3.94 \(\int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

Optimal. Leaf size=152 \[ \frac{a^8 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac{2 i a^2 \cos ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac{2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d} \]

[Out]

(a^8*ArcTanh[Sin[c + d*x]])/d + (((2*I)/5)*a^3*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/d - (((2*I)/7)*a*Cos[c
+ d*x]^7*(a + I*a*Tan[c + d*x])^7)/d - (((2*I)/3)*a^2*Cos[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x])^3)/d + ((2*I)
*Cos[c + d*x]*(a^8 + I*a^8*Tan[c + d*x]))/d

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Rubi [A]  time = 0.159918, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3496, 3770} \[ \frac{a^8 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac{2 i a^2 \cos ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac{2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*ArcTanh[Sin[c + d*x]])/d + (((2*I)/5)*a^3*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/d - (((2*I)/7)*a*Cos[c
+ d*x]^7*(a + I*a*Tan[c + d*x])^7)/d - (((2*I)/3)*a^2*Cos[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x])^3)/d + ((2*I)
*Cos[c + d*x]*(a^8 + I*a^8*Tan[c + d*x]))/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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}-a^2 \int \cos ^5(c+d x) (a+i a \tan (c+d x))^6 \, dx\\ &=\frac{2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}+a^4 \int \cos ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx\\ &=-\frac{2 i a^5 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac{2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}-a^6 \int \cos (c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=-\frac{2 i a^5 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac{2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}+\frac{2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d}+a^8 \int \sec (c+d x) \, dx\\ &=\frac{a^8 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 i a^5 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac{2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}+\frac{2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d}\\ \end{align*}

Mathematica [B]  time = 2.23424, size = 305, normalized size = 2.01 \[ \frac{a^8 \left (\cos \left (\frac{1}{2} (7 c+23 d x)\right )+i \sin \left (\frac{1}{2} (7 c+23 d x)\right )\right ) \left (-70 \sin \left (\frac{1}{2} (c+d x)\right )-42 \sin \left (\frac{3}{2} (c+d x)\right )+210 \sin \left (\frac{5}{2} (c+d x)\right )+30 \sin \left (\frac{7}{2} (c+d x)\right )-70 i \cos \left (\frac{1}{2} (c+d x)\right )+42 i \cos \left (\frac{3}{2} (c+d x)\right )+210 i \cos \left (\frac{5}{2} (c+d x)\right )-30 i \cos \left (\frac{7}{2} (c+d x)\right )-105 \cos \left (\frac{7}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+105 \cos \left (\frac{7}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+105 i \sin \left (\frac{7}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-105 i \sin \left (\frac{7}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{105 d (\cos (d x)+i \sin (d x))^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*((-70*I)*Cos[(c + d*x)/2] + (42*I)*Cos[(3*(c + d*x))/2] + (210*I)*Cos[(5*(c + d*x))/2] - (30*I)*Cos[(7*(c
+ d*x))/2] - 105*Cos[(7*(c + d*x))/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*Cos[(7*(c + d*x))/2]*Log
[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 70*Sin[(c + d*x)/2] - 42*Sin[(3*(c + d*x))/2] + 210*Sin[(5*(c + d*x))/
2] + 30*Sin[(7*(c + d*x))/2] + (105*I)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[(7*(c + d*x))/2] - (105*I)
*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[(7*(c + d*x))/2])*(Cos[(7*c + 23*d*x)/2] + I*Sin[(7*c + 23*d*x)/
2]))/(105*d*(Cos[d*x] + I*Sin[d*x])^8)

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Maple [B]  time = 0.082, size = 385, normalized size = 2.5 \begin{align*} -{\frac{29\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{139\,{a}^{8}\sin \left ( dx+c \right ) }{105\,d}}+{\frac{{a}^{8}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{\frac{48\,i}{35}}{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{{\frac{16\,i}{5}}{a}^{8} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d}}-{\frac{8\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{{\frac{64\,i}{15}}{a}^{8} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{{\frac{32\,i}{5}}{a}^{8} \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{{\frac{64\,i}{35}}{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{8\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d}}+{\frac{{\frac{8\,i}{7}}{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d}}-10\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d}}-{\frac{232\,{a}^{8}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35\,d}}+{\frac{122\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{8}}{105\,d}}-{\frac{{\frac{8\,i}{7}}{a}^{8} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d}}+{\frac{29\,{a}^{8}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7\,d}}+{\frac{{\frac{128\,i}{35}}{a}^{8}\cos \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^8,x)

[Out]

-29/7*a^8*sin(d*x+c)^7/d-1/5*a^8*sin(d*x+c)^5/d-1/3*a^8*sin(d*x+c)^3/d+139/105*a^8*sin(d*x+c)/d+1/d*a^8*ln(sec
(d*x+c)+tan(d*x+c))+48/35*I/d*a^8*cos(d*x+c)*sin(d*x+c)^4+16/5*I/d*a^8*cos(d*x+c)^5-8*I/d*a^8*sin(d*x+c)^4*cos
(d*x+c)^3-64/15*I/d*a^8*cos(d*x+c)^3-32/5*I/d*a^8*cos(d*x+c)^3*sin(d*x+c)^2+64/35*I/d*a^8*cos(d*x+c)*sin(d*x+c
)^2+8*I/d*a^8*sin(d*x+c)^2*cos(d*x+c)^5+8/7*I/d*a^8*cos(d*x+c)*sin(d*x+c)^6-10/d*a^8*sin(d*x+c)^3*cos(d*x+c)^4
-232/35/d*a^8*sin(d*x+c)*cos(d*x+c)^4+122/105/d*sin(d*x+c)*cos(d*x+c)^2*a^8-8/7*I/d*a^8*cos(d*x+c)^7+29/7/d*a^
8*sin(d*x+c)*cos(d*x+c)^6+128/35*I/d*a^8*cos(d*x+c)

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Maxima [B]  time = 1.03751, size = 417, normalized size = 2.74 \begin{align*} -\frac{240 i \, a^{8} \cos \left (d x + c\right )^{7} + 840 \, a^{8} \sin \left (d x + c\right )^{7} + 112 i \,{\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 336 i \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 48 i \,{\left (5 \, \cos \left (d x + c\right )^{7} - 21 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )\right )} a^{8} +{\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a^{8} + 56 \,{\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{8} + 420 \,{\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} a^{8} + 6 \,{\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{8}}{210 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/210*(240*I*a^8*cos(d*x + c)^7 + 840*a^8*sin(d*x + c)^7 + 112*I*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*
cos(d*x + c)^3)*a^8 + 336*I*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^8 + 48*I*(5*cos(d*x + c)^7 - 21*cos(d*x +
c)^5 + 35*cos(d*x + c)^3 - 35*cos(d*x + c))*a^8 + (30*sin(d*x + c)^7 + 42*sin(d*x + c)^5 + 70*sin(d*x + c)^3 -
105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1) + 210*sin(d*x + c))*a^8 + 56*(15*sin(d*x + c)^7 - 42*si
n(d*x + c)^5 + 35*sin(d*x + c)^3)*a^8 + 420*(5*sin(d*x + c)^7 - 7*sin(d*x + c)^5)*a^8 + 6*(5*sin(d*x + c)^7 -
21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*a^8)/d

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Fricas [A]  time = 1.91751, size = 271, normalized size = 1.78 \begin{align*} \frac{-30 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} + 42 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 70 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} + 210 i \, a^{8} e^{\left (i \, d x + i \, c\right )} + 105 \, a^{8} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \, a^{8} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/105*(-30*I*a^8*e^(7*I*d*x + 7*I*c) + 42*I*a^8*e^(5*I*d*x + 5*I*c) - 70*I*a^8*e^(3*I*d*x + 3*I*c) + 210*I*a^8
*e^(I*d*x + I*c) + 105*a^8*log(e^(I*d*x + I*c) + I) - 105*a^8*log(e^(I*d*x + I*c) - I))/d

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Sympy [A]  time = 1.51796, size = 189, normalized size = 1.24 \begin{align*} \frac{a^{8} \left (- \log{\left (e^{i d x} - i e^{- i c} \right )} + \log{\left (e^{i d x} + i e^{- i c} \right )}\right )}{d} + \begin{cases} \frac{- 30 i a^{8} d^{3} e^{7 i c} e^{7 i d x} + 42 i a^{8} d^{3} e^{5 i c} e^{5 i d x} - 70 i a^{8} d^{3} e^{3 i c} e^{3 i d x} + 210 i a^{8} d^{3} e^{i c} e^{i d x}}{105 d^{4}} & \text{for}\: 105 d^{4} \neq 0 \\x \left (2 a^{8} e^{7 i c} - 2 a^{8} e^{5 i c} + 2 a^{8} e^{3 i c} - 2 a^{8} e^{i c}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7*(a+I*a*tan(d*x+c))**8,x)

[Out]

a**8*(-log(exp(I*d*x) - I*exp(-I*c)) + log(exp(I*d*x) + I*exp(-I*c)))/d + Piecewise(((-30*I*a**8*d**3*exp(7*I*
c)*exp(7*I*d*x) + 42*I*a**8*d**3*exp(5*I*c)*exp(5*I*d*x) - 70*I*a**8*d**3*exp(3*I*c)*exp(3*I*d*x) + 210*I*a**8
*d**3*exp(I*c)*exp(I*d*x))/(105*d**4), Ne(105*d**4, 0)), (x*(2*a**8*exp(7*I*c) - 2*a**8*exp(5*I*c) + 2*a**8*ex
p(3*I*c) - 2*a**8*exp(I*c)), True))

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Giac [B]  time = 3.46648, size = 3865, normalized size = 25.43 \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)^7*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/55050240*(1635552135*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 22897729890*a^8*e^(26*I*d*x + 12
*I*c)*log(I*e^(I*d*x + I*c) + 1) + 148835244285*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 5953409
77140*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1637187687135*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*
d*x + I*c) + 1) + 3274375374270*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 4911563061405*a^8*e^(16*
I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 4911563061405*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1)
+ 3274375374270*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1637187687135*a^8*e^(8*I*d*x - 6*I*c)*lo
g(I*e^(I*d*x + I*c) + 1) + 595340977140*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 148835244285*a^8*
e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 22897729890*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) +
1) + 5613214927320*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 1635552135*a^8*e^(-14*I*c)*log(I*e^(I*d*x +
I*c) + 1) + 1690450650*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) - 1) + 23666309100*a^8*e^(26*I*d*x + 12
*I*c)*log(I*e^(I*d*x + I*c) - 1) + 153831009150*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 6153240
36600*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1692141100650*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*
d*x + I*c) - 1) + 3384282201300*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5076423301950*a^8*e^(16*
I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5076423301950*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1)
+ 3384282201300*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1692141100650*a^8*e^(8*I*d*x - 6*I*c)*lo
g(I*e^(I*d*x + I*c) - 1) + 615324036600*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 153831009150*a^8*
e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 23666309100*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) -
1) + 5801626630800*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) - 1) + 1690450650*a^8*e^(-14*I*c)*log(I*e^(I*d*x +
I*c) - 1) - 1635552135*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 22897729890*a^8*e^(26*I*d*x + 1
2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 148835244285*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 5953
40977140*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1637187687135*a^8*e^(20*I*d*x + 6*I*c)*log(-I*
e^(I*d*x + I*c) + 1) - 3274375374270*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 4911563061405*a^8*
e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 4911563061405*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*
c) + 1) - 3274375374270*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1637187687135*a^8*e^(8*I*d*x -
6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 595340977140*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 148835
244285*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 22897729890*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I
*d*x + I*c) + 1) - 5613214927320*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) - 1635552135*a^8*e^(-14*I*c)*log
(-I*e^(I*d*x + I*c) + 1) - 1690450650*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 23666309100*a^8*
e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 153831009150*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I
*c) - 1) - 615324036600*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1692141100650*a^8*e^(20*I*d*x +
6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 3384282201300*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 507
6423301950*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 5076423301950*a^8*e^(12*I*d*x - 2*I*c)*log(-
I*e^(I*d*x + I*c) - 1) - 3384282201300*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1692141100650*a^
8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 615324036600*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c
) - 1) - 153831009150*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 23666309100*a^8*e^(2*I*d*x - 12*I
*c)*log(-I*e^(I*d*x + I*c) - 1) - 5801626630800*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 1690450650*a^8*
e^(-14*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 151725*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 21241
50*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 13806975*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x)
+ e^(-I*c)) - 55227900*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 151876725*a^8*e^(20*I*d*x + 6*I*
c)*log(I*e^(I*d*x) + e^(-I*c)) - 303753450*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 455630175*a^
8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 455630175*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x) + e^(-
I*c)) - 303753450*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 151876725*a^8*e^(8*I*d*x - 6*I*c)*log
(I*e^(I*d*x) + e^(-I*c)) - 55227900*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 13806975*a^8*e^(4*I*
d*x - 10*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 2124150*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 520
720200*a^8*e^(14*I*d*x)*log(I*e^(I*d*x) + e^(-I*c)) - 151725*a^8*e^(-14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 151
725*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 2124150*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x
) + e^(-I*c)) + 13806975*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 55227900*a^8*e^(22*I*d*x + 8
*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 151876725*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 3037534
50*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 455630175*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x)
+ e^(-I*c)) + 455630175*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 303753450*a^8*e^(10*I*d*x - 4
*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 151876725*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 55227900
*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 13806975*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x) + e
^(-I*c)) + 2124150*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 520720200*a^8*e^(14*I*d*x)*log(-I*e
^(I*d*x) + e^(-I*c)) + 151725*a^8*e^(-14*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 15728640*I*a^8*e^(35*I*d*x + 21*I
*c) - 198180864*I*a^8*e^(33*I*d*x + 19*I*c) - 1159725056*I*a^8*e^(31*I*d*x + 17*I*c) - 4125097984*I*a^8*e^(29*
I*d*x + 15*I*c) - 9527361536*I*a^8*e^(27*I*d*x + 13*I*c) - 12786335744*I*a^8*e^(25*I*d*x + 11*I*c) + 190840832
*I*a^8*e^(23*I*d*x + 9*I*c) + 48882515968*I*a^8*e^(21*I*d*x + 7*I*c) + 138550444032*I*a^8*e^(19*I*d*x + 5*I*c)
+ 239314403328*I*a^8*e^(17*I*d*x + 3*I*c) + 295994130432*I*a^8*e^(15*I*d*x + I*c) + 273474912256*I*a^8*e^(13*
I*d*x - I*c) + 190268309504*I*a^8*e^(11*I*d*x - 3*I*c) + 98635350016*I*a^8*e^(9*I*d*x - 5*I*c) + 37029412864*I
*a^8*e^(7*I*d*x - 7*I*c) + 9527361536*I*a^8*e^(5*I*d*x - 9*I*c) + 1504706560*I*a^8*e^(3*I*d*x - 11*I*c) + 1101
00480*I*a^8*e^(I*d*x - 13*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))