3.265 \(\int \frac{(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\)

Optimal. Leaf size=231 \[ -\frac{3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{a d^3}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac{3 i f (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{4 a d^2}-\frac{6 f^3 \cosh (c+d x)}{a d^4}+\frac{3 i f^3 \sinh (c+d x) \cosh (c+d x)}{8 a d^4}-\frac{i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}+\frac{(e+f x)^3 \sinh (c+d x)}{a d}-\frac{3 i f^3 x}{8 a d^3}-\frac{i (e+f x)^3}{4 a d} \]

[Out]

(((-3*I)/8)*f^3*x)/(a*d^3) - ((I/4)*(e + f*x)^3)/(a*d) - (6*f^3*Cosh[c + d*x])/(a*d^4) - (3*f*(e + f*x)^2*Cosh
[c + d*x])/(a*d^2) + (6*f^2*(e + f*x)*Sinh[c + d*x])/(a*d^3) + ((e + f*x)^3*Sinh[c + d*x])/(a*d) + (((3*I)/8)*
f^3*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^4) + (((3*I)/4)*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^2) - (((
3*I)/4)*f^2*(e + f*x)*Sinh[c + d*x]^2)/(a*d^3) - ((I/2)*(e + f*x)^3*Sinh[c + d*x]^2)/(a*d)

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Rubi [A]  time = 0.260774, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {5563, 3296, 2638, 5446, 3311, 32, 2635, 8} \[ -\frac{3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{a d^3}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac{3 i f (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{4 a d^2}-\frac{6 f^3 \cosh (c+d x)}{a d^4}+\frac{3 i f^3 \sinh (c+d x) \cosh (c+d x)}{8 a d^4}-\frac{i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}+\frac{(e+f x)^3 \sinh (c+d x)}{a d}-\frac{3 i f^3 x}{8 a d^3}-\frac{i (e+f x)^3}{4 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-3*I)/8)*f^3*x)/(a*d^3) - ((I/4)*(e + f*x)^3)/(a*d) - (6*f^3*Cosh[c + d*x])/(a*d^4) - (3*f*(e + f*x)^2*Cosh
[c + d*x])/(a*d^2) + (6*f^2*(e + f*x)*Sinh[c + d*x])/(a*d^3) + ((e + f*x)^3*Sinh[c + d*x])/(a*d) + (((3*I)/8)*
f^3*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^4) + (((3*I)/4)*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^2) - (((
3*I)/4)*f^2*(e + f*x)*Sinh[c + d*x]^2)/(a*d^3) - ((I/2)*(e + f*x)^3*Sinh[c + d*x]^2)/(a*d)

Rule 5563

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(n -
2)*Sinh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 + b^2, 0]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

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

Rule 5446

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c
+ d*x)^m*Sinh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{a}+\frac{\int (e+f x)^3 \cosh (c+d x) \, dx}{a}\\ &=\frac{(e+f x)^3 \sinh (c+d x)}{a d}-\frac{i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}+\frac{(3 i f) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 a d}-\frac{(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a d}\\ &=-\frac{3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac{(e+f x)^3 \sinh (c+d x)}{a d}+\frac{3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac{3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac{i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}-\frac{(3 i f) \int (e+f x)^2 \, dx}{4 a d}+\frac{\left (6 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a d^2}+\frac{\left (3 i f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 a d^3}\\ &=-\frac{i (e+f x)^3}{4 a d}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac{(e+f x)^3 \sinh (c+d x)}{a d}+\frac{3 i f^3 \cosh (c+d x) \sinh (c+d x)}{8 a d^4}+\frac{3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac{3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac{i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}-\frac{\left (3 i f^3\right ) \int 1 \, dx}{8 a d^3}-\frac{\left (6 f^3\right ) \int \sinh (c+d x) \, dx}{a d^3}\\ &=-\frac{3 i f^3 x}{8 a d^3}-\frac{i (e+f x)^3}{4 a d}-\frac{6 f^3 \cosh (c+d x)}{a d^4}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac{(e+f x)^3 \sinh (c+d x)}{a d}+\frac{3 i f^3 \cosh (c+d x) \sinh (c+d x)}{8 a d^4}+\frac{3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac{3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac{i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}\\ \end{align*}

Mathematica [A]  time = 1.30041, size = 134, normalized size = 0.58 \[ \frac{-96 f \cosh (c+d x) \left (d^2 (e+f x)^2+2 f^2\right )-4 i d (e+f x) \cosh (2 (c+d x)) \left (2 d^2 (e+f x)^2+3 f^2\right )+4 \sinh (c+d x) \left (8 d (e+f x) \left (d^2 (e+f x)^2+6 f^2\right )+3 i f \cosh (c+d x) \left (2 d^2 (e+f x)^2+f^2\right )\right )}{32 a d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-96*f*(2*f^2 + d^2*(e + f*x)^2)*Cosh[c + d*x] - (4*I)*d*(e + f*x)*(3*f^2 + 2*d^2*(e + f*x)^2)*Cosh[2*(c + d*x
)] + 4*(8*d*(e + f*x)*(6*f^2 + d^2*(e + f*x)^2) + (3*I)*f*(f^2 + 2*d^2*(e + f*x)^2)*Cosh[c + d*x])*Sinh[c + d*
x])/(32*a*d^4)

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Maple [B]  time = 0.16, size = 429, normalized size = 1.9 \begin{align*}{\frac{-{\frac{i}{32}} \left ( 4\,{f}^{3}{x}^{3}{d}^{3}+12\,{d}^{3}e{f}^{2}{x}^{2}+12\,{d}^{3}{e}^{2}fx-6\,{d}^{2}{f}^{3}{x}^{2}+4\,{d}^{3}{e}^{3}-12\,{d}^{2}e{f}^{2}x-6\,{d}^{2}{e}^{2}f+6\,d{f}^{3}x+6\,de{f}^{2}-3\,{f}^{3} \right ){{\rm e}^{2\,dx+2\,c}}}{a{d}^{4}}}+{\frac{ \left ({f}^{3}{x}^{3}{d}^{3}+3\,{d}^{3}e{f}^{2}{x}^{2}+3\,{d}^{3}{e}^{2}fx-3\,{d}^{2}{f}^{3}{x}^{2}+{d}^{3}{e}^{3}-6\,{d}^{2}e{f}^{2}x-3\,{d}^{2}{e}^{2}f+6\,d{f}^{3}x+6\,de{f}^{2}-6\,{f}^{3} \right ){{\rm e}^{dx+c}}}{2\,a{d}^{4}}}-{\frac{ \left ({f}^{3}{x}^{3}{d}^{3}+3\,{d}^{3}e{f}^{2}{x}^{2}+3\,{d}^{3}{e}^{2}fx+3\,{d}^{2}{f}^{3}{x}^{2}+{d}^{3}{e}^{3}+6\,{d}^{2}e{f}^{2}x+3\,{d}^{2}{e}^{2}f+6\,d{f}^{3}x+6\,de{f}^{2}+6\,{f}^{3} \right ){{\rm e}^{-dx-c}}}{2\,a{d}^{4}}}-{\frac{{\frac{i}{32}} \left ( 4\,{f}^{3}{x}^{3}{d}^{3}+12\,{d}^{3}e{f}^{2}{x}^{2}+12\,{d}^{3}{e}^{2}fx+6\,{d}^{2}{f}^{3}{x}^{2}+4\,{d}^{3}{e}^{3}+12\,{d}^{2}e{f}^{2}x+6\,{d}^{2}{e}^{2}f+6\,d{f}^{3}x+6\,de{f}^{2}+3\,{f}^{3} \right ){{\rm e}^{-2\,dx-2\,c}}}{a{d}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*I*(4*d^3*f^3*x^3+12*d^3*e*f^2*x^2+12*d^3*e^2*f*x-6*d^2*f^3*x^2+4*d^3*e^3-12*d^2*e*f^2*x-6*d^2*e^2*f+6*d*
f^3*x+6*d*e*f^2-3*f^3)/a/d^4*exp(2*d*x+2*c)+1/2*(d^3*f^3*x^3+3*d^3*e*f^2*x^2+3*d^3*e^2*f*x-3*d^2*f^3*x^2+d^3*e
^3-6*d^2*e*f^2*x-3*d^2*e^2*f+6*d*f^3*x+6*d*e*f^2-6*f^3)/a/d^4*exp(d*x+c)-1/2*(d^3*f^3*x^3+3*d^3*e*f^2*x^2+3*d^
3*e^2*f*x+3*d^2*f^3*x^2+d^3*e^3+6*d^2*e*f^2*x+3*d^2*e^2*f+6*d*f^3*x+6*d*e*f^2+6*f^3)/a/d^4*exp(-d*x-c)-1/32*I*
(4*d^3*f^3*x^3+12*d^3*e*f^2*x^2+12*d^3*e^2*f*x+6*d^2*f^3*x^2+4*d^3*e^3+12*d^2*e*f^2*x+6*d^2*e^2*f+6*d*f^3*x+6*
d*e*f^2+3*f^3)/a/d^4*exp(-2*d*x-2*c)

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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((f*x+e)^3*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.21988, size = 906, normalized size = 3.92 \begin{align*} \frac{{\left (-4 i \, d^{3} f^{3} x^{3} - 4 i \, d^{3} e^{3} - 6 i \, d^{2} e^{2} f - 6 i \, d e f^{2} - 3 i \, f^{3} +{\left (-12 i \, d^{3} e f^{2} - 6 i \, d^{2} f^{3}\right )} x^{2} +{\left (-12 i \, d^{3} e^{2} f - 12 i \, d^{2} e f^{2} - 6 i \, d f^{3}\right )} x +{\left (-4 i \, d^{3} f^{3} x^{3} - 4 i \, d^{3} e^{3} + 6 i \, d^{2} e^{2} f - 6 i \, d e f^{2} + 3 i \, f^{3} +{\left (-12 i \, d^{3} e f^{2} + 6 i \, d^{2} f^{3}\right )} x^{2} +{\left (-12 i \, d^{3} e^{2} f + 12 i \, d^{2} e f^{2} - 6 i \, d f^{3}\right )} x\right )} e^{\left (4 \, d x + 4 \, c\right )} + 16 \,{\left (d^{3} f^{3} x^{3} + d^{3} e^{3} - 3 \, d^{2} e^{2} f + 6 \, d e f^{2} - 6 \, f^{3} + 3 \,{\left (d^{3} e f^{2} - d^{2} f^{3}\right )} x^{2} + 3 \,{\left (d^{3} e^{2} f - 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - 16 \,{\left (d^{3} f^{3} x^{3} + d^{3} e^{3} + 3 \, d^{2} e^{2} f + 6 \, d e f^{2} + 6 \, f^{3} + 3 \,{\left (d^{3} e f^{2} + d^{2} f^{3}\right )} x^{2} + 3 \,{\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} x\right )} e^{\left (d x + c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{32 \, a d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(-4*I*d^3*f^3*x^3 - 4*I*d^3*e^3 - 6*I*d^2*e^2*f - 6*I*d*e*f^2 - 3*I*f^3 + (-12*I*d^3*e*f^2 - 6*I*d^2*f^3)
*x^2 + (-12*I*d^3*e^2*f - 12*I*d^2*e*f^2 - 6*I*d*f^3)*x + (-4*I*d^3*f^3*x^3 - 4*I*d^3*e^3 + 6*I*d^2*e^2*f - 6*
I*d*e*f^2 + 3*I*f^3 + (-12*I*d^3*e*f^2 + 6*I*d^2*f^3)*x^2 + (-12*I*d^3*e^2*f + 12*I*d^2*e*f^2 - 6*I*d*f^3)*x)*
e^(4*d*x + 4*c) + 16*(d^3*f^3*x^3 + d^3*e^3 - 3*d^2*e^2*f + 6*d*e*f^2 - 6*f^3 + 3*(d^3*e*f^2 - d^2*f^3)*x^2 +
3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*x)*e^(3*d*x + 3*c) - 16*(d^3*f^3*x^3 + d^3*e^3 + 3*d^2*e^2*f + 6*d*e*f^2
+ 6*f^3 + 3*(d^3*e*f^2 + d^2*f^3)*x^2 + 3*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*x)*e^(d*x + c))*e^(-2*d*x - 2*c
)/(a*d^4)

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Sympy [A]  time = 4.77379, size = 1061, normalized size = 4.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*cosh(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)

[Out]

Piecewise((((-2048*a**9*d**27*e**3*exp(8*c) - 6144*a**9*d**27*e**2*f*x*exp(8*c) - 6144*a**9*d**27*e*f**2*x**2*
exp(8*c) - 2048*a**9*d**27*f**3*x**3*exp(8*c) - 6144*a**9*d**26*e**2*f*exp(8*c) - 12288*a**9*d**26*e*f**2*x*ex
p(8*c) - 6144*a**9*d**26*f**3*x**2*exp(8*c) - 12288*a**9*d**25*e*f**2*exp(8*c) - 12288*a**9*d**25*f**3*x*exp(8
*c) - 12288*a**9*d**24*f**3*exp(8*c))*exp(-d*x) + (2048*a**9*d**27*e**3*exp(10*c) + 6144*a**9*d**27*e**2*f*x*e
xp(10*c) + 6144*a**9*d**27*e*f**2*x**2*exp(10*c) + 2048*a**9*d**27*f**3*x**3*exp(10*c) - 6144*a**9*d**26*e**2*
f*exp(10*c) - 12288*a**9*d**26*e*f**2*x*exp(10*c) - 6144*a**9*d**26*f**3*x**2*exp(10*c) + 12288*a**9*d**25*e*f
**2*exp(10*c) + 12288*a**9*d**25*f**3*x*exp(10*c) - 12288*a**9*d**24*f**3*exp(10*c))*exp(d*x) + (-512*I*a**9*d
**27*e**3*exp(7*c) - 1536*I*a**9*d**27*e**2*f*x*exp(7*c) - 1536*I*a**9*d**27*e*f**2*x**2*exp(7*c) - 512*I*a**9
*d**27*f**3*x**3*exp(7*c) - 768*I*a**9*d**26*e**2*f*exp(7*c) - 1536*I*a**9*d**26*e*f**2*x*exp(7*c) - 768*I*a**
9*d**26*f**3*x**2*exp(7*c) - 768*I*a**9*d**25*e*f**2*exp(7*c) - 768*I*a**9*d**25*f**3*x*exp(7*c) - 384*I*a**9*
d**24*f**3*exp(7*c))*exp(-2*d*x) + (-512*I*a**9*d**27*e**3*exp(11*c) - 1536*I*a**9*d**27*e**2*f*x*exp(11*c) -
1536*I*a**9*d**27*e*f**2*x**2*exp(11*c) - 512*I*a**9*d**27*f**3*x**3*exp(11*c) + 768*I*a**9*d**26*e**2*f*exp(1
1*c) + 1536*I*a**9*d**26*e*f**2*x*exp(11*c) + 768*I*a**9*d**26*f**3*x**2*exp(11*c) - 768*I*a**9*d**25*e*f**2*e
xp(11*c) - 768*I*a**9*d**25*f**3*x*exp(11*c) + 384*I*a**9*d**24*f**3*exp(11*c))*exp(2*d*x))*exp(-9*c)/(4096*a*
*10*d**28), Ne(4096*a**10*d**28*exp(9*c), 0)), (-x**4*(I*f**3*exp(4*c) - 2*f**3*exp(3*c) - 2*f**3*exp(c) - I*f
**3)*exp(-2*c)/(16*a) - x**3*(I*e*f**2*exp(4*c) - 2*e*f**2*exp(3*c) - 2*e*f**2*exp(c) - I*e*f**2)*exp(-2*c)/(4
*a) - x**2*(3*I*e**2*f*exp(4*c) - 6*e**2*f*exp(3*c) - 6*e**2*f*exp(c) - 3*I*e**2*f)*exp(-2*c)/(8*a) - x*(I*e**
3*exp(4*c) - 2*e**3*exp(3*c) - 2*e**3*exp(c) - I*e**3)*exp(-2*c)/(4*a), True))

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Giac [B]  time = 1.28698, size = 1357, normalized size = 5.87 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-4*I*d^3*f^3*x^3*e^(5*d*x + 6*c) + 12*d^3*f^3*x^3*e^(4*d*x + 5*c) - 16*I*d^3*f^3*x^3*e^(3*d*x + 4*c) - 16*d^3
*f^3*x^3*e^(2*d*x + 3*c) + 12*I*d^3*f^3*x^3*e^(d*x + 2*c) - 4*d^3*f^3*x^3*e^c - 12*I*d^3*f^2*x^2*e^(5*d*x + 6*
c + 1) + 6*I*d^2*f^3*x^2*e^(5*d*x + 6*c) + 36*d^3*f^2*x^2*e^(4*d*x + 5*c + 1) - 42*d^2*f^3*x^2*e^(4*d*x + 5*c)
- 48*I*d^3*f^2*x^2*e^(3*d*x + 4*c + 1) + 48*I*d^2*f^3*x^2*e^(3*d*x + 4*c) - 48*d^3*f^2*x^2*e^(2*d*x + 3*c + 1
) - 48*d^2*f^3*x^2*e^(2*d*x + 3*c) + 36*I*d^3*f^2*x^2*e^(d*x + 2*c + 1) + 42*I*d^2*f^3*x^2*e^(d*x + 2*c) - 12*
d^3*f^2*x^2*e^(c + 1) - 6*d^2*f^3*x^2*e^c - 12*I*d^3*f*x*e^(5*d*x + 6*c + 2) + 12*I*d^2*f^2*x*e^(5*d*x + 6*c +
1) - 6*I*d*f^3*x*e^(5*d*x + 6*c) + 36*d^3*f*x*e^(4*d*x + 5*c + 2) - 84*d^2*f^2*x*e^(4*d*x + 5*c + 1) + 90*d*f
^3*x*e^(4*d*x + 5*c) - 48*I*d^3*f*x*e^(3*d*x + 4*c + 2) + 96*I*d^2*f^2*x*e^(3*d*x + 4*c + 1) - 96*I*d*f^3*x*e^
(3*d*x + 4*c) - 48*d^3*f*x*e^(2*d*x + 3*c + 2) - 96*d^2*f^2*x*e^(2*d*x + 3*c + 1) - 96*d*f^3*x*e^(2*d*x + 3*c)
+ 36*I*d^3*f*x*e^(d*x + 2*c + 2) + 84*I*d^2*f^2*x*e^(d*x + 2*c + 1) + 90*I*d*f^3*x*e^(d*x + 2*c) - 12*d^3*f*x
*e^(c + 2) - 12*d^2*f^2*x*e^(c + 1) - 6*d*f^3*x*e^c - 4*I*d^3*e^(5*d*x + 6*c + 3) + 6*I*d^2*f*e^(5*d*x + 6*c +
2) - 6*I*d*f^2*e^(5*d*x + 6*c + 1) + 3*I*f^3*e^(5*d*x + 6*c) + 12*d^3*e^(4*d*x + 5*c + 3) - 42*d^2*f*e^(4*d*x
+ 5*c + 2) + 90*d*f^2*e^(4*d*x + 5*c + 1) - 93*f^3*e^(4*d*x + 5*c) - 16*I*d^3*e^(3*d*x + 4*c + 3) + 48*I*d^2*
f*e^(3*d*x + 4*c + 2) - 96*I*d*f^2*e^(3*d*x + 4*c + 1) + 96*I*f^3*e^(3*d*x + 4*c) - 16*d^3*e^(2*d*x + 3*c + 3)
- 48*d^2*f*e^(2*d*x + 3*c + 2) - 96*d*f^2*e^(2*d*x + 3*c + 1) - 96*f^3*e^(2*d*x + 3*c) + 12*I*d^3*e^(d*x + 2*
c + 3) + 42*I*d^2*f*e^(d*x + 2*c + 2) + 90*I*d*f^2*e^(d*x + 2*c + 1) + 93*I*f^3*e^(d*x + 2*c) - 4*d^3*e^(c + 3
) - 6*d^2*f*e^(c + 2) - 6*d*f^2*e^(c + 1) - 3*f^3*e^c)/(32*a*d^4*e^(3*d*x + 4*c) - 32*I*a*d^4*e^(2*d*x + 3*c))