3.76 \(\int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^5 \, dx\)
Optimal. Leaf size=159 \[ -\frac{5 a^5 \sin ^7(c+d x)}{231 d}+\frac{a^5 \sin ^5(c+d x)}{11 d}-\frac{5 a^5 \sin ^3(c+d x)}{33 d}+\frac{5 a^5 \sin (c+d x)}{33 d}-\frac{5 i a^5 \cos ^7(c+d x)}{231 d}-\frac{2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac{2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d} \]
[Out]
(((-5*I)/231)*a^5*Cos[c + d*x]^7)/d + (5*a^5*Sin[c + d*x])/(33*d) - (5*a^5*Sin[c + d*x]^3)/(33*d) + (a^5*Sin[c
+ d*x]^5)/(11*d) - (5*a^5*Sin[c + d*x]^7)/(231*d) - (((2*I)/33)*a^3*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^2)/
d - (((2*I)/11)*a*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^4)/d
________________________________________________________________________________________
Rubi [A] time = 0.125985, antiderivative size = 159, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{3496, 3486, 2633} \[ -\frac{5 a^5 \sin ^7(c+d x)}{231 d}+\frac{a^5 \sin ^5(c+d x)}{11 d}-\frac{5 a^5 \sin ^3(c+d x)}{33 d}+\frac{5 a^5 \sin (c+d x)}{33 d}-\frac{5 i a^5 \cos ^7(c+d x)}{231 d}-\frac{2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac{2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^5,x]
[Out]
(((-5*I)/231)*a^5*Cos[c + d*x]^7)/d + (5*a^5*Sin[c + d*x])/(33*d) - (5*a^5*Sin[c + d*x]^3)/(33*d) + (a^5*Sin[c
+ d*x]^5)/(11*d) - (5*a^5*Sin[c + d*x]^7)/(231*d) - (((2*I)/33)*a^3*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^2)/
d - (((2*I)/11)*a*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^4)/d
Rule 3496
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(2*b*(
d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*m), x] - Dist[(b^2*(m + 2*n - 2))/(d^2*m), Int[(d*Sec[e + f
*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n,
1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILt
Q[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]
Rule 3486
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])
Rule 2633
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Rubi steps
\begin{align*} \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac{2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d}+\frac{1}{11} \left (3 a^2\right ) \int \cos ^9(c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac{2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac{2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d}+\frac{1}{33} \left (5 a^4\right ) \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac{5 i a^5 \cos ^7(c+d x)}{231 d}-\frac{2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac{2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d}+\frac{1}{33} \left (5 a^5\right ) \int \cos ^7(c+d x) \, dx\\ &=-\frac{5 i a^5 \cos ^7(c+d x)}{231 d}-\frac{2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac{2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d}-\frac{\left (5 a^5\right ) \operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{33 d}\\ &=-\frac{5 i a^5 \cos ^7(c+d x)}{231 d}+\frac{5 a^5 \sin (c+d x)}{33 d}-\frac{5 a^5 \sin ^3(c+d x)}{33 d}+\frac{a^5 \sin ^5(c+d x)}{11 d}-\frac{5 a^5 \sin ^7(c+d x)}{231 d}-\frac{2 i a^3 \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{33 d}-\frac{2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^4}{11 d}\\ \end{align*}
Mathematica [A] time = 1.19917, size = 118, normalized size = 0.74 \[ \frac{i a^5 (330 i \sin (2 (c+d x))+616 i \sin (4 (c+d x))-126 i \sin (6 (c+d x))-825 \cos (2 (c+d x))-770 \cos (4 (c+d x))+105 \cos (6 (c+d x))-462) (\cos (5 (c+2 d x))+i \sin (5 (c+2 d x)))}{7392 d (\cos (d x)+i \sin (d x))^5} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^5,x]
[Out]
((I/7392)*a^5*(-462 - 825*Cos[2*(c + d*x)] - 770*Cos[4*(c + d*x)] + 105*Cos[6*(c + d*x)] + (330*I)*Sin[2*(c +
d*x)] + (616*I)*Sin[4*(c + d*x)] - (126*I)*Sin[6*(c + d*x)])*(Cos[5*(c + 2*d*x)] + I*Sin[5*(c + 2*d*x)]))/(d*(
Cos[d*x] + I*Sin[d*x])^5)
________________________________________________________________________________________
Maple [B] time = 0.121, size = 317, normalized size = 2. \begin{align*}{\frac{1}{d} \left ( i{a}^{5} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{11}}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{99}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{693}} \right ) +5\,{a}^{5} \left ( -1/11\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{8}-1/33\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{\sin \left ( dx+c \right ) }{231} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+6/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+8/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) } \right ) -10\,i{a}^{5} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{9}}{11}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{9}}{99}} \right ) -10\,{a}^{5} \left ( -1/11\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{10}+{\frac{\sin \left ( dx+c \right ) }{99} \left ({\frac{128}{35}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7}}+{\frac{48\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{64\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) } \right ) -{\frac{5\,i}{11}}{a}^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{11}+{\frac{{a}^{5}\sin \left ( dx+c \right ) }{11} \left ({\frac{256}{63}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{10}+{\frac{10\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{9}}+{\frac{80\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{63}}+{\frac{32\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{21}}+{\frac{128\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{63}} \right ) } \right ) } \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))^5,x)
[Out]
1/d*(I*a^5*(-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)+5*a^5*(-1/11*si
n(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))-10*I*a^5*(-1/11*sin(d*x+c)^2*cos(d*x+c)^9-2/99*cos(d*x+c)^9)-10*a^5*(-1/11*sin(d*x+c)*cos(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))-5/11*I*a^5*
cos(d*x+c)^11+1/11*a^5*(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+128/63*co
s(d*x+c)^2)*sin(d*x+c))
________________________________________________________________________________________
Maxima [A] time = 1.13572, size = 332, normalized size = 2.09 \begin{align*} -\frac{315 i \, a^{5} \cos \left (d x + c\right )^{11} + 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^{5} + 70 i \,{\left (9 \, \cos \left (d x + c\right )^{11} - 11 \, \cos \left (d x + c\right )^{9}\right )} a^{5} + 2 \,{\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^{5} + 3 \,{\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^{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^{5}}{693 \, 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))^5,x, algorithm="maxima")
[Out]
-1/693*(315*I*a^5*cos(d*x + c)^11 + I*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^5 + 70*I
*(9*cos(d*x + c)^11 - 11*cos(d*x + c)^9)*a^5 + 2*(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^5 + 3*(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^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^5)/d
________________________________________________________________________________________
Fricas [A] time = 1.55928, size = 321, normalized size = 2.02 \begin{align*} \frac{{\left (-21 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} - 154 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} - 495 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 924 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 1155 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 1386 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 231 i \, a^{5}\right )} e^{\left (-i \, d x - i \, c\right )}}{14784 \, 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))^5,x, algorithm="fricas")
[Out]
1/14784*(-21*I*a^5*e^(12*I*d*x + 12*I*c) - 154*I*a^5*e^(10*I*d*x + 10*I*c) - 495*I*a^5*e^(8*I*d*x + 8*I*c) - 9
24*I*a^5*e^(6*I*d*x + 6*I*c) - 1155*I*a^5*e^(4*I*d*x + 4*I*c) - 1386*I*a^5*e^(2*I*d*x + 2*I*c) + 231*I*a^5)*e^
(-I*d*x - I*c)/d
________________________________________________________________________________________
Sympy [A] time = 1.45324, size = 267, normalized size = 1.68 \begin{align*} \begin{cases} \frac{\left (- 90194313216 i a^{5} d^{6} e^{12 i c} e^{11 i d x} - 661424963584 i a^{5} d^{6} e^{10 i c} e^{9 i d x} - 2126008811520 i a^{5} d^{6} e^{8 i c} e^{7 i d x} - 3968549781504 i a^{5} d^{6} e^{6 i c} e^{5 i d x} - 4960687226880 i a^{5} d^{6} e^{4 i c} e^{3 i d x} - 5952824672256 i a^{5} d^{6} e^{2 i c} e^{i d x} + 992137445376 i a^{5} d^{6} e^{- i d x}\right ) e^{- i c}}{63496796504064 d^{7}} & \text{for}\: 63496796504064 d^{7} e^{i c} \neq 0 \\\frac{x \left (a^{5} e^{12 i c} + 6 a^{5} e^{10 i c} + 15 a^{5} e^{8 i c} + 20 a^{5} e^{6 i c} + 15 a^{5} e^{4 i c} + 6 a^{5} e^{2 i c} + a^{5}\right ) e^{- i c}}{64} & \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))**5,x)
[Out]
Piecewise(((-90194313216*I*a**5*d**6*exp(12*I*c)*exp(11*I*d*x) - 661424963584*I*a**5*d**6*exp(10*I*c)*exp(9*I*
d*x) - 2126008811520*I*a**5*d**6*exp(8*I*c)*exp(7*I*d*x) - 3968549781504*I*a**5*d**6*exp(6*I*c)*exp(5*I*d*x) -
4960687226880*I*a**5*d**6*exp(4*I*c)*exp(3*I*d*x) - 5952824672256*I*a**5*d**6*exp(2*I*c)*exp(I*d*x) + 9921374
45376*I*a**5*d**6*exp(-I*d*x))*exp(-I*c)/(63496796504064*d**7), Ne(63496796504064*d**7*exp(I*c), 0)), (x*(a**5
*exp(12*I*c) + 6*a**5*exp(10*I*c) + 15*a**5*exp(8*I*c) + 20*a**5*exp(6*I*c) + 15*a**5*exp(4*I*c) + 6*a**5*exp(
2*I*c) + a**5)*exp(-I*c)/64, True))
________________________________________________________________________________________
Giac [B] time = 2.30674, size = 2439, normalized size = 15.34 \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))^5,x, algorithm="giac")
[Out]
-1/121110528*(168111405*a^5*e^(17*I*d*x + 9*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1344891240*a^5*e^(15*I*d*x + 7*I
*c)*log(I*e^(I*d*x + I*c) + 1) + 4707119340*a^5*e^(13*I*d*x + 5*I*c)*log(I*e^(I*d*x + I*c) + 1) + 9414238680*a
^5*e^(11*I*d*x + 3*I*c)*log(I*e^(I*d*x + I*c) + 1) + 11767798350*a^5*e^(9*I*d*x + I*c)*log(I*e^(I*d*x + I*c) +
1) + 9414238680*a^5*e^(7*I*d*x - I*c)*log(I*e^(I*d*x + I*c) + 1) + 4707119340*a^5*e^(5*I*d*x - 3*I*c)*log(I*e
^(I*d*x + I*c) + 1) + 1344891240*a^5*e^(3*I*d*x - 5*I*c)*log(I*e^(I*d*x + I*c) + 1) + 168111405*a^5*e^(I*d*x -
7*I*c)*log(I*e^(I*d*x + I*c) + 1) + 170251620*a^5*e^(17*I*d*x + 9*I*c)*log(I*e^(I*d*x + I*c) - 1) + 136201296
0*a^5*e^(15*I*d*x + 7*I*c)*log(I*e^(I*d*x + I*c) - 1) + 4767045360*a^5*e^(13*I*d*x + 5*I*c)*log(I*e^(I*d*x + I
*c) - 1) + 9534090720*a^5*e^(11*I*d*x + 3*I*c)*log(I*e^(I*d*x + I*c) - 1) + 11917613400*a^5*e^(9*I*d*x + I*c)*
log(I*e^(I*d*x + I*c) - 1) + 9534090720*a^5*e^(7*I*d*x - I*c)*log(I*e^(I*d*x + I*c) - 1) + 4767045360*a^5*e^(5
*I*d*x - 3*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1362012960*a^5*e^(3*I*d*x - 5*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1
70251620*a^5*e^(I*d*x - 7*I*c)*log(I*e^(I*d*x + I*c) - 1) - 168111405*a^5*e^(17*I*d*x + 9*I*c)*log(-I*e^(I*d*x
+ I*c) + 1) - 1344891240*a^5*e^(15*I*d*x + 7*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 4707119340*a^5*e^(13*I*d*x +
5*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 9414238680*a^5*e^(11*I*d*x + 3*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1176779
8350*a^5*e^(9*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) + 1) - 9414238680*a^5*e^(7*I*d*x - I*c)*log(-I*e^(I*d*x + I*
c) + 1) - 4707119340*a^5*e^(5*I*d*x - 3*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1344891240*a^5*e^(3*I*d*x - 5*I*c)*
log(-I*e^(I*d*x + I*c) + 1) - 168111405*a^5*e^(I*d*x - 7*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 170251620*a^5*e^(1
7*I*d*x + 9*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1362012960*a^5*e^(15*I*d*x + 7*I*c)*log(-I*e^(I*d*x + I*c) - 1)
- 4767045360*a^5*e^(13*I*d*x + 5*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 9534090720*a^5*e^(11*I*d*x + 3*I*c)*log(-
I*e^(I*d*x + I*c) - 1) - 11917613400*a^5*e^(9*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) - 1) - 9534090720*a^5*e^(7*I
*d*x - I*c)*log(-I*e^(I*d*x + I*c) - 1) - 4767045360*a^5*e^(5*I*d*x - 3*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 136
2012960*a^5*e^(3*I*d*x - 5*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 170251620*a^5*e^(I*d*x - 7*I*c)*log(-I*e^(I*d*x
+ I*c) - 1) + 2140215*a^5*e^(17*I*d*x + 9*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 17121720*a^5*e^(15*I*d*x + 7*I*c)
*log(I*e^(I*d*x) + e^(-I*c)) + 59926020*a^5*e^(13*I*d*x + 5*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 119852040*a^5*e
^(11*I*d*x + 3*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 149815050*a^5*e^(9*I*d*x + I*c)*log(I*e^(I*d*x) + e^(-I*c))
+ 119852040*a^5*e^(7*I*d*x - I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 59926020*a^5*e^(5*I*d*x - 3*I*c)*log(I*e^(I*d*
x) + e^(-I*c)) + 17121720*a^5*e^(3*I*d*x - 5*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 2140215*a^5*e^(I*d*x - 7*I*c)*
log(I*e^(I*d*x) + e^(-I*c)) - 2140215*a^5*e^(17*I*d*x + 9*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 17121720*a^5*e^(
15*I*d*x + 7*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 59926020*a^5*e^(13*I*d*x + 5*I*c)*log(-I*e^(I*d*x) + e^(-I*c)
) - 119852040*a^5*e^(11*I*d*x + 3*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 149815050*a^5*e^(9*I*d*x + I*c)*log(-I*e
^(I*d*x) + e^(-I*c)) - 119852040*a^5*e^(7*I*d*x - I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 59926020*a^5*e^(5*I*d*x
- 3*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 17121720*a^5*e^(3*I*d*x - 5*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 214021
5*a^5*e^(I*d*x - 7*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 172032*I*a^5*e^(28*I*d*x + 20*I*c) + 2637824*I*a^5*e^(2
6*I*d*x + 18*I*c) + 18964480*I*a^5*e^(24*I*d*x + 16*I*c) + 84967424*I*a^5*e^(22*I*d*x + 14*I*c) + 266248192*I*
a^5*e^(20*I*d*x + 12*I*c) + 624017408*I*a^5*e^(18*I*d*x + 10*I*c) + 1137074176*I*a^5*e^(16*I*d*x + 8*I*c) + 16
26275840*I*a^5*e^(14*I*d*x + 6*I*c) + 1792860160*I*a^5*e^(12*I*d*x + 4*I*c) + 1464320000*I*a^5*e^(10*I*d*x + 2
*I*c) + 295206912*I*a^5*e^(6*I*d*x - 2*I*c) + 47308800*I*a^5*e^(4*I*d*x - 4*I*c) - 3784704*I*a^5*e^(2*I*d*x -
6*I*c) + 832905216*I*a^5*e^(8*I*d*x) - 1892352*I*a^5*e^(-8*I*c))/(d*e^(17*I*d*x + 9*I*c) + 8*d*e^(15*I*d*x + 7
*I*c) + 28*d*e^(13*I*d*x + 5*I*c) + 56*d*e^(11*I*d*x + 3*I*c) + 70*d*e^(9*I*d*x + I*c) + 56*d*e^(7*I*d*x - I*c
) + 28*d*e^(5*I*d*x - 3*I*c) + 8*d*e^(3*I*d*x - 5*I*c) + d*e^(I*d*x - 7*I*c))