3.90 \(\int \cos ^{18}(c+d x) (a+i a \tan (c+d x))^8 \, dx\)
Optimal. Leaf size=279 \[ -\frac{i a^{17}}{36 d (a-i a \tan (c+d x))^9}-\frac{i a^{16}}{32 d (a-i a \tan (c+d x))^8}-\frac{3 i a^{15}}{112 d (a-i a \tan (c+d x))^7}-\frac{i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac{i a^{13}}{64 d (a-i a \tan (c+d x))^5}-\frac{3 i a^{12}}{256 d (a-i a \tan (c+d x))^4}-\frac{7 i a^{11}}{768 d (a-i a \tan (c+d x))^3}-\frac{i a^{10}}{128 d (a-i a \tan (c+d x))^2}-\frac{9 i a^9}{1024 d (a-i a \tan (c+d x))}+\frac{i a^9}{1024 d (a+i a \tan (c+d x))}+\frac{5 a^8 x}{512} \]
[Out]
(5*a^8*x)/512 - ((I/36)*a^17)/(d*(a - I*a*Tan[c + d*x])^9) - ((I/32)*a^16)/(d*(a - I*a*Tan[c + d*x])^8) - (((3
*I)/112)*a^15)/(d*(a - I*a*Tan[c + d*x])^7) - ((I/48)*a^14)/(d*(a - I*a*Tan[c + d*x])^6) - ((I/64)*a^13)/(d*(a
- I*a*Tan[c + d*x])^5) - (((3*I)/256)*a^12)/(d*(a - I*a*Tan[c + d*x])^4) - (((7*I)/768)*a^11)/(d*(a - I*a*Tan
[c + d*x])^3) - ((I/128)*a^10)/(d*(a - I*a*Tan[c + d*x])^2) - (((9*I)/1024)*a^9)/(d*(a - I*a*Tan[c + d*x])) +
((I/1024)*a^9)/(d*(a + I*a*Tan[c + d*x]))
________________________________________________________________________________________
Rubi [A] time = 0.155992, antiderivative size = 279, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{3487, 44, 206} \[ -\frac{i a^{17}}{36 d (a-i a \tan (c+d x))^9}-\frac{i a^{16}}{32 d (a-i a \tan (c+d x))^8}-\frac{3 i a^{15}}{112 d (a-i a \tan (c+d x))^7}-\frac{i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac{i a^{13}}{64 d (a-i a \tan (c+d x))^5}-\frac{3 i a^{12}}{256 d (a-i a \tan (c+d x))^4}-\frac{7 i a^{11}}{768 d (a-i a \tan (c+d x))^3}-\frac{i a^{10}}{128 d (a-i a \tan (c+d x))^2}-\frac{9 i a^9}{1024 d (a-i a \tan (c+d x))}+\frac{i a^9}{1024 d (a+i a \tan (c+d x))}+\frac{5 a^8 x}{512} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^18*(a + I*a*Tan[c + d*x])^8,x]
[Out]
(5*a^8*x)/512 - ((I/36)*a^17)/(d*(a - I*a*Tan[c + d*x])^9) - ((I/32)*a^16)/(d*(a - I*a*Tan[c + d*x])^8) - (((3
*I)/112)*a^15)/(d*(a - I*a*Tan[c + d*x])^7) - ((I/48)*a^14)/(d*(a - I*a*Tan[c + d*x])^6) - ((I/64)*a^13)/(d*(a
- I*a*Tan[c + d*x])^5) - (((3*I)/256)*a^12)/(d*(a - I*a*Tan[c + d*x])^4) - (((7*I)/768)*a^11)/(d*(a - I*a*Tan
[c + d*x])^3) - ((I/128)*a^10)/(d*(a - I*a*Tan[c + d*x])^2) - (((9*I)/1024)*a^9)/(d*(a - I*a*Tan[c + d*x])) +
((I/1024)*a^9)/(d*(a + I*a*Tan[c + d*x]))
Rule 3487
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
Rule 44
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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \cos ^{18}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{\left (i a^{19}\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^{10} (a+x)^2} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^{19}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{4 a^2 (a-x)^{10}}+\frac{1}{4 a^3 (a-x)^9}+\frac{3}{16 a^4 (a-x)^8}+\frac{1}{8 a^5 (a-x)^7}+\frac{5}{64 a^6 (a-x)^6}+\frac{3}{64 a^7 (a-x)^5}+\frac{7}{256 a^8 (a-x)^4}+\frac{1}{64 a^9 (a-x)^3}+\frac{9}{1024 a^{10} (a-x)^2}+\frac{1}{1024 a^{10} (a+x)^2}+\frac{5}{512 a^{10} \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i a^{17}}{36 d (a-i a \tan (c+d x))^9}-\frac{i a^{16}}{32 d (a-i a \tan (c+d x))^8}-\frac{3 i a^{15}}{112 d (a-i a \tan (c+d x))^7}-\frac{i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac{i a^{13}}{64 d (a-i a \tan (c+d x))^5}-\frac{3 i a^{12}}{256 d (a-i a \tan (c+d x))^4}-\frac{7 i a^{11}}{768 d (a-i a \tan (c+d x))^3}-\frac{i a^{10}}{128 d (a-i a \tan (c+d x))^2}-\frac{9 i a^9}{1024 d (a-i a \tan (c+d x))}+\frac{i a^9}{1024 d (a+i a \tan (c+d x))}-\frac{\left (5 i a^9\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{512 d}\\ &=\frac{5 a^8 x}{512}-\frac{i a^{17}}{36 d (a-i a \tan (c+d x))^9}-\frac{i a^{16}}{32 d (a-i a \tan (c+d x))^8}-\frac{3 i a^{15}}{112 d (a-i a \tan (c+d x))^7}-\frac{i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac{i a^{13}}{64 d (a-i a \tan (c+d x))^5}-\frac{3 i a^{12}}{256 d (a-i a \tan (c+d x))^4}-\frac{7 i a^{11}}{768 d (a-i a \tan (c+d x))^3}-\frac{i a^{10}}{128 d (a-i a \tan (c+d x))^2}-\frac{9 i a^9}{1024 d (a-i a \tan (c+d x))}+\frac{i a^9}{1024 d (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 6.71641, size = 188, normalized size = 0.67 \[ \frac{a^8 (-7056 \sin (2 (c+d x))-10080 \sin (4 (c+d x))-9720 \sin (6 (c+d x))-5040 i d x \sin (8 (c+d x))+315 \sin (8 (c+d x))+280 \sin (10 (c+d x))-28224 i \cos (2 (c+d x))-20160 i \cos (4 (c+d x))-12960 i \cos (6 (c+d x))+5040 d x \cos (8 (c+d x))-315 i \cos (8 (c+d x))+224 i \cos (10 (c+d x))-15876 i) (\cos (8 (c+2 d x))+i \sin (8 (c+2 d x)))}{516096 d (\cos (d x)+i \sin (d x))^8} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^18*(a + I*a*Tan[c + d*x])^8,x]
[Out]
(a^8*(-15876*I - (28224*I)*Cos[2*(c + d*x)] - (20160*I)*Cos[4*(c + d*x)] - (12960*I)*Cos[6*(c + d*x)] - (315*I
)*Cos[8*(c + d*x)] + 5040*d*x*Cos[8*(c + d*x)] + (224*I)*Cos[10*(c + d*x)] - 7056*Sin[2*(c + d*x)] - 10080*Sin
[4*(c + d*x)] - 9720*Sin[6*(c + d*x)] + 315*Sin[8*(c + d*x)] - (5040*I)*d*x*Sin[8*(c + d*x)] + 280*Sin[10*(c +
d*x)])*(Cos[8*(c + 2*d*x)] + I*Sin[8*(c + 2*d*x)]))/(516096*d*(Cos[d*x] + I*Sin[d*x])^8)
________________________________________________________________________________________
Maple [B] time = 0.13, size = 789, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^18*(a+I*a*tan(d*x+c))^8,x)
[Out]
1/d*(a^8*(-1/18*sin(d*x+c)^7*cos(d*x+c)^11-7/288*sin(d*x+c)^5*cos(d*x+c)^11-5/576*sin(d*x+c)^3*cos(d*x+c)^11-5
/2304*sin(d*x+c)*cos(d*x+c)^11+1/4608*(cos(d*x+c)^9+9/8*cos(d*x+c)^7+21/16*cos(d*x+c)^5+105/64*cos(d*x+c)^3+31
5/128*cos(d*x+c))*sin(d*x+c)+35/65536*d*x+35/65536*c)-8*I*a^8*(-1/18*sin(d*x+c)^6*cos(d*x+c)^12-1/48*sin(d*x+c
)^4*cos(d*x+c)^12-1/168*sin(d*x+c)^2*cos(d*x+c)^12-1/1008*cos(d*x+c)^12)-28*a^8*(-1/18*sin(d*x+c)^5*cos(d*x+c)
^13-5/288*sin(d*x+c)^3*cos(d*x+c)^13-5/1344*sin(d*x+c)*cos(d*x+c)^13+5/16128*(cos(d*x+c)^11+11/10*cos(d*x+c)^9
+99/80*cos(d*x+c)^7+231/160*cos(d*x+c)^5+231/128*cos(d*x+c)^3+693/256*cos(d*x+c))*sin(d*x+c)+55/65536*d*x+55/6
5536*c)+56*I*a^8*(-1/18*sin(d*x+c)^4*cos(d*x+c)^14-1/72*sin(d*x+c)^2*cos(d*x+c)^14-1/504*cos(d*x+c)^14)+70*a^8
*(-1/18*sin(d*x+c)^3*cos(d*x+c)^15-1/96*sin(d*x+c)*cos(d*x+c)^15+1/1344*(cos(d*x+c)^13+13/12*cos(d*x+c)^11+143
/120*cos(d*x+c)^9+429/320*cos(d*x+c)^7+1001/640*cos(d*x+c)^5+1001/512*cos(d*x+c)^3+3003/1024*cos(d*x+c))*sin(d
*x+c)+143/65536*d*x+143/65536*c)-56*I*a^8*(-1/18*sin(d*x+c)^2*cos(d*x+c)^16-1/144*cos(d*x+c)^16)-28*a^8*(-1/18
*sin(d*x+c)*cos(d*x+c)^17+1/288*(cos(d*x+c)^15+15/14*cos(d*x+c)^13+65/56*cos(d*x+c)^11+143/112*cos(d*x+c)^9+12
87/896*cos(d*x+c)^7+429/256*cos(d*x+c)^5+2145/1024*cos(d*x+c)^3+6435/2048*cos(d*x+c))*sin(d*x+c)+715/65536*d*x
+715/65536*c)-4/9*I*a^8*cos(d*x+c)^18+a^8*(1/18*(cos(d*x+c)^17+17/16*cos(d*x+c)^15+255/224*cos(d*x+c)^13+1105/
896*cos(d*x+c)^11+2431/1792*cos(d*x+c)^9+21879/14336*cos(d*x+c)^7+7293/4096*cos(d*x+c)^5+36465/16384*cos(d*x+c
)^3+109395/32768*cos(d*x+c))*sin(d*x+c)+12155/65536*d*x+12155/65536*c))
________________________________________________________________________________________
Maxima [A] time = 1.76884, size = 363, normalized size = 1.3 \begin{align*} \frac{40320 \,{\left (d x + c\right )} a^{8} + \frac{40320 \, a^{8} \tan \left (d x + c\right )^{17} + 349440 \, a^{8} \tan \left (d x + c\right )^{15} + 1338624 \, a^{8} \tan \left (d x + c\right )^{13} + 2969856 \, a^{8} \tan \left (d x + c\right )^{11} + 4194304 \, a^{8} \tan \left (d x + c\right )^{9} + 3518208 \, a^{8} \tan \left (d x + c\right )^{7} + 2752512 i \, a^{8} \tan \left (d x + c\right )^{6} + 11047680 \, a^{8} \tan \left (d x + c\right )^{5} - 15335424 i \, a^{8} \tan \left (d x + c\right )^{4} - 15488256 \, a^{8} \tan \left (d x + c\right )^{3} + 10616832 i \, a^{8} \tan \left (d x + c\right )^{2} + 4088448 \, a^{8} \tan \left (d x + c\right ) - 655360 i \, a^{8}}{\tan \left (d x + c\right )^{18} + 9 \, \tan \left (d x + c\right )^{16} + 36 \, \tan \left (d x + c\right )^{14} + 84 \, \tan \left (d x + c\right )^{12} + 126 \, \tan \left (d x + c\right )^{10} + 126 \, \tan \left (d x + c\right )^{8} + 84 \, \tan \left (d x + c\right )^{6} + 36 \, \tan \left (d x + c\right )^{4} + 9 \, \tan \left (d x + c\right )^{2} + 1}}{4128768 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^18*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
[Out]
1/4128768*(40320*(d*x + c)*a^8 + (40320*a^8*tan(d*x + c)^17 + 349440*a^8*tan(d*x + c)^15 + 1338624*a^8*tan(d*x
+ c)^13 + 2969856*a^8*tan(d*x + c)^11 + 4194304*a^8*tan(d*x + c)^9 + 3518208*a^8*tan(d*x + c)^7 + 2752512*I*a
^8*tan(d*x + c)^6 + 11047680*a^8*tan(d*x + c)^5 - 15335424*I*a^8*tan(d*x + c)^4 - 15488256*a^8*tan(d*x + c)^3
+ 10616832*I*a^8*tan(d*x + c)^2 + 4088448*a^8*tan(d*x + c) - 655360*I*a^8)/(tan(d*x + c)^18 + 9*tan(d*x + c)^1
6 + 36*tan(d*x + c)^14 + 84*tan(d*x + c)^12 + 126*tan(d*x + c)^10 + 126*tan(d*x + c)^8 + 84*tan(d*x + c)^6 + 3
6*tan(d*x + c)^4 + 9*tan(d*x + c)^2 + 1))/d
________________________________________________________________________________________
Fricas [A] time = 2.82542, size = 528, normalized size = 1.89 \begin{align*} \frac{{\left (5040 \, a^{8} d x e^{\left (2 i \, d x + 2 i \, c\right )} - 28 i \, a^{8} e^{\left (20 i \, d x + 20 i \, c\right )} - 315 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} - 1620 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 5040 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 10584 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 15876 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 17640 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 15120 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 11340 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 252 i \, a^{8}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{516096 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^18*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
[Out]
1/516096*(5040*a^8*d*x*e^(2*I*d*x + 2*I*c) - 28*I*a^8*e^(20*I*d*x + 20*I*c) - 315*I*a^8*e^(18*I*d*x + 18*I*c)
- 1620*I*a^8*e^(16*I*d*x + 16*I*c) - 5040*I*a^8*e^(14*I*d*x + 14*I*c) - 10584*I*a^8*e^(12*I*d*x + 12*I*c) - 15
876*I*a^8*e^(10*I*d*x + 10*I*c) - 17640*I*a^8*e^(8*I*d*x + 8*I*c) - 15120*I*a^8*e^(6*I*d*x + 6*I*c) - 11340*I*
a^8*e^(4*I*d*x + 4*I*c) + 252*I*a^8)*e^(-2*I*d*x - 2*I*c)/d
________________________________________________________________________________________
Sympy [A] time = 2.87144, size = 415, normalized size = 1.49 \begin{align*} \frac{5 a^{8} x}{512} + \begin{cases} \frac{\left (- 277298568799925181577403826176 i a^{8} d^{9} e^{20 i c} e^{18 i d x} - 3119608898999158292745793044480 i a^{8} d^{9} e^{18 i c} e^{16 i d x} - 16043702909138528362692649943040 i a^{8} d^{9} e^{16 i c} e^{14 i d x} - 49913742383986532683932688711680 i a^{8} d^{9} e^{14 i c} e^{12 i d x} - 104818859006371718636258646294528 i a^{8} d^{9} e^{12 i c} e^{10 i d x} - 157228288509557577954387969441792 i a^{8} d^{9} e^{10 i c} e^{8 i d x} - 174698098343952864393764410490880 i a^{8} d^{9} e^{8 i c} e^{6 i d x} - 149741227151959598051798066135040 i a^{8} d^{9} e^{6 i c} e^{4 i d x} - 112305920363969698538848549601280 i a^{8} d^{9} e^{4 i c} e^{2 i d x} + 2495687119199326634196634435584 i a^{8} d^{9} e^{- 2 i d x}\right ) e^{- 2 i c}}{5111167220120220946834707324076032 d^{10}} & \text{for}\: 5111167220120220946834707324076032 d^{10} e^{2 i c} \neq 0 \\x \left (- \frac{5 a^{8}}{512} + \frac{\left (a^{8} e^{20 i c} + 10 a^{8} e^{18 i c} + 45 a^{8} e^{16 i c} + 120 a^{8} e^{14 i c} + 210 a^{8} e^{12 i c} + 252 a^{8} e^{10 i c} + 210 a^{8} e^{8 i c} + 120 a^{8} e^{6 i c} + 45 a^{8} e^{4 i c} + 10 a^{8} e^{2 i c} + a^{8}\right ) e^{- 2 i c}}{1024}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**18*(a+I*a*tan(d*x+c))**8,x)
[Out]
5*a**8*x/512 + Piecewise(((-277298568799925181577403826176*I*a**8*d**9*exp(20*I*c)*exp(18*I*d*x) - 31196088989
99158292745793044480*I*a**8*d**9*exp(18*I*c)*exp(16*I*d*x) - 16043702909138528362692649943040*I*a**8*d**9*exp(
16*I*c)*exp(14*I*d*x) - 49913742383986532683932688711680*I*a**8*d**9*exp(14*I*c)*exp(12*I*d*x) - 1048188590063
71718636258646294528*I*a**8*d**9*exp(12*I*c)*exp(10*I*d*x) - 157228288509557577954387969441792*I*a**8*d**9*exp
(10*I*c)*exp(8*I*d*x) - 174698098343952864393764410490880*I*a**8*d**9*exp(8*I*c)*exp(6*I*d*x) - 14974122715195
9598051798066135040*I*a**8*d**9*exp(6*I*c)*exp(4*I*d*x) - 112305920363969698538848549601280*I*a**8*d**9*exp(4*
I*c)*exp(2*I*d*x) + 2495687119199326634196634435584*I*a**8*d**9*exp(-2*I*d*x))*exp(-2*I*c)/(511116722012022094
6834707324076032*d**10), Ne(5111167220120220946834707324076032*d**10*exp(2*I*c), 0)), (x*(-5*a**8/512 + (a**8*
exp(20*I*c) + 10*a**8*exp(18*I*c) + 45*a**8*exp(16*I*c) + 120*a**8*exp(14*I*c) + 210*a**8*exp(12*I*c) + 252*a*
*8*exp(10*I*c) + 210*a**8*exp(8*I*c) + 120*a**8*exp(6*I*c) + 45*a**8*exp(4*I*c) + 10*a**8*exp(2*I*c) + a**8)*e
xp(-2*I*c)/1024), True))
________________________________________________________________________________________
Giac [B] time = 3.35362, size = 2044, normalized size = 7.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^18*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
[Out]
1/330301440*(3225600*a^8*d*x*e^(30*I*d*x + 16*I*c) + 45158400*a^8*d*x*e^(28*I*d*x + 14*I*c) + 293529600*a^8*d*
x*e^(26*I*d*x + 12*I*c) + 1174118400*a^8*d*x*e^(24*I*d*x + 10*I*c) + 3228825600*a^8*d*x*e^(22*I*d*x + 8*I*c) +
6457651200*a^8*d*x*e^(20*I*d*x + 6*I*c) + 9686476800*a^8*d*x*e^(18*I*d*x + 4*I*c) + 11070259200*a^8*d*x*e^(16
*I*d*x + 2*I*c) + 6457651200*a^8*d*x*e^(12*I*d*x - 2*I*c) + 3228825600*a^8*d*x*e^(10*I*d*x - 4*I*c) + 11741184
00*a^8*d*x*e^(8*I*d*x - 6*I*c) + 293529600*a^8*d*x*e^(6*I*d*x - 8*I*c) + 45158400*a^8*d*x*e^(4*I*d*x - 10*I*c)
+ 3225600*a^8*d*x*e^(2*I*d*x - 12*I*c) + 9686476800*a^8*d*x*e^(14*I*d*x) - 1515780*I*a^8*e^(30*I*d*x + 16*I*c
)*log(e^(2*I*d*x + 2*I*c) + 1) - 21220920*I*a^8*e^(28*I*d*x + 14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 137935980
*I*a^8*e^(26*I*d*x + 12*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 551743920*I*a^8*e^(24*I*d*x + 10*I*c)*log(e^(2*I*d
*x + 2*I*c) + 1) - 1517295780*I*a^8*e^(22*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 3034591560*I*a^8*e^(20
*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 4551887340*I*a^8*e^(18*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) +
1) - 5202156960*I*a^8*e^(16*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 3034591560*I*a^8*e^(12*I*d*x - 2*I*
c)*log(e^(2*I*d*x + 2*I*c) + 1) - 1517295780*I*a^8*e^(10*I*d*x - 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 5517439
20*I*a^8*e^(8*I*d*x - 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 137935980*I*a^8*e^(6*I*d*x - 8*I*c)*log(e^(2*I*d*x
+ 2*I*c) + 1) - 21220920*I*a^8*e^(4*I*d*x - 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 1515780*I*a^8*e^(2*I*d*x -
12*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 4551887340*I*a^8*e^(14*I*d*x)*log(e^(2*I*d*x + 2*I*c) + 1) + 1515780*I
*a^8*e^(30*I*d*x + 16*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 21220920*I*a^8*e^(28*I*d*x + 14*I*c)*log(e^(2*I*d*x
) + e^(-2*I*c)) + 137935980*I*a^8*e^(26*I*d*x + 12*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 551743920*I*a^8*e^(24*
I*d*x + 10*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 1517295780*I*a^8*e^(22*I*d*x + 8*I*c)*log(e^(2*I*d*x) + e^(-2*
I*c)) + 3034591560*I*a^8*e^(20*I*d*x + 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 4551887340*I*a^8*e^(18*I*d*x + 4
*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 5202156960*I*a^8*e^(16*I*d*x + 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 30
34591560*I*a^8*e^(12*I*d*x - 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 1517295780*I*a^8*e^(10*I*d*x - 4*I*c)*log(
e^(2*I*d*x) + e^(-2*I*c)) + 551743920*I*a^8*e^(8*I*d*x - 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 137935980*I*a^
8*e^(6*I*d*x - 8*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 21220920*I*a^8*e^(4*I*d*x - 10*I*c)*log(e^(2*I*d*x) + e^
(-2*I*c)) + 1515780*I*a^8*e^(2*I*d*x - 12*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 4551887340*I*a^8*e^(14*I*d*x)*l
og(e^(2*I*d*x) + e^(-2*I*c)) - 17920*I*a^8*e^(48*I*d*x + 34*I*c) - 452480*I*a^8*e^(46*I*d*x + 32*I*c) - 548992
0*I*a^8*e^(44*I*d*x + 30*I*c) - 42609280*I*a^8*e^(42*I*d*x + 28*I*c) - 237601280*I*a^8*e^(40*I*d*x + 26*I*c) -
1013595520*I*a^8*e^(38*I*d*x + 24*I*c) - 3439322880*I*a^8*e^(36*I*d*x + 22*I*c) - 9529403520*I*a^8*e^(34*I*d*
x + 20*I*c) - 21965959680*I*a^8*e^(32*I*d*x + 18*I*c) - 42709532800*I*a^8*e^(30*I*d*x + 16*I*c) - 70772038400*
I*a^8*e^(28*I*d*x + 14*I*c) - 100658185600*I*a^8*e^(26*I*d*x + 12*I*c) - 123309222400*I*a^8*e^(24*I*d*x + 10*I
*c) - 129974633600*I*a^8*e^(22*I*d*x + 8*I*c) - 117140020480*I*a^8*e^(20*I*d*x + 6*I*c) - 89191105920*I*a^8*e^
(18*I*d*x + 4*I*c) - 56345172480*I*a^8*e^(16*I*d*x + 2*I*c) - 11479265280*I*a^8*e^(12*I*d*x - 2*I*c) - 3367687
680*I*a^8*e^(10*I*d*x - 4*I*c) - 645765120*I*a^8*e^(8*I*d*x - 6*I*c) - 52577280*I*a^8*e^(6*I*d*x - 8*I*c) + 74
18880*I*a^8*e^(4*I*d*x - 10*I*c) + 2257920*I*a^8*e^(2*I*d*x - 12*I*c) - 28794769920*I*a^8*e^(14*I*d*x) + 16128
0*I*a^8*e^(-14*I*c))/(d*e^(30*I*d*x + 16*I*c) + 14*d*e^(28*I*d*x + 14*I*c) + 91*d*e^(26*I*d*x + 12*I*c) + 364*
d*e^(24*I*d*x + 10*I*c) + 1001*d*e^(22*I*d*x + 8*I*c) + 2002*d*e^(20*I*d*x + 6*I*c) + 3003*d*e^(18*I*d*x + 4*I
*c) + 3432*d*e^(16*I*d*x + 2*I*c) + 2002*d*e^(12*I*d*x - 2*I*c) + 1001*d*e^(10*I*d*x - 4*I*c) + 364*d*e^(8*I*d
*x - 6*I*c) + 91*d*e^(6*I*d*x - 8*I*c) + 14*d*e^(4*I*d*x - 10*I*c) + d*e^(2*I*d*x - 12*I*c) + 3003*d*e^(14*I*d
*x))