3.96 \(\int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

Optimal. Leaf size=136 \[ -\frac{2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac{2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{1155 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}-\frac{i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d} \]

[Out]

(((-2*I)/1155)*a^3*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/d - (((2*I)/231)*a^2*Cos[c + d*x]^7*(a + I*a*Tan[c
+ d*x])^6)/d - ((I/33)*a*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^7)/d - ((I/11)*Cos[c + d*x]^11*(a + I*a*Tan[c
+ d*x])^8)/d

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Rubi [A]  time = 0.155986, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3497, 3488} \[ -\frac{2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac{2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{1155 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}-\frac{i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(((-2*I)/1155)*a^3*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/d - (((2*I)/231)*a^2*Cos[c + d*x]^7*(a + I*a*Tan[c
+ d*x])^6)/d - ((I/33)*a*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^7)/d - ((I/11)*Cos[c + d*x]^11*(a + I*a*Tan[c
+ d*x])^8)/d

Rule 3497

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d*
Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] + Dist[(a*(m + n))/(m*d^2), Int[(d*Sec[e + f*x])^(m + 2)*(
a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && LtQ[m, -
1] && IntegersQ[2*m, 2*n]

Rule 3488

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]

Rubi steps

\begin{align*} \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}+\frac{1}{11} (3 a) \int \cos ^9(c+d x) (a+i a \tan (c+d x))^7 \, dx\\ &=-\frac{i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}+\frac{1}{33} \left (2 a^2\right ) \int \cos ^7(c+d x) (a+i a \tan (c+d x))^6 \, dx\\ &=-\frac{2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac{i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}+\frac{1}{231} \left (2 a^3\right ) \int \cos ^5(c+d x) (a+i a \tan (c+d x))^5 \, dx\\ &=-\frac{2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{1155 d}-\frac{2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac{i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}\\ \end{align*}

Mathematica [A]  time = 1.27438, size = 73, normalized size = 0.54 \[ \frac{a^8 (-i (55 \sin (c+d x)+63 \sin (3 (c+d x)))+440 \cos (c+d x)+168 \cos (3 (c+d x))) (\sin (8 (c+d x))-i \cos (8 (c+d x)))}{4620 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*(440*Cos[c + d*x] + 168*Cos[3*(c + d*x)] - I*(55*Sin[c + d*x] + 63*Sin[3*(c + d*x)]))*((-I)*Cos[8*(c + d*
x)] + Sin[8*(c + d*x)]))/(4620*d)

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Maple [B]  time = 0.094, size = 567, normalized size = 4.2 \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)^11*(a+I*a*tan(d*x+c))^8,x)

[Out]

1/d*(a^8*(-1/11*sin(d*x+c)^7*cos(d*x+c)^4-7/99*sin(d*x+c)^5*cos(d*x+c)^4-5/99*sin(d*x+c)^3*cos(d*x+c)^4-1/33*s
in(d*x+c)*cos(d*x+c)^4+1/99*(2+cos(d*x+c)^2)*sin(d*x+c))-8*I*a^8*(-1/11*sin(d*x+c)^6*cos(d*x+c)^5-2/33*sin(d*x
+c)^4*cos(d*x+c)^5-8/231*sin(d*x+c)^2*cos(d*x+c)^5-16/1155*cos(d*x+c)^5)-28*a^8*(-1/11*sin(d*x+c)^5*cos(d*x+c)
^6-5/99*sin(d*x+c)^3*cos(d*x+c)^6-5/231*cos(d*x+c)^6*sin(d*x+c)+1/231*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(
d*x+c))+56*I*a^8*(-1/11*sin(d*x+c)^4*cos(d*x+c)^7-4/99*sin(d*x+c)^2*cos(d*x+c)^7-8/693*cos(d*x+c)^7)+70*a^8*(-
1/11*sin(d*x+c)^3*cos(d*x+c)^8-1/33*sin(d*x+c)*cos(d*x+c)^8+1/231*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(
d*x+c)^2)*sin(d*x+c))-56*I*a^8*(-1/11*sin(d*x+c)^2*cos(d*x+c)^9-2/99*cos(d*x+c)^9)-28*a^8*(-1/11*sin(d*x+c)*co
s(d*x+c)^10+1/99*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35*cos(d*x+c)^4+64/35*cos(d*x+c)^2)*sin(d*x+c))-8/11
*I*a^8*cos(d*x+c)^11+1/11*a^8*(256/63+cos(d*x+c)^10+10/9*cos(d*x+c)^8+80/63*cos(d*x+c)^6+32/21*cos(d*x+c)^4+12
8/63*cos(d*x+c)^2)*sin(d*x+c))

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Maxima [B]  time = 1.17487, size = 479, normalized size = 3.52 \begin{align*} -\frac{2520 i \, a^{8} \cos \left (d x + c\right )^{11} + 24 i \,{\left (105 \, \cos \left (d x + c\right )^{11} - 385 \, \cos \left (d x + c\right )^{9} + 495 \, \cos \left (d x + c\right )^{7} - 231 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 280 i \,{\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{8} + 1960 i \,{\left (9 \, \cos \left (d x + c\right )^{11} - 11 \, \cos \left (d x + c\right )^{9}\right )} a^{8} + 28 \,{\left (315 \, \sin \left (d x + c\right )^{11} - 1540 \, \sin \left (d x + c\right )^{9} + 2970 \, \sin \left (d x + c\right )^{7} - 2772 \, \sin \left (d x + c\right )^{5} + 1155 \, \sin \left (d x + c\right )^{3}\right )} a^{8} + 210 \,{\left (105 \, \sin \left (d x + c\right )^{11} - 385 \, \sin \left (d x + c\right )^{9} + 495 \, \sin \left (d x + c\right )^{7} - 231 \, \sin \left (d x + c\right )^{5}\right )} a^{8} + 140 \,{\left (63 \, \sin \left (d x + c\right )^{11} - 154 \, \sin \left (d x + c\right )^{9} + 99 \, \sin \left (d x + c\right )^{7}\right )} a^{8} + 5 \,{\left (63 \, \sin \left (d x + c\right )^{11} - 385 \, \sin \left (d x + c\right )^{9} + 990 \, \sin \left (d x + c\right )^{7} - 1386 \, \sin \left (d x + c\right )^{5} + 1155 \, \sin \left (d x + c\right )^{3} - 693 \, \sin \left (d x + c\right )\right )} a^{8} + 35 \,{\left (9 \, \sin \left (d x + c\right )^{11} - 11 \, \sin \left (d x + c\right )^{9}\right )} a^{8}}{3465 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/3465*(2520*I*a^8*cos(d*x + c)^11 + 24*I*(105*cos(d*x + c)^11 - 385*cos(d*x + c)^9 + 495*cos(d*x + c)^7 - 23
1*cos(d*x + c)^5)*a^8 + 280*I*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^8 + 1960*I*(9*co
s(d*x + c)^11 - 11*cos(d*x + c)^9)*a^8 + 28*(315*sin(d*x + c)^11 - 1540*sin(d*x + c)^9 + 2970*sin(d*x + c)^7 -
2772*sin(d*x + c)^5 + 1155*sin(d*x + c)^3)*a^8 + 210*(105*sin(d*x + c)^11 - 385*sin(d*x + c)^9 + 495*sin(d*x
+ c)^7 - 231*sin(d*x + c)^5)*a^8 + 140*(63*sin(d*x + c)^11 - 154*sin(d*x + c)^9 + 99*sin(d*x + c)^7)*a^8 + 5*(
63*sin(d*x + c)^11 - 385*sin(d*x + c)^9 + 990*sin(d*x + c)^7 - 1386*sin(d*x + c)^5 + 1155*sin(d*x + c)^3 - 693
*sin(d*x + c))*a^8 + 35*(9*sin(d*x + c)^11 - 11*sin(d*x + c)^9)*a^8)/d

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Fricas [A]  time = 2.09096, size = 190, normalized size = 1.4 \begin{align*} \frac{-105 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} - 385 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 495 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 231 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )}}{9240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/9240*(-105*I*a^8*e^(11*I*d*x + 11*I*c) - 385*I*a^8*e^(9*I*d*x + 9*I*c) - 495*I*a^8*e^(7*I*d*x + 7*I*c) - 231
*I*a^8*e^(5*I*d*x + 5*I*c))/d

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Sympy [A]  time = 1.57391, size = 163, normalized size = 1.2 \begin{align*} \begin{cases} \frac{- 53760 i a^{8} d^{3} e^{11 i c} e^{11 i d x} - 197120 i a^{8} d^{3} e^{9 i c} e^{9 i d x} - 253440 i a^{8} d^{3} e^{7 i c} e^{7 i d x} - 118272 i a^{8} d^{3} e^{5 i c} e^{5 i d x}}{4730880 d^{4}} & \text{for}\: 4730880 d^{4} \neq 0 \\x \left (\frac{a^{8} e^{11 i c}}{8} + \frac{3 a^{8} e^{9 i c}}{8} + \frac{3 a^{8} e^{7 i c}}{8} + \frac{a^{8} e^{5 i c}}{8}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**11*(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise(((-53760*I*a**8*d**3*exp(11*I*c)*exp(11*I*d*x) - 197120*I*a**8*d**3*exp(9*I*c)*exp(9*I*d*x) - 253440
*I*a**8*d**3*exp(7*I*c)*exp(7*I*d*x) - 118272*I*a**8*d**3*exp(5*I*c)*exp(5*I*d*x))/(4730880*d**4), Ne(4730880*
d**4, 0)), (x*(a**8*exp(11*I*c)/8 + 3*a**8*exp(9*I*c)/8 + 3*a**8*exp(7*I*c)/8 + a**8*exp(5*I*c)/8), True))

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Giac [B]  time = 3.81121, size = 3865, normalized size = 28.42 \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)^11*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/4844421120*(82027951005*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1148391314070*a^8*e^(26*I*d*x
+ 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 7464543541455*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 2
9858174165820*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 82109978956005*a^8*e^(20*I*d*x + 6*I*c)*lo
g(I*e^(I*d*x + I*c) + 1) + 164219957912010*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 2463299368680
15*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 246329936868015*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d
*x + I*c) + 1) + 164219957912010*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 82109978956005*a^8*e^(8
*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 29858174165820*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) + 1)
+ 7464543541455*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1148391314070*a^8*e^(2*I*d*x - 12*I*c)*
log(I*e^(I*d*x + I*c) + 1) + 281519927849160*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 82027951005*a^8*e^(
-14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 82004266575*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) - 1) + 11480
59732050*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 7462388258325*a^8*e^(24*I*d*x + 10*I*c)*log(I*
e^(I*d*x + I*c) - 1) + 29849553033300*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 82086270841575*a^8
*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 164172541683150*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I
*c) - 1) + 246258812524725*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 246258812524725*a^8*e^(12*I*d
*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 164172541683150*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) +
82086270841575*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 29849553033300*a^8*e^(6*I*d*x - 8*I*c)*lo
g(I*e^(I*d*x + I*c) - 1) + 7462388258325*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1148059732050*a
^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 281438642885400*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) -
1) + 82004266575*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*c) - 1) - 82027951005*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(
I*d*x + I*c) + 1) - 1148391314070*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 7464543541455*a^8*e^
(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 29858174165820*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*
c) + 1) - 82109978956005*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 164219957912010*a^8*e^(18*I*d*
x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 246329936868015*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1)
- 246329936868015*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 164219957912010*a^8*e^(10*I*d*x - 4*I
*c)*log(-I*e^(I*d*x + I*c) + 1) - 82109978956005*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 2985817
4165820*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 7464543541455*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^
(I*d*x + I*c) + 1) - 1148391314070*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 281519927849160*a^8*
e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) - 82027951005*a^8*e^(-14*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 820042665
75*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1148059732050*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I
*d*x + I*c) - 1) - 7462388258325*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 29849553033300*a^8*e^
(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 82086270841575*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c
) - 1) - 164172541683150*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 246258812524725*a^8*e^(16*I*d*
x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 246258812524725*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) - 1)
- 164172541683150*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 82086270841575*a^8*e^(8*I*d*x - 6*I*c
)*log(-I*e^(I*d*x + I*c) - 1) - 29849553033300*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 746238825
8325*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1148059732050*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I
*d*x + I*c) - 1) - 281438642885400*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 82004266575*a^8*e^(-14*I*c)*
log(-I*e^(I*d*x + I*c) - 1) - 23684430*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 331582020*a^8*e
^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 2155283130*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x) + e^(-
I*c)) - 8621132520*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 23708114430*a^8*e^(20*I*d*x + 6*I*c)
*log(I*e^(I*d*x) + e^(-I*c)) - 47416228860*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 71124343290*
a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 71124343290*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x) +
e^(-I*c)) - 47416228860*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 23708114430*a^8*e^(8*I*d*x - 6*
I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 8621132520*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 2155283130
*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 331582020*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x) + e
^(-I*c)) - 81284963760*a^8*e^(14*I*d*x)*log(I*e^(I*d*x) + e^(-I*c)) - 23684430*a^8*e^(-14*I*c)*log(I*e^(I*d*x)
+ e^(-I*c)) + 23684430*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 331582020*a^8*e^(26*I*d*x + 1
2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 2155283130*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 8621
132520*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 23708114430*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(
I*d*x) + e^(-I*c)) + 47416228860*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 71124343290*a^8*e^(16
*I*d*x + 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 71124343290*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c
)) + 47416228860*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 23708114430*a^8*e^(8*I*d*x - 6*I*c)*l
og(-I*e^(I*d*x) + e^(-I*c)) + 8621132520*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 2155283130*a^8
*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 331582020*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x) + e^(
-I*c)) + 81284963760*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x) + e^(-I*c)) + 23684430*a^8*e^(-14*I*c)*log(-I*e^(I*d*x)
+ e^(-I*c)) - 55050240*I*a^8*e^(39*I*d*x + 25*I*c) - 972554240*I*a^8*e^(37*I*d*x + 23*I*c) - 8095006720*I*a^8
*e^(35*I*d*x + 21*I*c) - 42161143808*I*a^8*e^(33*I*d*x + 19*I*c) - 153891110912*I*a^8*e^(31*I*d*x + 17*I*c) -
417750581248*I*a^8*e^(29*I*d*x + 15*I*c) - 873287647232*I*a^8*e^(27*I*d*x + 13*I*c) - 1435886419968*I*a^8*e^(2
5*I*d*x + 11*I*c) - 1879877615616*I*a^8*e^(23*I*d*x + 9*I*c) - 1970745114624*I*a^8*e^(21*I*d*x + 7*I*c) - 1654
208331776*I*a^8*e^(19*I*d*x + 5*I*c) - 1105350098944*I*a^8*e^(17*I*d*x + 3*I*c) - 580728651776*I*a^8*e^(15*I*d
*x + I*c) - 234836983808*I*a^8*e^(13*I*d*x - I*c) - 70581747712*I*a^8*e^(11*I*d*x - 3*I*c) - 14856224768*I*a^8
*e^(9*I*d*x - 5*I*c) - 1955069952*I*a^8*e^(7*I*d*x - 7*I*c) - 121110528*I*a^8*e^(5*I*d*x - 9*I*c))/(d*e^(28*I*
d*x + 14*I*c) + 14*d*e^(26*I*d*x + 12*I*c) + 91*d*e^(24*I*d*x + 10*I*c) + 364*d*e^(22*I*d*x + 8*I*c) + 1001*d*
e^(20*I*d*x + 6*I*c) + 2002*d*e^(18*I*d*x + 4*I*c) + 3003*d*e^(16*I*d*x + 2*I*c) + 3003*d*e^(12*I*d*x - 2*I*c)
+ 2002*d*e^(10*I*d*x - 4*I*c) + 1001*d*e^(8*I*d*x - 6*I*c) + 364*d*e^(6*I*d*x - 8*I*c) + 91*d*e^(4*I*d*x - 10
*I*c) + 14*d*e^(2*I*d*x - 12*I*c) + 3432*d*e^(14*I*d*x) + d*e^(-14*I*c))