Integrand size = 23, antiderivative size = 270 \[ \int \frac {(c+d x)^3}{(a+i a \cot (e+f x))^2} \, dx=\frac {3 d^3 e^{2 i e+2 i f x}}{16 a^2 f^4}-\frac {3 d^3 e^{4 i e+4 i f x}}{512 a^2 f^4}-\frac {3 i d^2 e^{2 i e+2 i f x} (c+d x)}{8 a^2 f^3}+\frac {3 i d^2 e^{4 i e+4 i f x} (c+d x)}{128 a^2 f^3}-\frac {3 d e^{2 i e+2 i f x} (c+d x)^2}{8 a^2 f^2}+\frac {3 d e^{4 i e+4 i f x} (c+d x)^2}{64 a^2 f^2}+\frac {i e^{2 i e+2 i f x} (c+d x)^3}{4 a^2 f}-\frac {i e^{4 i e+4 i f x} (c+d x)^3}{16 a^2 f}+\frac {(c+d x)^4}{16 a^2 d} \]
3/16*d^3*exp(2*I*e+2*I*f*x)/a^2/f^4-3/512*d^3*exp(4*I*e+4*I*f*x)/a^2/f^4-3 /8*I*d^2*exp(2*I*e+2*I*f*x)*(d*x+c)/a^2/f^3+3/128*I*d^2*exp(4*I*e+4*I*f*x) *(d*x+c)/a^2/f^3-3/8*d*exp(2*I*e+2*I*f*x)*(d*x+c)^2/a^2/f^2+3/64*d*exp(4*I *e+4*I*f*x)*(d*x+c)^2/a^2/f^2+1/4*I*exp(2*I*e+2*I*f*x)*(d*x+c)^3/a^2/f-1/1 6*I*exp(4*I*e+4*I*f*x)*(d*x+c)^3/a^2/f+1/16*(d*x+c)^4/a^2/d
Time = 2.17 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.34 \[ \int \frac {(c+d x)^3}{(a+i a \cot (e+f x))^2} \, dx=\frac {(\cos (2 (e+f x))+i \sin (2 (e+f x))) \left (\left (32 c^3 f^3 (-i+4 f x)+24 c^2 d f^2 \left (1-4 i f x+8 f^2 x^2\right )+4 c d^2 f \left (3 i+12 f x-24 i f^2 x^2+32 f^3 x^3\right )+d^3 \left (-3+12 i f x+24 f^2 x^2-32 i f^3 x^3+32 f^4 x^4\right )\right ) \cos (2 (e+f x))-i \left (-32 \left (4 c^3 f^3+6 c^2 d f^2 (i+2 f x)+6 c d^2 f \left (-1+2 i f x+2 f^2 x^2\right )+d^3 \left (-3 i-6 f x+6 i f^2 x^2+4 f^3 x^3\right )\right )+\left (32 c^3 f^3 (i+4 f x)+24 c^2 d f^2 \left (-1+4 i f x+8 f^2 x^2\right )+4 c d^2 f \left (-3 i-12 f x+24 i f^2 x^2+32 f^3 x^3\right )+d^3 \left (3-12 i f x-24 f^2 x^2+32 i f^3 x^3+32 f^4 x^4\right )\right ) \sin (2 (e+f x))\right )\right )}{512 a^2 f^4} \]
((Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*((32*c^3*f^3*(-I + 4*f*x) + 24*c^ 2*d*f^2*(1 - (4*I)*f*x + 8*f^2*x^2) + 4*c*d^2*f*(3*I + 12*f*x - (24*I)*f^2 *x^2 + 32*f^3*x^3) + d^3*(-3 + (12*I)*f*x + 24*f^2*x^2 - (32*I)*f^3*x^3 + 32*f^4*x^4))*Cos[2*(e + f*x)] - I*(-32*(4*c^3*f^3 + 6*c^2*d*f^2*(I + 2*f*x ) + 6*c*d^2*f*(-1 + (2*I)*f*x + 2*f^2*x^2) + d^3*(-3*I - 6*f*x + (6*I)*f^2 *x^2 + 4*f^3*x^3)) + (32*c^3*f^3*(I + 4*f*x) + 24*c^2*d*f^2*(-1 + (4*I)*f* x + 8*f^2*x^2) + 4*c*d^2*f*(-3*I - 12*f*x + (24*I)*f^2*x^2 + 32*f^3*x^3) + d^3*(3 - (12*I)*f*x - 24*f^2*x^2 + (32*I)*f^3*x^3 + 32*f^4*x^4))*Sin[2*(e + f*x)])))/(512*a^2*f^4)
Time = 0.50 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3042, 4212, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c+d x)^3}{(a+i a \cot (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d x)^3}{\left (a-i a \tan \left (e+f x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4212 |
| \(\displaystyle \int \left (-\frac {(c+d x)^3 e^{2 i e+2 i f x}}{2 a^2}+\frac {(c+d x)^3 e^{4 i e+4 i f x}}{4 a^2}+\frac {(c+d x)^3}{4 a^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 i d^2 (c+d x) e^{2 i e+2 i f x}}{8 a^2 f^3}+\frac {3 i d^2 (c+d x) e^{4 i e+4 i f x}}{128 a^2 f^3}-\frac {3 d (c+d x)^2 e^{2 i e+2 i f x}}{8 a^2 f^2}+\frac {3 d (c+d x)^2 e^{4 i e+4 i f x}}{64 a^2 f^2}+\frac {i (c+d x)^3 e^{2 i e+2 i f x}}{4 a^2 f}-\frac {i (c+d x)^3 e^{4 i e+4 i f x}}{16 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}+\frac {3 d^3 e^{2 i e+2 i f x}}{16 a^2 f^4}-\frac {3 d^3 e^{4 i e+4 i f x}}{512 a^2 f^4}\) |
(3*d^3*E^((2*I)*e + (2*I)*f*x))/(16*a^2*f^4) - (3*d^3*E^((4*I)*e + (4*I)*f *x))/(512*a^2*f^4) - (((3*I)/8)*d^2*E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(a^ 2*f^3) + (((3*I)/128)*d^2*E^((4*I)*e + (4*I)*f*x)*(c + d*x))/(a^2*f^3) - ( 3*d*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^2)/(8*a^2*f^2) + (3*d*E^((4*I)*e + ( 4*I)*f*x)*(c + d*x)^2)/(64*a^2*f^2) + ((I/4)*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^3)/(a^2*f) - ((I/16)*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^3)/(a^2*f) + ( c + d*x)^4/(16*a^2*d)
3.1.22.3.1 Defintions of rubi rules used
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/b)*(e + f* x))/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 + b^2 , 0] && ILtQ[n, 0]
Time = 0.70 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(\frac {d^{3} x^{4}}{16 a^{2}}+\frac {d^{2} c \,x^{3}}{4 a^{2}}+\frac {3 d \,c^{2} x^{2}}{8 a^{2}}+\frac {c^{3} x}{4 a^{2}}+\frac {c^{4}}{16 a^{2} d}-\frac {i \left (32 d^{3} x^{3} f^{3}+96 c \,d^{2} f^{3} x^{2}+24 i d^{3} f^{2} x^{2}+96 c^{2} d \,f^{3} x +48 i c \,d^{2} f^{2} x +32 c^{3} f^{3}+24 i c^{2} d \,f^{2}-12 d^{3} f x -12 c \,d^{2} f -3 i d^{3}\right ) {\mathrm e}^{4 i \left (f x +e \right )}}{512 a^{2} f^{4}}+\frac {i \left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+6 i d^{3} f^{2} x^{2}+12 c^{2} d \,f^{3} x +12 i c \,d^{2} f^{2} x +4 c^{3} f^{3}+6 i c^{2} d \,f^{2}-6 d^{3} f x -6 c \,d^{2} f -3 i d^{3}\right ) {\mathrm e}^{2 i \left (f x +e \right )}}{16 a^{2} f^{4}}\) | \(283\) |
| parallelrisch | \(\frac {-120 f \left (-\frac {4 \left (\frac {d x}{2}+c \right ) \left (\frac {1}{2} d^{2} x^{2}+c d x +c^{2}\right ) f^{3}}{15}+i d \left (\frac {1}{3} d^{2} x^{2}+c d x +c^{2}\right ) f^{2}-\frac {9 \left (\frac {d x}{2}+c \right ) d^{2} f}{10}-\frac {17 i d^{3}}{40}\right ) x \tan \left (f x +e \right )^{2}+\left (64 i \left (\frac {d x}{2}+c \right ) \left (\frac {1}{2} d^{2} x^{2}+c d x +c^{2}\right ) x \,f^{4}+16 \left (-d^{3} x^{3}-3 c \,d^{2} x^{2}-3 c^{2} d x -6 c^{3}\right ) f^{3}-120 i d \left (\frac {1}{10} d^{2} x^{2}+\frac {1}{5} c d x +c^{2}\right ) f^{2}+6 \left (d^{3} x +18 c \,d^{2}\right ) f +51 i d^{3}\right ) \tan \left (f x +e \right )+8 \left (-d^{3} x^{4}-4 d^{2} x^{3} c -6 d \,x^{2} c^{2}-4 x \,c^{3}\right ) f^{4}+8 i \left (-3 d^{3} x^{3}-9 c \,d^{2} x^{2}-9 c^{2} d x -8 c^{3}\right ) f^{3}+6 \left (7 d^{3} x^{2}+14 d^{2} x c +16 c^{2} d \right ) f^{2}+96 i d^{2} \left (\frac {15 d x}{32}+c \right ) f -48 d^{3}}{128 f^{4} a^{2} \left (-1+2 i \tan \left (f x +e \right )+\tan \left (f x +e \right )^{2}\right )}\) | \(348\) |
1/16/a^2*d^3*x^4+1/4/a^2*d^2*c*x^3+3/8/a^2*d*c^2*x^2+1/4/a^2*c^3*x+1/16/a^ 2/d*c^4-1/512*I*(32*d^3*x^3*f^3+24*I*d^3*f^2*x^2+96*c*d^2*f^3*x^2+48*I*c*d ^2*f^2*x+96*c^2*d*f^3*x+24*I*c^2*d*f^2+32*c^3*f^3-12*d^3*f*x-3*I*d^3-12*c* d^2*f)/a^2/f^4*exp(4*I*(f*x+e))+1/16*I*(4*d^3*x^3*f^3+6*I*d^3*f^2*x^2+12*c *d^2*f^3*x^2+12*I*c*d^2*f^2*x+12*c^2*d*f^3*x+6*I*c^2*d*f^2+4*c^3*f^3-6*d^3 *f*x-3*I*d^3-6*c*d^2*f)/a^2/f^4*exp(2*I*(f*x+e))
Time = 0.26 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^3}{(a+i a \cot (e+f x))^2} \, dx=\frac {32 \, d^{3} f^{4} x^{4} + 128 \, c d^{2} f^{4} x^{3} + 192 \, c^{2} d f^{4} x^{2} + 128 \, c^{3} f^{4} x + {\left (-32 i \, d^{3} f^{3} x^{3} - 32 i \, c^{3} f^{3} + 24 \, c^{2} d f^{2} + 12 i \, c d^{2} f - 3 \, d^{3} - 24 \, {\left (4 i \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} - 12 \, {\left (8 i \, c^{2} d f^{3} - 4 \, c d^{2} f^{2} - i \, d^{3} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 32 \, {\left (-4 i \, d^{3} f^{3} x^{3} - 4 i \, c^{3} f^{3} + 6 \, c^{2} d f^{2} + 6 i \, c d^{2} f - 3 \, d^{3} + 6 \, {\left (-2 i \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + 6 \, {\left (-2 i \, c^{2} d f^{3} + 2 \, c d^{2} f^{2} + i \, d^{3} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{512 \, a^{2} f^{4}} \]
1/512*(32*d^3*f^4*x^4 + 128*c*d^2*f^4*x^3 + 192*c^2*d*f^4*x^2 + 128*c^3*f^ 4*x + (-32*I*d^3*f^3*x^3 - 32*I*c^3*f^3 + 24*c^2*d*f^2 + 12*I*c*d^2*f - 3* d^3 - 24*(4*I*c*d^2*f^3 - d^3*f^2)*x^2 - 12*(8*I*c^2*d*f^3 - 4*c*d^2*f^2 - I*d^3*f)*x)*e^(4*I*f*x + 4*I*e) - 32*(-4*I*d^3*f^3*x^3 - 4*I*c^3*f^3 + 6* c^2*d*f^2 + 6*I*c*d^2*f - 3*d^3 + 6*(-2*I*c*d^2*f^3 + d^3*f^2)*x^2 + 6*(-2 *I*c^2*d*f^3 + 2*c*d^2*f^2 + I*d^3*f)*x)*e^(2*I*f*x + 2*I*e))/(a^2*f^4)
Time = 0.31 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.41 \[ \int \frac {(c+d x)^3}{(a+i a \cot (e+f x))^2} \, dx=\begin {cases} \frac {\left (2048 i a^{2} c^{3} f^{7} e^{2 i e} + 6144 i a^{2} c^{2} d f^{7} x e^{2 i e} - 3072 a^{2} c^{2} d f^{6} e^{2 i e} + 6144 i a^{2} c d^{2} f^{7} x^{2} e^{2 i e} - 6144 a^{2} c d^{2} f^{6} x e^{2 i e} - 3072 i a^{2} c d^{2} f^{5} e^{2 i e} + 2048 i a^{2} d^{3} f^{7} x^{3} e^{2 i e} - 3072 a^{2} d^{3} f^{6} x^{2} e^{2 i e} - 3072 i a^{2} d^{3} f^{5} x e^{2 i e} + 1536 a^{2} d^{3} f^{4} e^{2 i e}\right ) e^{2 i f x} + \left (- 512 i a^{2} c^{3} f^{7} e^{4 i e} - 1536 i a^{2} c^{2} d f^{7} x e^{4 i e} + 384 a^{2} c^{2} d f^{6} e^{4 i e} - 1536 i a^{2} c d^{2} f^{7} x^{2} e^{4 i e} + 768 a^{2} c d^{2} f^{6} x e^{4 i e} + 192 i a^{2} c d^{2} f^{5} e^{4 i e} - 512 i a^{2} d^{3} f^{7} x^{3} e^{4 i e} + 384 a^{2} d^{3} f^{6} x^{2} e^{4 i e} + 192 i a^{2} d^{3} f^{5} x e^{4 i e} - 48 a^{2} d^{3} f^{4} e^{4 i e}\right ) e^{4 i f x}}{8192 a^{4} f^{8}} & \text {for}\: a^{4} f^{8} \neq 0 \\\frac {x^{4} \left (d^{3} e^{4 i e} - 2 d^{3} e^{2 i e}\right )}{16 a^{2}} + \frac {x^{3} \left (c d^{2} e^{4 i e} - 2 c d^{2} e^{2 i e}\right )}{4 a^{2}} + \frac {x^{2} \cdot \left (3 c^{2} d e^{4 i e} - 6 c^{2} d e^{2 i e}\right )}{8 a^{2}} + \frac {x \left (c^{3} e^{4 i e} - 2 c^{3} e^{2 i e}\right )}{4 a^{2}} & \text {otherwise} \end {cases} + \frac {c^{3} x}{4 a^{2}} + \frac {3 c^{2} d x^{2}}{8 a^{2}} + \frac {c d^{2} x^{3}}{4 a^{2}} + \frac {d^{3} x^{4}}{16 a^{2}} \]
Piecewise((((2048*I*a**2*c**3*f**7*exp(2*I*e) + 6144*I*a**2*c**2*d*f**7*x* exp(2*I*e) - 3072*a**2*c**2*d*f**6*exp(2*I*e) + 6144*I*a**2*c*d**2*f**7*x* *2*exp(2*I*e) - 6144*a**2*c*d**2*f**6*x*exp(2*I*e) - 3072*I*a**2*c*d**2*f* *5*exp(2*I*e) + 2048*I*a**2*d**3*f**7*x**3*exp(2*I*e) - 3072*a**2*d**3*f** 6*x**2*exp(2*I*e) - 3072*I*a**2*d**3*f**5*x*exp(2*I*e) + 1536*a**2*d**3*f* *4*exp(2*I*e))*exp(2*I*f*x) + (-512*I*a**2*c**3*f**7*exp(4*I*e) - 1536*I*a **2*c**2*d*f**7*x*exp(4*I*e) + 384*a**2*c**2*d*f**6*exp(4*I*e) - 1536*I*a* *2*c*d**2*f**7*x**2*exp(4*I*e) + 768*a**2*c*d**2*f**6*x*exp(4*I*e) + 192*I *a**2*c*d**2*f**5*exp(4*I*e) - 512*I*a**2*d**3*f**7*x**3*exp(4*I*e) + 384* a**2*d**3*f**6*x**2*exp(4*I*e) + 192*I*a**2*d**3*f**5*x*exp(4*I*e) - 48*a* *2*d**3*f**4*exp(4*I*e))*exp(4*I*f*x))/(8192*a**4*f**8), Ne(a**4*f**8, 0)) , (x**4*(d**3*exp(4*I*e) - 2*d**3*exp(2*I*e))/(16*a**2) + x**3*(c*d**2*exp (4*I*e) - 2*c*d**2*exp(2*I*e))/(4*a**2) + x**2*(3*c**2*d*exp(4*I*e) - 6*c* *2*d*exp(2*I*e))/(8*a**2) + x*(c**3*exp(4*I*e) - 2*c**3*exp(2*I*e))/(4*a** 2), True)) + c**3*x/(4*a**2) + 3*c**2*d*x**2/(8*a**2) + c*d**2*x**3/(4*a** 2) + d**3*x**4/(16*a**2)
Exception generated. \[ \int \frac {(c+d x)^3}{(a+i a \cot (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (204) = 408\).
Time = 0.32 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.53 \[ \int \frac {(c+d x)^3}{(a+i a \cot (e+f x))^2} \, dx=\frac {32 \, d^{3} f^{4} x^{4} + 128 \, c d^{2} f^{4} x^{3} - 32 i \, d^{3} f^{3} x^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 128 i \, d^{3} f^{3} x^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 192 \, c^{2} d f^{4} x^{2} - 96 i \, c d^{2} f^{3} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 384 i \, c d^{2} f^{3} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 128 \, c^{3} f^{4} x - 96 i \, c^{2} d f^{3} x e^{\left (4 i \, f x + 4 i \, e\right )} + 24 \, d^{3} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 384 i \, c^{2} d f^{3} x e^{\left (2 i \, f x + 2 i \, e\right )} - 192 \, d^{3} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 32 i \, c^{3} f^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 48 \, c d^{2} f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 128 i \, c^{3} f^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 384 \, c d^{2} f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 \, c^{2} d f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 12 i \, d^{3} f x e^{\left (4 i \, f x + 4 i \, e\right )} - 192 \, c^{2} d f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 192 i \, d^{3} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 12 i \, c d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )} - 192 i \, c d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - 3 \, d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )}}{512 \, a^{2} f^{4}} \]
1/512*(32*d^3*f^4*x^4 + 128*c*d^2*f^4*x^3 - 32*I*d^3*f^3*x^3*e^(4*I*f*x + 4*I*e) + 128*I*d^3*f^3*x^3*e^(2*I*f*x + 2*I*e) + 192*c^2*d*f^4*x^2 - 96*I* c*d^2*f^3*x^2*e^(4*I*f*x + 4*I*e) + 384*I*c*d^2*f^3*x^2*e^(2*I*f*x + 2*I*e ) + 128*c^3*f^4*x - 96*I*c^2*d*f^3*x*e^(4*I*f*x + 4*I*e) + 24*d^3*f^2*x^2* e^(4*I*f*x + 4*I*e) + 384*I*c^2*d*f^3*x*e^(2*I*f*x + 2*I*e) - 192*d^3*f^2* x^2*e^(2*I*f*x + 2*I*e) - 32*I*c^3*f^3*e^(4*I*f*x + 4*I*e) + 48*c*d^2*f^2* x*e^(4*I*f*x + 4*I*e) + 128*I*c^3*f^3*e^(2*I*f*x + 2*I*e) - 384*c*d^2*f^2* x*e^(2*I*f*x + 2*I*e) + 24*c^2*d*f^2*e^(4*I*f*x + 4*I*e) + 12*I*d^3*f*x*e^ (4*I*f*x + 4*I*e) - 192*c^2*d*f^2*e^(2*I*f*x + 2*I*e) - 192*I*d^3*f*x*e^(2 *I*f*x + 2*I*e) + 12*I*c*d^2*f*e^(4*I*f*x + 4*I*e) - 192*I*c*d^2*f*e^(2*I* f*x + 2*I*e) - 3*d^3*e^(4*I*f*x + 4*I*e) + 96*d^3*e^(2*I*f*x + 2*I*e))/(a^ 2*f^4)
Time = 0.92 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d x)^3}{(a+i a \cot (e+f x))^2} \, dx={\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (-\frac {\left (-4\,c^3\,f^3-c^2\,d\,f^2\,6{}\mathrm {i}+6\,c\,d^2\,f+d^3\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a^2\,f^4}+\frac {d^3\,x^3\,1{}\mathrm {i}}{4\,a^2\,f}+\frac {d\,x\,\left (2\,c^2\,f^2+c\,d\,f\,2{}\mathrm {i}-d^2\right )\,3{}\mathrm {i}}{8\,a^2\,f^3}+\frac {d^2\,x^2\,\left (2\,c\,f+d\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,a^2\,f^2}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (-\frac {\left (-32\,c^3\,f^3-c^2\,d\,f^2\,24{}\mathrm {i}+12\,c\,d^2\,f+d^3\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{512\,a^2\,f^4}+\frac {d^3\,x^3\,1{}\mathrm {i}}{16\,a^2\,f}+\frac {d\,x\,\left (8\,c^2\,f^2+c\,d\,f\,4{}\mathrm {i}-d^2\right )\,3{}\mathrm {i}}{128\,a^2\,f^3}+\frac {d^2\,x^2\,\left (4\,c\,f+d\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,a^2\,f^2}\right )+\frac {c^3\,x}{4\,a^2}+\frac {d^3\,x^4}{16\,a^2}+\frac {3\,c^2\,d\,x^2}{8\,a^2}+\frac {c\,d^2\,x^3}{4\,a^2} \]
exp(e*2i + f*x*2i)*((d^3*x^3*1i)/(4*a^2*f) - ((d^3*3i - 4*c^3*f^3 - c^2*d* f^2*6i + 6*c*d^2*f)*1i)/(16*a^2*f^4) + (d*x*(2*c^2*f^2 - d^2 + c*d*f*2i)*3 i)/(8*a^2*f^3) + (d^2*x^2*(d*1i + 2*c*f)*3i)/(8*a^2*f^2)) - exp(e*4i + f*x *4i)*((d^3*x^3*1i)/(16*a^2*f) - ((d^3*3i - 32*c^3*f^3 - c^2*d*f^2*24i + 12 *c*d^2*f)*1i)/(512*a^2*f^4) + (d*x*(8*c^2*f^2 - d^2 + c*d*f*4i)*3i)/(128*a ^2*f^3) + (d^2*x^2*(d*1i + 4*c*f)*3i)/(64*a^2*f^2)) + (c^3*x)/(4*a^2) + (d ^3*x^4)/(16*a^2) + (3*c^2*d*x^2)/(8*a^2) + (c*d^2*x^3)/(4*a^2)