3.89 \(\int \cos ^{16}(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

Optimal. Leaf size=225 \[ -\frac{i a^{16}}{16 d (a-i a \tan (c+d x))^8}-\frac{i a^{15}}{28 d (a-i a \tan (c+d x))^7}-\frac{i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac{i a^{13}}{80 d (a-i a \tan (c+d x))^5}-\frac{i a^{12}}{128 d (a-i a \tan (c+d x))^4}-\frac{i a^{11}}{192 d (a-i a \tan (c+d x))^3}-\frac{i a^{10}}{256 d (a-i a \tan (c+d x))^2}-\frac{i a^9}{256 d (a-i a \tan (c+d x))}+\frac{a^8 x}{256} \]

[Out]

(a^8*x)/256 - ((I/16)*a^16)/(d*(a - I*a*Tan[c + d*x])^8) - ((I/28)*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/80)*a^13)/(d*(a - I*a*Tan[c + d*x])^5) - ((I/128)*a^12)/(d*(a - I*a
*Tan[c + d*x])^4) - ((I/192)*a^11)/(d*(a - I*a*Tan[c + d*x])^3) - ((I/256)*a^10)/(d*(a - I*a*Tan[c + d*x])^2)
- ((I/256)*a^9)/(d*(a - I*a*Tan[c + d*x]))

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Rubi [A]  time = 0.115079, antiderivative size = 225, 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} \[ -\frac{i a^{16}}{16 d (a-i a \tan (c+d x))^8}-\frac{i a^{15}}{28 d (a-i a \tan (c+d x))^7}-\frac{i a^{14}}{48 d (a-i a \tan (c+d x))^6}-\frac{i a^{13}}{80 d (a-i a \tan (c+d x))^5}-\frac{i a^{12}}{128 d (a-i a \tan (c+d x))^4}-\frac{i a^{11}}{192 d (a-i a \tan (c+d x))^3}-\frac{i a^{10}}{256 d (a-i a \tan (c+d x))^2}-\frac{i a^9}{256 d (a-i a \tan (c+d x))}+\frac{a^8 x}{256} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^16*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*x)/256 - ((I/16)*a^16)/(d*(a - I*a*Tan[c + d*x])^8) - ((I/28)*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/80)*a^13)/(d*(a - I*a*Tan[c + d*x])^5) - ((I/128)*a^12)/(d*(a - I*a
*Tan[c + d*x])^4) - ((I/192)*a^11)/(d*(a - I*a*Tan[c + d*x])^3) - ((I/256)*a^10)/(d*(a - I*a*Tan[c + d*x])^2)
- ((I/256)*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 ^{16}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{\left (i a^{17}\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^9 (a+x)} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^{17}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 a (a-x)^9}+\frac{1}{4 a^2 (a-x)^8}+\frac{1}{8 a^3 (a-x)^7}+\frac{1}{16 a^4 (a-x)^6}+\frac{1}{32 a^5 (a-x)^5}+\frac{1}{64 a^6 (a-x)^4}+\frac{1}{128 a^7 (a-x)^3}+\frac{1}{256 a^8 (a-x)^2}+\frac{1}{256 a^8 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i a^{16}}{16 d (a-i a \tan (c+d x))^8}-\frac{i a^{15}}{28 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}}{80 d (a-i a \tan (c+d x))^5}-\frac{i a^{12}}{128 d (a-i a \tan (c+d x))^4}-\frac{i a^{11}}{192 d (a-i a \tan (c+d x))^3}-\frac{i a^{10}}{256 d (a-i a \tan (c+d x))^2}-\frac{i a^9}{256 d (a-i a \tan (c+d x))}-\frac{\left (i a^9\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{256 d}\\ &=\frac{a^8 x}{256}-\frac{i a^{16}}{16 d (a-i a \tan (c+d x))^8}-\frac{i a^{15}}{28 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}}{80 d (a-i a \tan (c+d x))^5}-\frac{i a^{12}}{128 d (a-i a \tan (c+d x))^4}-\frac{i a^{11}}{192 d (a-i a \tan (c+d x))^3}-\frac{i a^{10}}{256 d (a-i a \tan (c+d x))^2}-\frac{i a^9}{256 d (a-i a \tan (c+d x))}\\ \end{align*}

Mathematica [A]  time = 5.3514, size = 166, normalized size = 0.74 \[ \frac{a^8 (-6272 \sin (2 (c+d x))-7840 \sin (4 (c+d x))-5760 \sin (6 (c+d x))-1680 i d x \sin (8 (c+d x))+105 \sin (8 (c+d x))-25088 i \cos (2 (c+d x))-15680 i \cos (4 (c+d x))-7680 i \cos (6 (c+d x))+1680 d x \cos (8 (c+d x))-105 i \cos (8 (c+d x))-14700 i) (\cos (8 (c+2 d x))+i \sin (8 (c+2 d x)))}{430080 d (\cos (d x)+i \sin (d x))^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^16*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*(-14700*I - (25088*I)*Cos[2*(c + d*x)] - (15680*I)*Cos[4*(c + d*x)] - (7680*I)*Cos[6*(c + d*x)] - (105*I)
*Cos[8*(c + d*x)] + 1680*d*x*Cos[8*(c + d*x)] - 6272*Sin[2*(c + d*x)] - 7840*Sin[4*(c + d*x)] - 5760*Sin[6*(c
+ d*x)] + 105*Sin[8*(c + d*x)] - (1680*I)*d*x*Sin[8*(c + d*x)])*(Cos[8*(c + 2*d*x)] + I*Sin[8*(c + 2*d*x)]))/(
430080*d*(Cos[d*x] + I*Sin[d*x])^8)

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Maple [B]  time = 0.141, size = 739, normalized size = 3.3 \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)^16*(a+I*a*tan(d*x+c))^8,x)

[Out]

1/d*(a^8*(-1/16*sin(d*x+c)^7*cos(d*x+c)^9-1/32*sin(d*x+c)^5*cos(d*x+c)^9-5/384*sin(d*x+c)^3*cos(d*x+c)^9-1/256
*sin(d*x+c)*cos(d*x+c)^9+1/2048*(cos(d*x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos(d*x+c))*sin(d*x+c)
+35/32768*d*x+35/32768*c)-8*I*a^8*(-1/16*sin(d*x+c)^6*cos(d*x+c)^10-3/112*sin(d*x+c)^4*cos(d*x+c)^10-1/112*sin
(d*x+c)^2*cos(d*x+c)^10-1/560*cos(d*x+c)^10)-28*a^8*(-1/16*sin(d*x+c)^5*cos(d*x+c)^11-5/224*sin(d*x+c)^3*cos(d
*x+c)^11-5/896*sin(d*x+c)*cos(d*x+c)^11+1/1792*(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+315/128*cos(d*x+c))*sin(d*x+c)+45/32768*d*x+45/32768*c)+56*I*a^8*(-1/16*sin(d*x+c)^4*cos(d*x+c)^12-1/56
*sin(d*x+c)^2*cos(d*x+c)^12-1/336*cos(d*x+c)^12)+70*a^8*(-1/16*sin(d*x+c)^3*cos(d*x+c)^13-3/224*sin(d*x+c)*cos
(d*x+c)^13+1/896*(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)+99/32768*d*x+99/32768*c)-56*I*a^8*(-1/16*sin(d*x+c)^2*cos(d*x+c)^14-1/112*cos
(d*x+c)^14)-28*a^8*(-1/16*sin(d*x+c)*cos(d*x+c)^15+1/224*(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)+429/32768
*d*x+429/32768*c)-1/2*I*a^8*cos(d*x+c)^16+a^8*(1/16*(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+1287/896*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)+6435/32768*d*x+6435/32768*c))

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Maxima [A]  time = 1.81627, size = 332, normalized size = 1.48 \begin{align*} \frac{13440 \,{\left (d x + c\right )} a^{8} + \frac{13440 \, a^{8} \tan \left (d x + c\right )^{15} + 103040 \, a^{8} \tan \left (d x + c\right )^{13} + 343168 \, a^{8} \tan \left (d x + c\right )^{11} + 646784 \, a^{8} \tan \left (d x + c\right )^{9} + 369024 \, a^{8} \tan \left (d x + c\right )^{7} + 2752512 i \, a^{8} \tan \left (d x + c\right )^{6} + 9061248 \, a^{8} \tan \left (d x + c\right )^{5} - 14680064 i \, a^{8} \tan \left (d x + c\right )^{4} - 15012480 \, a^{8} \tan \left (d x + c\right )^{3} + 9568256 i \, a^{8} \tan \left (d x + c\right )^{2} + 3427200 \, a^{8} \tan \left (d x + c\right ) - 524288 i \, a^{8}}{\tan \left (d x + c\right )^{16} + 8 \, \tan \left (d x + c\right )^{14} + 28 \, \tan \left (d x + c\right )^{12} + 56 \, \tan \left (d x + c\right )^{10} + 70 \, \tan \left (d x + c\right )^{8} + 56 \, \tan \left (d x + c\right )^{6} + 28 \, \tan \left (d x + c\right )^{4} + 8 \, \tan \left (d x + c\right )^{2} + 1}}{3440640 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^16*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

1/3440640*(13440*(d*x + c)*a^8 + (13440*a^8*tan(d*x + c)^15 + 103040*a^8*tan(d*x + c)^13 + 343168*a^8*tan(d*x
+ c)^11 + 646784*a^8*tan(d*x + c)^9 + 369024*a^8*tan(d*x + c)^7 + 2752512*I*a^8*tan(d*x + c)^6 + 9061248*a^8*t
an(d*x + c)^5 - 14680064*I*a^8*tan(d*x + c)^4 - 15012480*a^8*tan(d*x + c)^3 + 9568256*I*a^8*tan(d*x + c)^2 + 3
427200*a^8*tan(d*x + c) - 524288*I*a^8)/(tan(d*x + c)^16 + 8*tan(d*x + c)^14 + 28*tan(d*x + c)^12 + 56*tan(d*x
+ c)^10 + 70*tan(d*x + c)^8 + 56*tan(d*x + c)^6 + 28*tan(d*x + c)^4 + 8*tan(d*x + c)^2 + 1))/d

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Fricas [A]  time = 2.60139, size = 405, normalized size = 1.8 \begin{align*} \frac{1680 \, a^{8} d x - 105 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 960 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 3920 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 9408 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 14700 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 15680 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 11760 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 6720 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )}}{430080 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^16*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/430080*(1680*a^8*d*x - 105*I*a^8*e^(16*I*d*x + 16*I*c) - 960*I*a^8*e^(14*I*d*x + 14*I*c) - 3920*I*a^8*e^(12*
I*d*x + 12*I*c) - 9408*I*a^8*e^(10*I*d*x + 10*I*c) - 14700*I*a^8*e^(8*I*d*x + 8*I*c) - 15680*I*a^8*e^(6*I*d*x
+ 6*I*c) - 11760*I*a^8*e^(4*I*d*x + 4*I*c) - 6720*I*a^8*e^(2*I*d*x + 2*I*c))/d

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Sympy [A]  time = 2.35049, size = 325, normalized size = 1.44 \begin{align*} \frac{a^{8} x}{256} + \begin{cases} \frac{- 354658470655426560 i a^{8} d^{7} e^{16 i c} e^{16 i d x} - 3242591731706757120 i a^{8} d^{7} e^{14 i c} e^{14 i d x} - 13240582904469258240 i a^{8} d^{7} e^{12 i c} e^{12 i d x} - 31777398970726219776 i a^{8} d^{7} e^{10 i c} e^{10 i d x} - 49652185891759718400 i a^{8} d^{7} e^{8 i c} e^{8 i d x} - 52962331617877032960 i a^{8} d^{7} e^{6 i c} e^{6 i d x} - 39721748713407774720 i a^{8} d^{7} e^{4 i c} e^{4 i d x} - 22698142121947299840 i a^{8} d^{7} e^{2 i c} e^{2 i d x}}{1452681095804627189760 d^{8}} & \text{for}\: 1452681095804627189760 d^{8} \neq 0 \\x \left (\frac{a^{8} e^{16 i c}}{256} + \frac{a^{8} e^{14 i c}}{32} + \frac{7 a^{8} e^{12 i c}}{64} + \frac{7 a^{8} e^{10 i c}}{32} + \frac{35 a^{8} e^{8 i c}}{128} + \frac{7 a^{8} e^{6 i c}}{32} + \frac{7 a^{8} e^{4 i c}}{64} + \frac{a^{8} e^{2 i c}}{32}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**16*(a+I*a*tan(d*x+c))**8,x)

[Out]

a**8*x/256 + Piecewise(((-354658470655426560*I*a**8*d**7*exp(16*I*c)*exp(16*I*d*x) - 3242591731706757120*I*a**
8*d**7*exp(14*I*c)*exp(14*I*d*x) - 13240582904469258240*I*a**8*d**7*exp(12*I*c)*exp(12*I*d*x) - 31777398970726
219776*I*a**8*d**7*exp(10*I*c)*exp(10*I*d*x) - 49652185891759718400*I*a**8*d**7*exp(8*I*c)*exp(8*I*d*x) - 5296
2331617877032960*I*a**8*d**7*exp(6*I*c)*exp(6*I*d*x) - 39721748713407774720*I*a**8*d**7*exp(4*I*c)*exp(4*I*d*x
) - 22698142121947299840*I*a**8*d**7*exp(2*I*c)*exp(2*I*d*x))/(1452681095804627189760*d**8), Ne(14526810958046
27189760*d**8, 0)), (x*(a**8*exp(16*I*c)/256 + a**8*exp(14*I*c)/32 + 7*a**8*exp(12*I*c)/64 + 7*a**8*exp(10*I*c
)/32 + 35*a**8*exp(8*I*c)/128 + 7*a**8*exp(6*I*c)/32 + 7*a**8*exp(4*I*c)/64 + a**8*exp(2*I*c)/32), True))

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Giac [B]  time = 3.07861, size = 1967, normalized size = 8.74 \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)^16*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/55050240*(215040*a^8*d*x*e^(28*I*d*x + 14*I*c) + 3010560*a^8*d*x*e^(26*I*d*x + 12*I*c) + 19568640*a^8*d*x*e^
(24*I*d*x + 10*I*c) + 78274560*a^8*d*x*e^(22*I*d*x + 8*I*c) + 215255040*a^8*d*x*e^(20*I*d*x + 6*I*c) + 4305100
80*a^8*d*x*e^(18*I*d*x + 4*I*c) + 645765120*a^8*d*x*e^(16*I*d*x + 2*I*c) + 645765120*a^8*d*x*e^(12*I*d*x - 2*I
*c) + 430510080*a^8*d*x*e^(10*I*d*x - 4*I*c) + 215255040*a^8*d*x*e^(8*I*d*x - 6*I*c) + 78274560*a^8*d*x*e^(6*I
*d*x - 8*I*c) + 19568640*a^8*d*x*e^(4*I*d*x - 10*I*c) + 3010560*a^8*d*x*e^(2*I*d*x - 12*I*c) + 738017280*a^8*d
*x*e^(14*I*d*x) + 215040*a^8*d*x*e^(-14*I*c) - 103740*I*a^8*e^(28*I*d*x + 14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1)
- 1452360*I*a^8*e^(26*I*d*x + 12*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 9440340*I*a^8*e^(24*I*d*x + 10*I*c)*log(
e^(2*I*d*x + 2*I*c) + 1) - 37761360*I*a^8*e^(22*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 103843740*I*a^8*
e^(20*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 207687480*I*a^8*e^(18*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*
c) + 1) - 311531220*I*a^8*e^(16*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 311531220*I*a^8*e^(12*I*d*x - 2*
I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 207687480*I*a^8*e^(10*I*d*x - 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 103843
740*I*a^8*e^(8*I*d*x - 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 37761360*I*a^8*e^(6*I*d*x - 8*I*c)*log(e^(2*I*d*x
+ 2*I*c) + 1) - 9440340*I*a^8*e^(4*I*d*x - 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 1452360*I*a^8*e^(2*I*d*x -
12*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 356035680*I*a^8*e^(14*I*d*x)*log(e^(2*I*d*x + 2*I*c) + 1) - 103740*I*a^
8*e^(-14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 103740*I*a^8*e^(28*I*d*x + 14*I*c)*log(e^(2*I*d*x) + e^(-2*I*c))
+ 1452360*I*a^8*e^(26*I*d*x + 12*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 9440340*I*a^8*e^(24*I*d*x + 10*I*c)*log(
e^(2*I*d*x) + e^(-2*I*c)) + 37761360*I*a^8*e^(22*I*d*x + 8*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 103843740*I*a^
8*e^(20*I*d*x + 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 207687480*I*a^8*e^(18*I*d*x + 4*I*c)*log(e^(2*I*d*x) +
e^(-2*I*c)) + 311531220*I*a^8*e^(16*I*d*x + 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 311531220*I*a^8*e^(12*I*d*x
- 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 207687480*I*a^8*e^(10*I*d*x - 4*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) +
103843740*I*a^8*e^(8*I*d*x - 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 37761360*I*a^8*e^(6*I*d*x - 8*I*c)*log(e^
(2*I*d*x) + e^(-2*I*c)) + 9440340*I*a^8*e^(4*I*d*x - 10*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 1452360*I*a^8*e^(
2*I*d*x - 12*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 356035680*I*a^8*e^(14*I*d*x)*log(e^(2*I*d*x) + e^(-2*I*c)) +
103740*I*a^8*e^(-14*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) - 13440*I*a^8*e^(44*I*d*x + 30*I*c) - 311040*I*a^8*e^(
42*I*d*x + 28*I*c) - 3445120*I*a^8*e^(40*I*d*x + 26*I*c) - 24303104*I*a^8*e^(38*I*d*x + 24*I*c) - 122582656*I*
a^8*e^(36*I*d*x + 22*I*c) - 470484224*I*a^8*e^(34*I*d*x + 20*I*c) - 1427794816*I*a^8*e^(32*I*d*x + 18*I*c) - 3
514563584*I*a^8*e^(30*I*d*x + 16*I*c) - 7142793088*I*a^8*e^(28*I*d*x + 14*I*c) - 12136447232*I*a^8*e^(26*I*d*x
+ 12*I*c) - 17387563648*I*a^8*e^(24*I*d*x + 10*I*c) - 21108086272*I*a^8*e^(22*I*d*x + 8*I*c) - 21740071808*I*
a^8*e^(20*I*d*x + 6*I*c) - 18942724864*I*a^8*e^(18*I*d*x + 4*I*c) - 13859732096*I*a^8*e^(16*I*d*x + 2*I*c) - 4
147974656*I*a^8*e^(12*I*d*x - 2*I*c) - 1619129344*I*a^8*e^(10*I*d*x - 4*I*c) - 480058880*I*a^8*e^(8*I*d*x - 6*
I*c) - 101355520*I*a^8*e^(6*I*d*x - 8*I*c) - 13547520*I*a^8*e^(4*I*d*x - 10*I*c) - 860160*I*a^8*e^(2*I*d*x - 1
2*I*c) - 8407312384*I*a^8*e^(14*I*d*x))/(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))