3.30 \(\int \frac{(c+d x)^2}{(a+i a \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=294 \[ \frac{3 d (c+d x) e^{-2 i e-2 i f x}}{16 a^3 f^2}+\frac{3 d (c+d x) e^{-4 i e-4 i f x}}{64 a^3 f^2}+\frac{d (c+d x) e^{-6 i e-6 i f x}}{144 a^3 f^2}+\frac{3 i (c+d x)^2 e^{-2 i e-2 i f x}}{16 a^3 f}+\frac{3 i (c+d x)^2 e^{-4 i e-4 i f x}}{32 a^3 f}+\frac{i (c+d x)^2 e^{-6 i e-6 i f x}}{48 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{3 i d^2 e^{-2 i e-2 i f x}}{32 a^3 f^3}-\frac{3 i d^2 e^{-4 i e-4 i f x}}{256 a^3 f^3}-\frac{i d^2 e^{-6 i e-6 i f x}}{864 a^3 f^3} \]

[Out]

(((-3*I)/32)*d^2*E^((-2*I)*e - (2*I)*f*x))/(a^3*f^3) - (((3*I)/256)*d^2*E^((-4*I)*e - (4*I)*f*x))/(a^3*f^3) -
((I/864)*d^2*E^((-6*I)*e - (6*I)*f*x))/(a^3*f^3) + (3*d*E^((-2*I)*e - (2*I)*f*x)*(c + d*x))/(16*a^3*f^2) + (3*
d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(64*a^3*f^2) + (d*E^((-6*I)*e - (6*I)*f*x)*(c + d*x))/(144*a^3*f^2) + ((
(3*I)/16)*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(a^3*f) + (((3*I)/32)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(a
^3*f) + ((I/48)*E^((-6*I)*e - (6*I)*f*x)*(c + d*x)^2)/(a^3*f) + (c + d*x)^3/(24*a^3*d)

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Rubi [A]  time = 0.265228, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3729, 2176, 2194} \[ \frac{3 d (c+d x) e^{-2 i e-2 i f x}}{16 a^3 f^2}+\frac{3 d (c+d x) e^{-4 i e-4 i f x}}{64 a^3 f^2}+\frac{d (c+d x) e^{-6 i e-6 i f x}}{144 a^3 f^2}+\frac{3 i (c+d x)^2 e^{-2 i e-2 i f x}}{16 a^3 f}+\frac{3 i (c+d x)^2 e^{-4 i e-4 i f x}}{32 a^3 f}+\frac{i (c+d x)^2 e^{-6 i e-6 i f x}}{48 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{3 i d^2 e^{-2 i e-2 i f x}}{32 a^3 f^3}-\frac{3 i d^2 e^{-4 i e-4 i f x}}{256 a^3 f^3}-\frac{i d^2 e^{-6 i e-6 i f x}}{864 a^3 f^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + I*a*Tan[e + f*x])^3,x]

[Out]

(((-3*I)/32)*d^2*E^((-2*I)*e - (2*I)*f*x))/(a^3*f^3) - (((3*I)/256)*d^2*E^((-4*I)*e - (4*I)*f*x))/(a^3*f^3) -
((I/864)*d^2*E^((-6*I)*e - (6*I)*f*x))/(a^3*f^3) + (3*d*E^((-2*I)*e - (2*I)*f*x)*(c + d*x))/(16*a^3*f^2) + (3*
d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(64*a^3*f^2) + (d*E^((-6*I)*e - (6*I)*f*x)*(c + d*x))/(144*a^3*f^2) + ((
(3*I)/16)*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(a^3*f) + (((3*I)/32)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(a
^3*f) + ((I/48)*E^((-6*I)*e - (6*I)*f*x)*(c + d*x)^2)/(a^3*f) + (c + d*x)^3/(24*a^3*d)

Rule 3729

Int[((c_.) + (d_.)*(x_))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(c
+ d*x)^m, (1/(2*a) + E^((2*a*(e + f*x))/b)/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
+ b^2, 0] && ILtQ[n, 0]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{(c+d x)^2}{(a+i a \tan (e+f x))^3} \, dx &=\int \left (\frac{(c+d x)^2}{8 a^3}+\frac{3 e^{-2 i e-2 i f x} (c+d x)^2}{8 a^3}+\frac{3 e^{-4 i e-4 i f x} (c+d x)^2}{8 a^3}+\frac{e^{-6 i e-6 i f x} (c+d x)^2}{8 a^3}\right ) \, dx\\ &=\frac{(c+d x)^3}{24 a^3 d}+\frac{\int e^{-6 i e-6 i f x} (c+d x)^2 \, dx}{8 a^3}+\frac{3 \int e^{-2 i e-2 i f x} (c+d x)^2 \, dx}{8 a^3}+\frac{3 \int e^{-4 i e-4 i f x} (c+d x)^2 \, dx}{8 a^3}\\ &=\frac{3 i e^{-2 i e-2 i f x} (c+d x)^2}{16 a^3 f}+\frac{3 i e^{-4 i e-4 i f x} (c+d x)^2}{32 a^3 f}+\frac{i e^{-6 i e-6 i f x} (c+d x)^2}{48 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{(i d) \int e^{-6 i e-6 i f x} (c+d x) \, dx}{24 a^3 f}-\frac{(3 i d) \int e^{-4 i e-4 i f x} (c+d x) \, dx}{16 a^3 f}-\frac{(3 i d) \int e^{-2 i e-2 i f x} (c+d x) \, dx}{8 a^3 f}\\ &=\frac{3 d e^{-2 i e-2 i f x} (c+d x)}{16 a^3 f^2}+\frac{3 d e^{-4 i e-4 i f x} (c+d x)}{64 a^3 f^2}+\frac{d e^{-6 i e-6 i f x} (c+d x)}{144 a^3 f^2}+\frac{3 i e^{-2 i e-2 i f x} (c+d x)^2}{16 a^3 f}+\frac{3 i e^{-4 i e-4 i f x} (c+d x)^2}{32 a^3 f}+\frac{i e^{-6 i e-6 i f x} (c+d x)^2}{48 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{d^2 \int e^{-6 i e-6 i f x} \, dx}{144 a^3 f^2}-\frac{\left (3 d^2\right ) \int e^{-4 i e-4 i f x} \, dx}{64 a^3 f^2}-\frac{\left (3 d^2\right ) \int e^{-2 i e-2 i f x} \, dx}{16 a^3 f^2}\\ &=-\frac{3 i d^2 e^{-2 i e-2 i f x}}{32 a^3 f^3}-\frac{3 i d^2 e^{-4 i e-4 i f x}}{256 a^3 f^3}-\frac{i d^2 e^{-6 i e-6 i f x}}{864 a^3 f^3}+\frac{3 d e^{-2 i e-2 i f x} (c+d x)}{16 a^3 f^2}+\frac{3 d e^{-4 i e-4 i f x} (c+d x)}{64 a^3 f^2}+\frac{d e^{-6 i e-6 i f x} (c+d x)}{144 a^3 f^2}+\frac{3 i e^{-2 i e-2 i f x} (c+d x)^2}{16 a^3 f}+\frac{3 i e^{-4 i e-4 i f x} (c+d x)^2}{32 a^3 f}+\frac{i e^{-6 i e-6 i f x} (c+d x)^2}{48 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}\\ \end{align*}

Mathematica [A]  time = 1.46337, size = 405, normalized size = 1.38 \[ \frac{i \sec ^3(e+f x) \left (81 \left (24 i c^2 f^2+4 c d f (5+12 i f x)+d^2 \left (24 i f^2 x^2+20 f x-9 i\right )\right ) \cos (e+f x)+8 \left (18 c^2 f^2 (6 f x+i)+6 c d f \left (18 f^2 x^2+6 i f x+1\right )+d^2 \left (36 f^3 x^3+18 i f^2 x^2+6 f x-i\right )\right ) \cos (3 (e+f x))+864 i c^2 f^3 x \sin (3 (e+f x))-648 c^2 f^2 \sin (e+f x)+144 c^2 f^2 \sin (3 (e+f x))+864 i c d f^3 x^2 \sin (3 (e+f x))-1296 c d f^2 x \sin (e+f x)+288 c d f^2 x \sin (3 (e+f x))+972 i c d f \sin (e+f x)-48 i c d f \sin (3 (e+f x))+288 i d^2 f^3 x^3 \sin (3 (e+f x))-648 d^2 f^2 x^2 \sin (e+f x)+144 d^2 f^2 x^2 \sin (3 (e+f x))+972 i d^2 f x \sin (e+f x)-48 i d^2 f x \sin (3 (e+f x))+567 d^2 \sin (e+f x)-8 d^2 \sin (3 (e+f x))\right )}{6912 a^3 f^3 (\tan (e+f x)-i)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + I*a*Tan[e + f*x])^3,x]

[Out]

((I/6912)*Sec[e + f*x]^3*(81*((24*I)*c^2*f^2 + 4*c*d*f*(5 + (12*I)*f*x) + d^2*(-9*I + 20*f*x + (24*I)*f^2*x^2)
)*Cos[e + f*x] + 8*(18*c^2*f^2*(I + 6*f*x) + 6*c*d*f*(1 + (6*I)*f*x + 18*f^2*x^2) + d^2*(-I + 6*f*x + (18*I)*f
^2*x^2 + 36*f^3*x^3))*Cos[3*(e + f*x)] + 567*d^2*Sin[e + f*x] + (972*I)*c*d*f*Sin[e + f*x] - 648*c^2*f^2*Sin[e
+ f*x] + (972*I)*d^2*f*x*Sin[e + f*x] - 1296*c*d*f^2*x*Sin[e + f*x] - 648*d^2*f^2*x^2*Sin[e + f*x] - 8*d^2*Si
n[3*(e + f*x)] - (48*I)*c*d*f*Sin[3*(e + f*x)] + 144*c^2*f^2*Sin[3*(e + f*x)] - (48*I)*d^2*f*x*Sin[3*(e + f*x)
] + 288*c*d*f^2*x*Sin[3*(e + f*x)] + (864*I)*c^2*f^3*x*Sin[3*(e + f*x)] + 144*d^2*f^2*x^2*Sin[3*(e + f*x)] + (
864*I)*c*d*f^3*x^2*Sin[3*(e + f*x)] + (288*I)*d^2*f^3*x^3*Sin[3*(e + f*x)]))/(a^3*f^3*(-I + Tan[e + f*x])^3)

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Maple [A]  time = 0.321, size = 227, normalized size = 0.8 \begin{align*}{\frac{{d}^{2}{x}^{3}}{24\,{a}^{3}}}+{\frac{cd{x}^{2}}{8\,{a}^{3}}}+{\frac{{c}^{2}x}{8\,{a}^{3}}}+{\frac{{\frac{3\,i}{32}} \left ( 2\,{d}^{2}{x}^{2}{f}^{2}-2\,i{d}^{2}fx+4\,cd{f}^{2}x-2\,icdf+2\,{c}^{2}{f}^{2}-{d}^{2} \right ){{\rm e}^{-2\,i \left ( fx+e \right ) }}}{{a}^{3}{f}^{3}}}+{\frac{{\frac{3\,i}{256}} \left ( 8\,{d}^{2}{x}^{2}{f}^{2}-4\,i{d}^{2}fx+16\,cd{f}^{2}x-4\,icdf+8\,{c}^{2}{f}^{2}-{d}^{2} \right ){{\rm e}^{-4\,i \left ( fx+e \right ) }}}{{a}^{3}{f}^{3}}}+{\frac{{\frac{i}{864}} \left ( 18\,{d}^{2}{x}^{2}{f}^{2}-6\,i{d}^{2}fx+36\,cd{f}^{2}x-6\,icdf+18\,{c}^{2}{f}^{2}-{d}^{2} \right ){{\rm e}^{-6\,i \left ( fx+e \right ) }}}{{a}^{3}{f}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/(a+I*a*tan(f*x+e))^3,x)

[Out]

1/24/a^3*d^2*x^3+1/8/a^3*c*d*x^2+1/8/a^3*c^2*x+3/32*I*(2*d^2*x^2*f^2-2*I*d^2*f*x+4*c*d*f^2*x-2*I*c*d*f+2*c^2*f
^2-d^2)/a^3/f^3*exp(-2*I*(f*x+e))+3/256*I*(8*d^2*x^2*f^2-4*I*d^2*f*x+16*c*d*f^2*x-4*I*c*d*f+8*c^2*f^2-d^2)/a^3
/f^3*exp(-4*I*(f*x+e))+1/864*I*(18*d^2*x^2*f^2-6*I*d^2*f*x+36*c*d*f^2*x-6*I*c*d*f+18*c^2*f^2-d^2)/a^3/f^3*exp(
-6*I*(f*x+e))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+I*a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.591, size = 581, normalized size = 1.98 \begin{align*} \frac{{\left (144 i \, d^{2} f^{2} x^{2} + 144 i \, c^{2} f^{2} + 48 \, c d f - 8 i \, d^{2} +{\left (288 i \, c d f^{2} + 48 \, d^{2} f\right )} x + 288 \,{\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} + 3 \, c^{2} f^{3} x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (1296 i \, d^{2} f^{2} x^{2} + 1296 i \, c^{2} f^{2} + 1296 \, c d f - 648 i \, d^{2} +{\left (2592 i \, c d f^{2} + 1296 \, d^{2} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (648 i \, d^{2} f^{2} x^{2} + 648 i \, c^{2} f^{2} + 324 \, c d f - 81 i \, d^{2} +{\left (1296 i \, c d f^{2} + 324 \, d^{2} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{6912 \, a^{3} f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+I*a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/6912*(144*I*d^2*f^2*x^2 + 144*I*c^2*f^2 + 48*c*d*f - 8*I*d^2 + (288*I*c*d*f^2 + 48*d^2*f)*x + 288*(d^2*f^3*x
^3 + 3*c*d*f^3*x^2 + 3*c^2*f^3*x)*e^(6*I*f*x + 6*I*e) + (1296*I*d^2*f^2*x^2 + 1296*I*c^2*f^2 + 1296*c*d*f - 64
8*I*d^2 + (2592*I*c*d*f^2 + 1296*d^2*f)*x)*e^(4*I*f*x + 4*I*e) + (648*I*d^2*f^2*x^2 + 648*I*c^2*f^2 + 324*c*d*
f - 81*I*d^2 + (1296*I*c*d*f^2 + 324*d^2*f)*x)*e^(2*I*f*x + 2*I*e))*e^(-6*I*f*x - 6*I*e)/(a^3*f^3)

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Sympy [A]  time = 2.21763, size = 590, normalized size = 2.01 \begin{align*} \begin{cases} \frac{\left (\left (147456 i a^{24} c^{2} f^{17} e^{30 i e} + 294912 i a^{24} c d f^{17} x e^{30 i e} + 49152 a^{24} c d f^{16} e^{30 i e} + 147456 i a^{24} d^{2} f^{17} x^{2} e^{30 i e} + 49152 a^{24} d^{2} f^{16} x e^{30 i e} - 8192 i a^{24} d^{2} f^{15} e^{30 i e}\right ) e^{- 6 i f x} + \left (663552 i a^{24} c^{2} f^{17} e^{32 i e} + 1327104 i a^{24} c d f^{17} x e^{32 i e} + 331776 a^{24} c d f^{16} e^{32 i e} + 663552 i a^{24} d^{2} f^{17} x^{2} e^{32 i e} + 331776 a^{24} d^{2} f^{16} x e^{32 i e} - 82944 i a^{24} d^{2} f^{15} e^{32 i e}\right ) e^{- 4 i f x} + \left (1327104 i a^{24} c^{2} f^{17} e^{34 i e} + 2654208 i a^{24} c d f^{17} x e^{34 i e} + 1327104 a^{24} c d f^{16} e^{34 i e} + 1327104 i a^{24} d^{2} f^{17} x^{2} e^{34 i e} + 1327104 a^{24} d^{2} f^{16} x e^{34 i e} - 663552 i a^{24} d^{2} f^{15} e^{34 i e}\right ) e^{- 2 i f x}\right ) e^{- 36 i e}}{7077888 a^{27} f^{18}} & \text{for}\: 7077888 a^{27} f^{18} e^{36 i e} \neq 0 \\\frac{x^{3} \left (3 d^{2} e^{4 i e} + 3 d^{2} e^{2 i e} + d^{2}\right ) e^{- 6 i e}}{24 a^{3}} + \frac{x^{2} \left (3 c d e^{4 i e} + 3 c d e^{2 i e} + c d\right ) e^{- 6 i e}}{8 a^{3}} + \frac{x \left (3 c^{2} e^{4 i e} + 3 c^{2} e^{2 i e} + c^{2}\right ) e^{- 6 i e}}{8 a^{3}} & \text{otherwise} \end{cases} + \frac{c^{2} x}{8 a^{3}} + \frac{c d x^{2}}{8 a^{3}} + \frac{d^{2} x^{3}}{24 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/(a+I*a*tan(f*x+e))**3,x)

[Out]

Piecewise((((147456*I*a**24*c**2*f**17*exp(30*I*e) + 294912*I*a**24*c*d*f**17*x*exp(30*I*e) + 49152*a**24*c*d*
f**16*exp(30*I*e) + 147456*I*a**24*d**2*f**17*x**2*exp(30*I*e) + 49152*a**24*d**2*f**16*x*exp(30*I*e) - 8192*I
*a**24*d**2*f**15*exp(30*I*e))*exp(-6*I*f*x) + (663552*I*a**24*c**2*f**17*exp(32*I*e) + 1327104*I*a**24*c*d*f*
*17*x*exp(32*I*e) + 331776*a**24*c*d*f**16*exp(32*I*e) + 663552*I*a**24*d**2*f**17*x**2*exp(32*I*e) + 331776*a
**24*d**2*f**16*x*exp(32*I*e) - 82944*I*a**24*d**2*f**15*exp(32*I*e))*exp(-4*I*f*x) + (1327104*I*a**24*c**2*f*
*17*exp(34*I*e) + 2654208*I*a**24*c*d*f**17*x*exp(34*I*e) + 1327104*a**24*c*d*f**16*exp(34*I*e) + 1327104*I*a*
*24*d**2*f**17*x**2*exp(34*I*e) + 1327104*a**24*d**2*f**16*x*exp(34*I*e) - 663552*I*a**24*d**2*f**15*exp(34*I*
e))*exp(-2*I*f*x))*exp(-36*I*e)/(7077888*a**27*f**18), Ne(7077888*a**27*f**18*exp(36*I*e), 0)), (x**3*(3*d**2*
exp(4*I*e) + 3*d**2*exp(2*I*e) + d**2)*exp(-6*I*e)/(24*a**3) + x**2*(3*c*d*exp(4*I*e) + 3*c*d*exp(2*I*e) + c*d
)*exp(-6*I*e)/(8*a**3) + x*(3*c**2*exp(4*I*e) + 3*c**2*exp(2*I*e) + c**2)*exp(-6*I*e)/(8*a**3), True)) + c**2*
x/(8*a**3) + c*d*x**2/(8*a**3) + d**2*x**3/(24*a**3)

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Giac [A]  time = 1.27947, size = 447, normalized size = 1.52 \begin{align*} \frac{{\left (288 \, d^{2} f^{3} x^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 864 \, c d f^{3} x^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 864 \, c^{2} f^{3} x e^{\left (6 i \, f x + 6 i \, e\right )} + 1296 i \, d^{2} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 648 i \, d^{2} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 144 i \, d^{2} f^{2} x^{2} + 2592 i \, c d f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 1296 i \, c d f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 288 i \, c d f^{2} x + 1296 i \, c^{2} f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 1296 \, d^{2} f x e^{\left (4 i \, f x + 4 i \, e\right )} + 648 i \, c^{2} f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 324 \, d^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 144 i \, c^{2} f^{2} + 48 \, d^{2} f x + 1296 \, c d f e^{\left (4 i \, f x + 4 i \, e\right )} + 324 \, c d f e^{\left (2 i \, f x + 2 i \, e\right )} + 48 \, c d f - 648 i \, d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 81 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 8 i \, d^{2}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{6912 \, a^{3} f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

1/6912*(288*d^2*f^3*x^3*e^(6*I*f*x + 6*I*e) + 864*c*d*f^3*x^2*e^(6*I*f*x + 6*I*e) + 864*c^2*f^3*x*e^(6*I*f*x +
6*I*e) + 1296*I*d^2*f^2*x^2*e^(4*I*f*x + 4*I*e) + 648*I*d^2*f^2*x^2*e^(2*I*f*x + 2*I*e) + 144*I*d^2*f^2*x^2 +
2592*I*c*d*f^2*x*e^(4*I*f*x + 4*I*e) + 1296*I*c*d*f^2*x*e^(2*I*f*x + 2*I*e) + 288*I*c*d*f^2*x + 1296*I*c^2*f^
2*e^(4*I*f*x + 4*I*e) + 1296*d^2*f*x*e^(4*I*f*x + 4*I*e) + 648*I*c^2*f^2*e^(2*I*f*x + 2*I*e) + 324*d^2*f*x*e^(
2*I*f*x + 2*I*e) + 144*I*c^2*f^2 + 48*d^2*f*x + 1296*c*d*f*e^(4*I*f*x + 4*I*e) + 324*c*d*f*e^(2*I*f*x + 2*I*e)
+ 48*c*d*f - 648*I*d^2*e^(4*I*f*x + 4*I*e) - 81*I*d^2*e^(2*I*f*x + 2*I*e) - 8*I*d^2)*e^(-6*I*f*x - 6*I*e)/(a^
3*f^3)