatorname {arcsec}\left (c x \right ) d^{2} x^{4} c^{4}}{4}-\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e^{2} x^{6}+168 c^{6} d e \,x^{4}+210 c^{6} d^{2} x^{2}+54 c^{4} e^{2} x^{4}+224 c^{4} d e \,x^{2}+420 c^{4} d^{2}+72 c^{2} e^{2} x^{2}+448 c^{2} d e +144 e^{2}\right )}{2520 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )}{c^{4}}\)	\(198\)
derivativedivides	\(\frac {-\frac {a \left (\frac {c^{2} d \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}-\frac {b \,c^{4} \operatorname {arcsec}\left (c x \right ) d^{4}}{24 e^{2}}+\frac {b \,\operatorname {arcsec}\left (c x \right ) d^{2} c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\operatorname {arcsec}\left (c x \right ) d \,x^{6}}{3}+\frac {b \,c^{4} e^{2} \operatorname {arcsec}\left (c x \right ) x^{8}}{8}-\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{24 e^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {b \left (c^{2} x^{2}-1\right ) c x \,d^{2}}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b c e \left (c^{2} x^{2}-1\right ) x^{3} d}{15 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b c \,e^{2} \left (c^{2} x^{2}-1\right ) x^{5}}{56 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {4 b e \left (c^{2} x^{2}-1\right ) x d}{45 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \,e^{2} \left (c^{2} x^{2}-1\right ) x^{3}}{140 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {8 b e \left (c^{2} x^{2}-1\right ) d}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{35 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {2 b \,e^{2} \left (c^{2} x^{2}-1\right )}{35 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{4}}\)	\(511\)
default	\(\frac {-\frac {a \left (\frac {c^{2} d \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}-\frac {b \,c^{4} \operatorname {arcsec}\left (c x \right ) d^{4}}{24 e^{2}}+\frac {b \,\operatorname {arcsec}\left (c x \right ) d^{2} c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\operatorname {arcsec}\left (c x \right ) d \,x^{6}}{3}+\frac {b \,c^{4} e^{2} \operatorname {arcsec}\left (c x \right ) x^{8}}{8}-\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{24 e^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {b \left (c^{2} x^{2}-1\right ) c x \,d^{2}}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b c e \left (c^{2} x^{2}-1\right ) x^{3} d}{15 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b c \,e^{2} \left (c^{2} x^{2}-1\right ) x^{5}}{56 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {4 b e \left (c^{2} x^{2}-1\right ) x d}{45 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \,e^{2} \left (c^{2} x^{2}-1\right ) x^{3}}{140 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {8 b e \left (c^{2} x^{2}-1\right ) d}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{35 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {2 b \,e^{2} \left (c^{2} x^{2}-1\right )}{35 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{4}}\)	\(511\)

[In]

int(x^3*(e*x^2+d)^2*(a+b*arcsec(c*x)),x,method=_RETURNVERBOSE)

[Out]

a*(1/8*e^2*x^8+1/3*d*e*x^6+1/4*x^4*d^2)+b/c^4*(1/8*c^4*arcsec(c*x)*e^2*x^8+1/3*c^4*arcsec(c*x)*d*e*x^6+1/4*arc
sec(c*x)*d^2*x^4*c^4-1/2520/c^5*(c^2*x^2-1)*(45*c^6*e^2*x^6+168*c^6*d*e*x^4+210*c^6*d^2*x^2+54*c^4*e^2*x^4+224
*c^4*d*e*x^2+420*c^4*d^2+72*c^2*e^2*x^2+448*c^2*d*e+144*e^2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.77 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {315 \, a c^{8} e^{2} x^{8} + 840 \, a c^{8} d e x^{6} + 630 \, a c^{8} d^{2} x^{4} + 105 \, {\left (3 \, b c^{8} e^{2} x^{8} + 8 \, b c^{8} d e x^{6} + 6 \, b c^{8} d^{2} x^{4}\right )} \operatorname {arcsec}\left (c x\right ) - {\left (45 \, b c^{6} e^{2} x^{6} + 420 \, b c^{4} d^{2} + 448 \, b c^{2} d e + 6 \, {\left (28 \, b c^{6} d e + 9 \, b c^{4} e^{2}\right )} x^{4} + 144 \, b e^{2} + 2 \, {\left (105 \, b c^{6} d^{2} + 112 \, b c^{4} d e + 36 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{2520 \, c^{8}} \]

[In]

integrate(x^3*(e*x^2+d)^2*(a+b*arcsec(c*x)),x, algorithm="fricas")

[Out]

1/2520*(315*a*c^8*e^2*x^8 + 840*a*c^8*d*e*x^6 + 630*a*c^8*d^2*x^4 + 105*(3*b*c^8*e^2*x^8 + 8*b*c^8*d*e*x^6 + 6
*b*c^8*d^2*x^4)*arcsec(c*x) - (45*b*c^6*e^2*x^6 + 420*b*c^4*d^2 + 448*b*c^2*d*e + 6*(28*b*c^6*d*e + 9*b*c^4*e^
2)*x^4 + 144*b*e^2 + 2*(105*b*c^6*d^2 + 112*b*c^4*d*e + 36*b*c^2*e^2)*x^2)*sqrt(c^2*x^2 - 1))/c^8

Sympy [A] (verification not implemented)

Time = 4.38 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.04 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} x^{4} \operatorname {asec}{\left (c x \right )}}{4} + \frac {b d e x^{6} \operatorname {asec}{\left (c x \right )}}{3} + \frac {b e^{2} x^{8} \operatorname {asec}{\left (c x \right )}}{8} - \frac {b d^{2} \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} - \frac {b d e \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{3 c} - \frac {b e^{2} \left (\begin {cases} \frac {x^{6} \sqrt {c^{2} x^{2} - 1}}{7 c} + \frac {6 x^{4} \sqrt {c^{2} x^{2} - 1}}{35 c^{3}} + \frac {8 x^{2} \sqrt {c^{2} x^{2} - 1}}{35 c^{5}} + \frac {16 \sqrt {c^{2} x^{2} - 1}}{35 c^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{6} \sqrt {- c^{2} x^{2} + 1}}{7 c} + \frac {6 i x^{4} \sqrt {- c^{2} x^{2} + 1}}{35 c^{3}} + \frac {8 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{5}} + \frac {16 i \sqrt {- c^{2} x^{2} + 1}}{35 c^{7}} & \text {otherwise} \end {cases}\right )}{8 c} \]

[In]

integrate(x**3*(e*x**2+d)**2*(a+b*asec(c*x)),x)

[Out]

a*d**2*x**4/4 + a*d*e*x**6/3 + a*e**2*x**8/8 + b*d**2*x**4*asec(c*x)/4 + b*d*e*x**6*asec(c*x)/3 + b*e**2*x**8*
asec(c*x)/8 - b*d**2*Piecewise((x**2*sqrt(c**2*x**2 - 1)/(3*c) + 2*sqrt(c**2*x**2 - 1)/(3*c**3), Abs(c**2*x**2
) > 1), (I*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 2*I*sqrt(-c**2*x**2 + 1)/(3*c**3), True))/(4*c) - b*d*e*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), True))/(3*c) - b*e**2*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*sqrt(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 = 0.20 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.06 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arcsec}\left (c x\right ) - \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d^{2} + \frac {1}{45} \, {\left (15 \, x^{6} \operatorname {arcsec}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d e + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arcsec}\left (c x\right ) - \frac {5 \, c^{6} x^{7} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} + 21 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 35 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e^{2} \]

[In]

integrate(x^3*(e*x^2+d)^2*(a+b*arcsec(c*x)),x, algorithm="maxima")

[Out]

1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/12*(3*x^4*arcsec(c*x) - (c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) +
3*x*sqrt(-1/(c^2*x^2) + 1))/c^3)*b*d^2 + 1/45*(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*e + 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^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17666 vs. \(2 (212) = 424\).

Time = 0.50 (sec) , antiderivative size = 17666, normalized size of antiderivative = 73.00 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]

[In]

integrate(x^3*(e*x^2+d)^2*(a+b*arcsec(c*x)),x, algorithm="giac")

[Out]

1/2520*(630*b*c^4*d^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) + 630*a*c^4*d^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) - 1260*b*c^4*d^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)) + 840*b*c^2
*d*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) - 2520*b*c^4*d^2*(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) + 7140*b*c^4*d^2*(-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) + 840*a*c^2*d*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) - 2520*a*c^4*d^2*(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) - 3360*b*c^2*d*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*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)) - 18060*b*c^4*d^2*(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) - 3360*a*c^2*d*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) + 315*b*e^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 + 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) + 3360*b*c^2*d*e*(1/(c^2*x^2) - 1)^2*arcco
s(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) + 3780*b*c^4*d^2*(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) - 27300*b*c^4*d^2*(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) + 7280*b*c^2*d*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) + 315*a*e^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) + 3360*a*c^2*d*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) + 3780*a*c^4*d^2*(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) - 2520*b*e^2*(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) + 3360*b*c^2*d*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) - 630*b*e^2*sqrt(-1/(c^2*x^2) + 1)/((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)) -
18256*b*c^2*d*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 + 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)^5)
- 27300*b*c^4*d^2*(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
) - 2520*a*e^2*(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) + 3360*a*c^2*d*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) + 8820*b*e^2*(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) - 8400*b*c^2*d*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) - 2520*b*c^4*d^2*(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) - 30128*b*c^2*d*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) - 18060*b*c^4*d^2*(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) + 1890*b*e^2*(-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^2*(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) -
8400*a*c^2*d*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) - 2520*a*c^4*d^2*(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 + 5
6*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^2*(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) + 3360*b*c^2*d*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) - 6678*b*e^2*(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) - 30128*b*c^2*d*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) - 7140*b*c^4*d^2*(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^2*(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) + 3360*a*c^2*d*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) + 22050*b*e^2*(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)^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)^8) +
3360*b*c^2*d*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) + 630*b
*c^4*d^2*(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 + 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) - 9234*b*e^2*
(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) - 18256*b*c^2*d*
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) - 1260*b*c^4*
d^2*(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) + 22050*a*e
^2*(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) + 3360*a*c^2*d*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) + 630*a*c^4*d^2*(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) - 17640*b*e^2*(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) - 3360*b*c^2*d*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) - 9234*b*e^2*(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) - 7280*b*c^2*d*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 + 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^2*(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) - 3360*
a*c^2*d*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) + 8820*b*e^2*(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) + 840*b*c^2*d*e*(1/(c^2*x^2) - 1)^8*a
rccos(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)^16) - 6678*b*e^2*(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) - 1680*b*c^2*d*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) + 8820*a*e^2*(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) + 840*a*c^2*d*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) - 2520*b*e^2*(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^2*(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) - 2520*a*e^2*(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^2*(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 + 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) - 630*b*e^2*(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) + 315*a*e^2*(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. \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int x^3\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

[In]

int(x^3*(d + e*x^2)^2*(a + b*acos(1/(c*x))),x)

[Out]

int(x^3*(d + e*x^2)^2*(a + b*acos(1/(c*x))), x)