3.75 \(\int \cos ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx\)
Optimal. Leaf size=141 \[ \frac{a^5 \sin ^5(c+d x)}{105 d}-\frac{2 a^5 \sin ^3(c+d x)}{63 d}+\frac{a^5 \sin (c+d x)}{21 d}-\frac{i a^5 \cos ^5(c+d x)}{105 d}-\frac{2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d} \]
[Out]
((-I/105)*a^5*Cos[c + d*x]^5)/d + (a^5*Sin[c + d*x])/(21*d) - (2*a^5*Sin[c + d*x]^3)/(63*d) + (a^5*Sin[c + d*x
]^5)/(105*d) - (((2*I)/63)*a^3*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2)/d - (((2*I)/9)*a*Cos[c + d*x]^9*(a + I
*a*Tan[c + d*x])^4)/d
________________________________________________________________________________________
Rubi [A] time = 0.121577, antiderivative size = 141, 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} \[ \frac{a^5 \sin ^5(c+d x)}{105 d}-\frac{2 a^5 \sin ^3(c+d x)}{63 d}+\frac{a^5 \sin (c+d x)}{21 d}-\frac{i a^5 \cos ^5(c+d x)}{105 d}-\frac{2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^5,x]
[Out]
((-I/105)*a^5*Cos[c + d*x]^5)/d + (a^5*Sin[c + d*x])/(21*d) - (2*a^5*Sin[c + d*x]^3)/(63*d) + (a^5*Sin[c + d*x
]^5)/(105*d) - (((2*I)/63)*a^3*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2)/d - (((2*I)/9)*a*Cos[c + d*x]^9*(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 ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}+\frac{1}{9} a^2 \int \cos ^7(c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac{2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}+\frac{1}{21} a^4 \int \cos ^5(c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac{i a^5 \cos ^5(c+d x)}{105 d}-\frac{2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}+\frac{1}{21} a^5 \int \cos ^5(c+d x) \, dx\\ &=-\frac{i a^5 \cos ^5(c+d x)}{105 d}-\frac{2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}-\frac{a^5 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{21 d}\\ &=-\frac{i a^5 \cos ^5(c+d x)}{105 d}+\frac{a^5 \sin (c+d x)}{21 d}-\frac{2 a^5 \sin ^3(c+d x)}{63 d}+\frac{a^5 \sin ^5(c+d x)}{105 d}-\frac{2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}\\ \end{align*}
Mathematica [A] time = 0.833622, size = 94, normalized size = 0.67 \[ \frac{a^5 (-120 i \sin (2 (c+d x))-140 i \sin (4 (c+d x))+300 \cos (2 (c+d x))+175 \cos (4 (c+d x))+189) (\sin (5 (c+2 d x))-i \cos (5 (c+2 d x)))}{2520 d (\cos (d x)+i \sin (d x))^5} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^5,x]
[Out]
(a^5*(189 + 300*Cos[2*(c + d*x)] + 175*Cos[4*(c + d*x)] - (120*I)*Sin[2*(c + d*x)] - (140*I)*Sin[4*(c + d*x)])
*((-I)*Cos[5*(c + 2*d*x)] + Sin[5*(c + 2*d*x)]))/(2520*d*(Cos[d*x] + I*Sin[d*x])^5)
________________________________________________________________________________________
Maple [B] time = 0.09, size = 287, 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 ) ^{5}}{9}}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{63}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{315}} \right ) +5\,{a}^{5} \left ( -1/9\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}-1/21\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) +{\frac{ \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{105}} \right ) -10\,i{a}^{5} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) -10\,{a}^{5} \left ( -1/9\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{\sin \left ( dx+c \right ) }{63} \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 ) -{\frac{5\,i}{9}}{a}^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{9}+{\frac{{a}^{5}\sin \left ( dx+c \right ) }{9} \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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^5,x)
[Out]
1/d*(I*a^5*(-1/9*sin(d*x+c)^4*cos(d*x+c)^5-4/63*sin(d*x+c)^2*cos(d*x+c)^5-8/315*cos(d*x+c)^5)+5*a^5*(-1/9*sin(
d*x+c)^3*cos(d*x+c)^6-1/21*cos(d*x+c)^6*sin(d*x+c)+1/105*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-10*I*
a^5*(-1/9*sin(d*x+c)^2*cos(d*x+c)^7-2/63*cos(d*x+c)^7)-10*a^5*(-1/9*sin(d*x+c)*cos(d*x+c)^8+1/63*(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))-5/9*I*a^5*cos(d*x+c)^9+1/9*a^5*(128/35+cos(d*x+c)^8+8/7*c
os(d*x+c)^6+48/35*cos(d*x+c)^4+64/35*cos(d*x+c)^2)*sin(d*x+c))
________________________________________________________________________________________
Maxima [A] time = 1.19079, size = 293, normalized size = 2.08 \begin{align*} -\frac{175 i \, a^{5} \cos \left (d x + c\right )^{9} + i \,{\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{5} + 50 i \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{5} - 5 \,{\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} a^{5} - 10 \,{\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{5} -{\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{5}}{315 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^5,x, algorithm="maxima")
[Out]
-1/315*(175*I*a^5*cos(d*x + c)^9 + I*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*a^5 + 50*I*(7
*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^5 - 5*(35*sin(d*x + c)^9 - 90*sin(d*x + c)^7 + 63*sin(d*x + c)^5)*a^5 -
10*(35*sin(d*x + c)^9 - 135*sin(d*x + c)^7 + 189*sin(d*x + c)^5 - 105*sin(d*x + c)^3)*a^5 - (35*sin(d*x + c)^9
- 180*sin(d*x + c)^7 + 378*sin(d*x + c)^5 - 420*sin(d*x + c)^3 + 315*sin(d*x + c))*a^5)/d
________________________________________________________________________________________
Fricas [A] time = 1.58065, size = 224, normalized size = 1.59 \begin{align*} \frac{-35 i \, a^{5} e^{\left (9 i \, d x + 9 i \, c\right )} - 180 i \, a^{5} e^{\left (7 i \, d x + 7 i \, c\right )} - 378 i \, a^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 420 i \, a^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - 315 i \, a^{5} e^{\left (i \, d x + i \, c\right )}}{5040 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^5,x, algorithm="fricas")
[Out]
1/5040*(-35*I*a^5*e^(9*I*d*x + 9*I*c) - 180*I*a^5*e^(7*I*d*x + 7*I*c) - 378*I*a^5*e^(5*I*d*x + 5*I*c) - 420*I*
a^5*e^(3*I*d*x + 3*I*c) - 315*I*a^5*e^(I*d*x + I*c))/d
________________________________________________________________________________________
Sympy [A] time = 1.11539, size = 194, normalized size = 1.38 \begin{align*} \begin{cases} \frac{- 215040 i a^{5} d^{4} e^{9 i c} e^{9 i d x} - 1105920 i a^{5} d^{4} e^{7 i c} e^{7 i d x} - 2322432 i a^{5} d^{4} e^{5 i c} e^{5 i d x} - 2580480 i a^{5} d^{4} e^{3 i c} e^{3 i d x} - 1935360 i a^{5} d^{4} e^{i c} e^{i d x}}{30965760 d^{5}} & \text{for}\: 30965760 d^{5} \neq 0 \\x \left (\frac{a^{5} e^{9 i c}}{16} + \frac{a^{5} e^{7 i c}}{4} + \frac{3 a^{5} e^{5 i c}}{8} + \frac{a^{5} e^{3 i c}}{4} + \frac{a^{5} e^{i c}}{16}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**9*(a+I*a*tan(d*x+c))**5,x)
[Out]
Piecewise(((-215040*I*a**5*d**4*exp(9*I*c)*exp(9*I*d*x) - 1105920*I*a**5*d**4*exp(7*I*c)*exp(7*I*d*x) - 232243
2*I*a**5*d**4*exp(5*I*c)*exp(5*I*d*x) - 2580480*I*a**5*d**4*exp(3*I*c)*exp(3*I*d*x) - 1935360*I*a**5*d**4*exp(
I*c)*exp(I*d*x))/(30965760*d**5), Ne(30965760*d**5, 0)), (x*(a**5*exp(9*I*c)/16 + a**5*exp(7*I*c)/4 + 3*a**5*e
xp(5*I*c)/8 + a**5*exp(3*I*c)/4 + a**5*exp(I*c)/16), True))
________________________________________________________________________________________
Giac [B] time = 2.18092, size = 2329, normalized size = 16.52 \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)^9*(a+I*a*tan(d*x+c))^5,x, algorithm="giac")
[Out]
-1/41287680*(69853455*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 558827640*a^5*e^(14*I*d*x + 6*I*c)
*log(I*e^(I*d*x + I*c) + 1) + 1955896740*a^5*e^(12*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 3911793480*a^5*
e^(10*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 3911793480*a^5*e^(6*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1
) + 1955896740*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 558827640*a^5*e^(2*I*d*x - 6*I*c)*log(I*e^
(I*d*x + I*c) + 1) + 4889741850*a^5*e^(8*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 69853455*a^5*e^(-8*I*c)*log(I*e^(
I*d*x + I*c) + 1) + 70703325*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 565626600*a^5*e^(14*I*d*x +
6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1979693100*a^5*e^(12*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 39593862
00*a^5*e^(10*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 3959386200*a^5*e^(6*I*d*x - 2*I*c)*log(I*e^(I*d*x + I
*c) - 1) + 1979693100*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 565626600*a^5*e^(2*I*d*x - 6*I*c)*l
og(I*e^(I*d*x + I*c) - 1) + 4949232750*a^5*e^(8*I*d*x)*log(I*e^(I*d*x + I*c) - 1) + 70703325*a^5*e^(-8*I*c)*lo
g(I*e^(I*d*x + I*c) - 1) - 69853455*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 558827640*a^5*e^(14
*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1955896740*a^5*e^(12*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1)
- 3911793480*a^5*e^(10*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 3911793480*a^5*e^(6*I*d*x - 2*I*c)*log(-I*
e^(I*d*x + I*c) + 1) - 1955896740*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 558827640*a^5*e^(2*I*d
*x - 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 4889741850*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) - 69853455*a^
5*e^(-8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 70703325*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 565
626600*a^5*e^(14*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1979693100*a^5*e^(12*I*d*x + 4*I*c)*log(-I*e^(I*
d*x + I*c) - 1) - 3959386200*a^5*e^(10*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 3959386200*a^5*e^(6*I*d*x
- 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1979693100*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 565626
600*a^5*e^(2*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 4949232750*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x + I*c) -
1) - 70703325*a^5*e^(-8*I*c)*log(-I*e^(I*d*x + I*c) - 1) + 849870*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e
^(-I*c)) + 6798960*a^5*e^(14*I*d*x + 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 23796360*a^5*e^(12*I*d*x + 4*I*c)*lo
g(I*e^(I*d*x) + e^(-I*c)) + 47592720*a^5*e^(10*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 47592720*a^5*e^(6*
I*d*x - 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 23796360*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 67
98960*a^5*e^(2*I*d*x - 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 59490900*a^5*e^(8*I*d*x)*log(I*e^(I*d*x) + e^(-I*c
)) + 849870*a^5*e^(-8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 849870*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x) + e^
(-I*c)) - 6798960*a^5*e^(14*I*d*x + 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 23796360*a^5*e^(12*I*d*x + 4*I*c)*lo
g(-I*e^(I*d*x) + e^(-I*c)) - 47592720*a^5*e^(10*I*d*x + 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 47592720*a^5*e^(
6*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 23796360*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c))
- 6798960*a^5*e^(2*I*d*x - 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 59490900*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x) + e
^(-I*c)) - 849870*a^5*e^(-8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 286720*I*a^5*e^(25*I*d*x + 17*I*c) + 3768320*I
*a^5*e^(23*I*d*x + 15*I*c) + 22921216*I*a^5*e^(21*I*d*x + 13*I*c) + 85557248*I*a^5*e^(19*I*d*x + 11*I*c) + 219
455488*I*a^5*e^(17*I*d*x + 9*I*c) + 409665536*I*a^5*e^(15*I*d*x + 7*I*c) + 572293120*I*a^5*e^(13*I*d*x + 5*I*c
) + 602341376*I*a^5*e^(11*I*d*x + 3*I*c) + 472096768*I*a^5*e^(9*I*d*x + I*c) + 267091968*I*a^5*e^(7*I*d*x - I*
c) + 102875136*I*a^5*e^(5*I*d*x - 3*I*c) + 24084480*I*a^5*e^(3*I*d*x - 5*I*c) + 2580480*I*a^5*e^(I*d*x - 7*I*c
))/(d*e^(16*I*d*x + 8*I*c) + 8*d*e^(14*I*d*x + 6*I*c) + 28*d*e^(12*I*d*x + 4*I*c) + 56*d*e^(10*I*d*x + 2*I*c)
+ 56*d*e^(6*I*d*x - 2*I*c) + 28*d*e^(4*I*d*x - 4*I*c) + 8*d*e^(2*I*d*x - 6*I*c) + 70*d*e^(8*I*d*x) + d*e^(-8*I
*c))