3.81 \(\int (a+i a \tan (c+d x))^8 \, dx\)

Optimal. Leaf size=200 \[ -\frac{64 a^8 \tan (c+d x)}{d}+\frac{4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac{i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac{16 i a^2 \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac{2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac{16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d}-\frac{128 i a^8 \log (\cos (c+d x))}{d}+128 a^8 x+\frac{i a (a+i a \tan (c+d x))^7}{7 d} \]

[Out]

128*a^8*x - ((128*I)*a^8*Log[Cos[c + d*x]])/d - (64*a^8*Tan[c + d*x])/d + (((4*I)/5)*a^3*(a + I*a*Tan[c + d*x]
)^5)/d + ((I/3)*a^2*(a + I*a*Tan[c + d*x])^6)/d + ((I/7)*a*(a + I*a*Tan[c + d*x])^7)/d + (((16*I)/3)*a^2*(a^2
+ I*a^2*Tan[c + d*x])^3)/d + ((2*I)*(a^2 + I*a^2*Tan[c + d*x])^4)/d + ((16*I)*(a^4 + I*a^4*Tan[c + d*x])^2)/d

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Rubi [A]  time = 0.138817, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3478, 3477, 3475} \[ -\frac{64 a^8 \tan (c+d x)}{d}+\frac{4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac{i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac{16 i a^2 \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac{2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac{16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d}-\frac{128 i a^8 \log (\cos (c+d x))}{d}+128 a^8 x+\frac{i a (a+i a \tan (c+d x))^7}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[c + d*x])^8,x]

[Out]

128*a^8*x - ((128*I)*a^8*Log[Cos[c + d*x]])/d - (64*a^8*Tan[c + d*x])/d + (((4*I)/5)*a^3*(a + I*a*Tan[c + d*x]
)^5)/d + ((I/3)*a^2*(a + I*a*Tan[c + d*x])^6)/d + ((I/7)*a*(a + I*a*Tan[c + d*x])^7)/d + (((16*I)/3)*a^2*(a^2
+ I*a^2*Tan[c + d*x])^3)/d + ((2*I)*(a^2 + I*a^2*Tan[c + d*x])^4)/d + ((16*I)*(a^4 + I*a^4*Tan[c + d*x])^2)/d

Rule 3478

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n - 1))/(d*(n - 1)
), x] + Dist[2*a, Int[(a + b*Tan[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && G
tQ[n, 1]

Rule 3477

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2)*x, x] + (Dist[2*a*b, Int[Tan[c + d
*x], x], x] + Simp[(b^2*Tan[c + d*x])/d, x]) /; FreeQ[{a, b, c, d}, x]

Rule 3475

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

Rubi steps

\begin{align*} \int (a+i a \tan (c+d x))^8 \, dx &=\frac{i a (a+i a \tan (c+d x))^7}{7 d}+(2 a) \int (a+i a \tan (c+d x))^7 \, dx\\ &=\frac{i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac{i a (a+i a \tan (c+d x))^7}{7 d}+\left (4 a^2\right ) \int (a+i a \tan (c+d x))^6 \, dx\\ &=\frac{4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac{i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac{i a (a+i a \tan (c+d x))^7}{7 d}+\left (8 a^3\right ) \int (a+i a \tan (c+d x))^5 \, dx\\ &=\frac{4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac{i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac{i a (a+i a \tan (c+d x))^7}{7 d}+\frac{2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\left (16 a^4\right ) \int (a+i a \tan (c+d x))^4 \, dx\\ &=\frac{16 i a^5 (a+i a \tan (c+d x))^3}{3 d}+\frac{4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac{i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac{i a (a+i a \tan (c+d x))^7}{7 d}+\frac{2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\left (32 a^5\right ) \int (a+i a \tan (c+d x))^3 \, dx\\ &=\frac{16 i a^5 (a+i a \tan (c+d x))^3}{3 d}+\frac{4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac{i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac{i a (a+i a \tan (c+d x))^7}{7 d}+\frac{2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac{16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d}+\left (64 a^6\right ) \int (a+i a \tan (c+d x))^2 \, dx\\ &=128 a^8 x-\frac{64 a^8 \tan (c+d x)}{d}+\frac{16 i a^5 (a+i a \tan (c+d x))^3}{3 d}+\frac{4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac{i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac{i a (a+i a \tan (c+d x))^7}{7 d}+\frac{2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac{16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d}+\left (128 i a^8\right ) \int \tan (c+d x) \, dx\\ &=128 a^8 x-\frac{128 i a^8 \log (\cos (c+d x))}{d}-\frac{64 a^8 \tan (c+d x)}{d}+\frac{16 i a^5 (a+i a \tan (c+d x))^3}{3 d}+\frac{4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac{i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac{i a (a+i a \tan (c+d x))^7}{7 d}+\frac{2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac{16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d}\\ \end{align*}

Mathematica [A]  time = 4.13016, size = 383, normalized size = 1.92 \[ \frac{a^8 \sec (c) \sec ^7(c+d x) \left (70 \cos (d x) \left (-105 i \log \left (\cos ^2(c+d x)\right )+210 d x-139 i\right )+70 \cos (2 c+d x) \left (-105 i \log \left (\cos ^2(c+d x)\right )+210 d x-139 i\right )+3 \left (5740 \sin (2 c+d x)-4963 \sin (2 c+3 d x)+2660 \sin (4 c+3 d x)-1981 \sin (4 c+5 d x)+560 \sin (6 c+5 d x)-363 \sin (6 c+7 d x)+980 d x \cos (4 c+5 d x)-420 i \cos (4 c+5 d x)+980 d x \cos (6 c+5 d x)-420 i \cos (6 c+5 d x)+140 d x \cos (6 c+7 d x)+140 d x \cos (8 c+7 d x)-490 i \cos (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )+70 \cos (2 c+3 d x) \left (-21 i \log \left (\cos ^2(c+d x)\right )+42 d x-25 i\right )+70 \cos (4 c+3 d x) \left (-21 i \log \left (\cos ^2(c+d x)\right )+42 d x-25 i\right )-490 i \cos (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )-70 i \cos (6 c+7 d x) \log \left (\cos ^2(c+d x)\right )-70 i \cos (8 c+7 d x) \log \left (\cos ^2(c+d x)\right )-6965 \sin (d x)\right )\right )}{420 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*Sec[c]*Sec[c + d*x]^7*(70*Cos[d*x]*(-139*I + 210*d*x - (105*I)*Log[Cos[c + d*x]^2]) + 70*Cos[2*c + d*x]*(
-139*I + 210*d*x - (105*I)*Log[Cos[c + d*x]^2]) + 3*((-420*I)*Cos[4*c + 5*d*x] + 980*d*x*Cos[4*c + 5*d*x] - (4
20*I)*Cos[6*c + 5*d*x] + 980*d*x*Cos[6*c + 5*d*x] + 140*d*x*Cos[6*c + 7*d*x] + 140*d*x*Cos[8*c + 7*d*x] + 70*C
os[2*c + 3*d*x]*(-25*I + 42*d*x - (21*I)*Log[Cos[c + d*x]^2]) + 70*Cos[4*c + 3*d*x]*(-25*I + 42*d*x - (21*I)*L
og[Cos[c + d*x]^2]) - (490*I)*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]^2] - (490*I)*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]
^2] - (70*I)*Cos[6*c + 7*d*x]*Log[Cos[c + d*x]^2] - (70*I)*Cos[8*c + 7*d*x]*Log[Cos[c + d*x]^2] - 6965*Sin[d*x
] + 5740*Sin[2*c + d*x] - 4963*Sin[2*c + 3*d*x] + 2660*Sin[4*c + 3*d*x] - 1981*Sin[4*c + 5*d*x] + 560*Sin[6*c
+ 5*d*x] - 363*Sin[6*c + 7*d*x])))/(420*d)

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Maple [A]  time = 0.006, size = 150, normalized size = 0.8 \begin{align*} -127\,{\frac{{a}^{8}\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{8} \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{{\frac{4\,i}{3}}{a}^{8} \left ( \tan \left ( dx+c \right ) \right ) ^{6}}{d}}-{\frac{29\,{a}^{8} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{16\,i{a}^{8} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{d}}+33\,{\frac{{a}^{8} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{60\,i{a}^{8} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{64\,i{a}^{8}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}+128\,{\frac{{a}^{8}\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-127*a^8*tan(d*x+c)/d+1/7/d*a^8*tan(d*x+c)^7-4/3*I/d*a^8*tan(d*x+c)^6-29/5*a^8*tan(d*x+c)^5/d+16*I/d*a^8*tan(d
*x+c)^4+33*a^8*tan(d*x+c)^3/d-60*I/d*a^8*tan(d*x+c)^2+64*I/d*a^8*ln(1+tan(d*x+c)^2)+128/d*a^8*arctan(tan(d*x+c
))

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Maxima [A]  time = 1.62787, size = 163, normalized size = 0.82 \begin{align*} \frac{15 \, a^{8} \tan \left (d x + c\right )^{7} - 140 i \, a^{8} \tan \left (d x + c\right )^{6} - 609 \, a^{8} \tan \left (d x + c\right )^{5} + 1680 i \, a^{8} \tan \left (d x + c\right )^{4} + 3465 \, a^{8} \tan \left (d x + c\right )^{3} - 6300 i \, a^{8} \tan \left (d x + c\right )^{2} + 13440 \,{\left (d x + c\right )} a^{8} + 6720 i \, a^{8} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 13335 \, a^{8} \tan \left (d x + c\right )}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(15*a^8*tan(d*x + c)^7 - 140*I*a^8*tan(d*x + c)^6 - 609*a^8*tan(d*x + c)^5 + 1680*I*a^8*tan(d*x + c)^4 +
3465*a^8*tan(d*x + c)^3 - 6300*I*a^8*tan(d*x + c)^2 + 13440*(d*x + c)*a^8 + 6720*I*a^8*log(tan(d*x + c)^2 + 1
) - 13335*a^8*tan(d*x + c))/d

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Fricas [A]  time = 1.4577, size = 976, normalized size = 4.88 \begin{align*} \frac{-94080 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 423360 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 862400 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 980000 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 644448 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 230496 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - 34848 i \, a^{8} +{\left (-13440 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 94080 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 282240 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 470400 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 470400 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 282240 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 94080 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - 13440 i \, a^{8}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{105 \,{\left (d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(-94080*I*a^8*e^(12*I*d*x + 12*I*c) - 423360*I*a^8*e^(10*I*d*x + 10*I*c) - 862400*I*a^8*e^(8*I*d*x + 8*I
*c) - 980000*I*a^8*e^(6*I*d*x + 6*I*c) - 644448*I*a^8*e^(4*I*d*x + 4*I*c) - 230496*I*a^8*e^(2*I*d*x + 2*I*c) -
34848*I*a^8 + (-13440*I*a^8*e^(14*I*d*x + 14*I*c) - 94080*I*a^8*e^(12*I*d*x + 12*I*c) - 282240*I*a^8*e^(10*I*
d*x + 10*I*c) - 470400*I*a^8*e^(8*I*d*x + 8*I*c) - 470400*I*a^8*e^(6*I*d*x + 6*I*c) - 282240*I*a^8*e^(4*I*d*x
+ 4*I*c) - 94080*I*a^8*e^(2*I*d*x + 2*I*c) - 13440*I*a^8)*log(e^(2*I*d*x + 2*I*c) + 1))/(d*e^(14*I*d*x + 14*I*
c) + 7*d*e^(12*I*d*x + 12*I*c) + 21*d*e^(10*I*d*x + 10*I*c) + 35*d*e^(8*I*d*x + 8*I*c) + 35*d*e^(6*I*d*x + 6*I
*c) + 21*d*e^(4*I*d*x + 4*I*c) + 7*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [A]  time = 13.3389, size = 313, normalized size = 1.56 \begin{align*} - \frac{128 i a^{8} \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{- \frac{896 i a^{8} e^{- 2 i c} e^{12 i d x}}{d} - \frac{4032 i a^{8} e^{- 4 i c} e^{10 i d x}}{d} - \frac{24640 i a^{8} e^{- 6 i c} e^{8 i d x}}{3 d} - \frac{28000 i a^{8} e^{- 8 i c} e^{6 i d x}}{3 d} - \frac{30688 i a^{8} e^{- 10 i c} e^{4 i d x}}{5 d} - \frac{10976 i a^{8} e^{- 12 i c} e^{2 i d x}}{5 d} - \frac{11616 i a^{8} e^{- 14 i c}}{35 d}}{e^{14 i d x} + 7 e^{- 2 i c} e^{12 i d x} + 21 e^{- 4 i c} e^{10 i d x} + 35 e^{- 6 i c} e^{8 i d x} + 35 e^{- 8 i c} e^{6 i d x} + 21 e^{- 10 i c} e^{4 i d x} + 7 e^{- 12 i c} e^{2 i d x} + e^{- 14 i c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-128*I*a**8*log(exp(2*I*d*x) + exp(-2*I*c))/d + (-896*I*a**8*exp(-2*I*c)*exp(12*I*d*x)/d - 4032*I*a**8*exp(-4*
I*c)*exp(10*I*d*x)/d - 24640*I*a**8*exp(-6*I*c)*exp(8*I*d*x)/(3*d) - 28000*I*a**8*exp(-8*I*c)*exp(6*I*d*x)/(3*
d) - 30688*I*a**8*exp(-10*I*c)*exp(4*I*d*x)/(5*d) - 10976*I*a**8*exp(-12*I*c)*exp(2*I*d*x)/(5*d) - 11616*I*a**
8*exp(-14*I*c)/(35*d))/(exp(14*I*d*x) + 7*exp(-2*I*c)*exp(12*I*d*x) + 21*exp(-4*I*c)*exp(10*I*d*x) + 35*exp(-6
*I*c)*exp(8*I*d*x) + 35*exp(-8*I*c)*exp(6*I*d*x) + 21*exp(-10*I*c)*exp(4*I*d*x) + 7*exp(-12*I*c)*exp(2*I*d*x)
+ exp(-14*I*c))

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Giac [B]  time = 1.212, size = 510, normalized size = 2.55 \begin{align*} \frac{-13440 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 94080 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 282240 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 470400 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 470400 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 282240 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 94080 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 94080 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 423360 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 862400 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 980000 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 644448 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 230496 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - 13440 i \, a^{8} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 34848 i \, a^{8}}{105 \,{\left (d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(-13440*I*a^8*e^(14*I*d*x + 14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 94080*I*a^8*e^(12*I*d*x + 12*I*c)*log
(e^(2*I*d*x + 2*I*c) + 1) - 282240*I*a^8*e^(10*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 470400*I*a^8*e^(
8*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 470400*I*a^8*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1)
- 282240*I*a^8*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 94080*I*a^8*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d
*x + 2*I*c) + 1) - 94080*I*a^8*e^(12*I*d*x + 12*I*c) - 423360*I*a^8*e^(10*I*d*x + 10*I*c) - 862400*I*a^8*e^(8*
I*d*x + 8*I*c) - 980000*I*a^8*e^(6*I*d*x + 6*I*c) - 644448*I*a^8*e^(4*I*d*x + 4*I*c) - 230496*I*a^8*e^(2*I*d*x
+ 2*I*c) - 13440*I*a^8*log(e^(2*I*d*x + 2*I*c) + 1) - 34848*I*a^8)/(d*e^(14*I*d*x + 14*I*c) + 7*d*e^(12*I*d*x
+ 12*I*c) + 21*d*e^(10*I*d*x + 10*I*c) + 35*d*e^(8*I*d*x + 8*I*c) + 35*d*e^(6*I*d*x + 6*I*c) + 21*d*e^(4*I*d*
x + 4*I*c) + 7*d*e^(2*I*d*x + 2*I*c) + d)