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

Optimal. Leaf size=108 \[ -\frac{6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac{3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}+\frac{6 i f^3 \sinh (c+d x)}{a d^4}-\frac{i (e+f x)^3 \cosh (c+d x)}{a d}+\frac{(e+f x)^4}{4 a f} \]

[Out]

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

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Rubi [A]  time = 0.163624, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {5563, 32, 3296, 2637} \[ -\frac{6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac{3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}+\frac{6 i f^3 \sinh (c+d x)}{a d^4}-\frac{i (e+f x)^3 \cosh (c+d x)}{a d}+\frac{(e+f x)^4}{4 a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.779149, size = 106, normalized size = 0.98 \[ \frac{12 i f \sinh (c+d x) \left (d^2 (e+f x)^2+2 f^2\right )-4 i d (e+f x) \cosh (c+d x) \left (d^2 (e+f x)^2+6 f^2\right )+d^4 x \left (6 e^2 f x+4 e^3+4 e f^2 x^2+f^3 x^3\right )}{4 a d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^4*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) - (4*I)*d*(e + f*x)*(6*f^2 + d^2*(e + f*x)^2)*Cosh[c + d*x]
+ (12*I)*f*(2*f^2 + d^2*(e + f*x)^2)*Sinh[c + d*x])/(4*a*d^4)

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Maple [B]  time = 0.045, size = 448, normalized size = 4.2 \begin{align*} -{\frac{1}{{d}^{4}a} \left ( i{d}^{3}{e}^{3}\cosh \left ( dx+c \right ) +3\,i{c}^{2}{f}^{2}ed\cosh \left ( dx+c \right ) +i{f}^{3} \left ( \left ( dx+c \right ) ^{3}\cosh \left ( dx+c \right ) -3\, \left ( dx+c \right ) ^{2}\sinh \left ( dx+c \right ) +6\, \left ( dx+c \right ) \cosh \left ( dx+c \right ) -6\,\sinh \left ( dx+c \right ) \right ) -6\,i{f}^{2}ced \left ( \left ( dx+c \right ) \cosh \left ( dx+c \right ) -\sinh \left ( dx+c \right ) \right ) +3\,i{f}^{2}ed \left ( \left ( dx+c \right ) ^{2}\cosh \left ( dx+c \right ) -2\, \left ( dx+c \right ) \sinh \left ( dx+c \right ) +2\,\cosh \left ( dx+c \right ) \right ) +3\,if{e}^{2}{d}^{2} \left ( \left ( dx+c \right ) \cosh \left ( dx+c \right ) -\sinh \left ( dx+c \right ) \right ) -3\,ic{f}^{3} \left ( \left ( dx+c \right ) ^{2}\cosh \left ( dx+c \right ) -2\, \left ( dx+c \right ) \sinh \left ( dx+c \right ) +2\,\cosh \left ( dx+c \right ) \right ) -i{f}^{3}{c}^{3}\cosh \left ( dx+c \right ) +3\,i{c}^{2}{f}^{3} \left ( \left ( dx+c \right ) \cosh \left ( dx+c \right ) -\sinh \left ( dx+c \right ) \right ) -3\,icf{e}^{2}{d}^{2}\cosh \left ( dx+c \right ) -{\frac{{f}^{3} \left ( dx+c \right ) ^{4}}{4}}+c{f}^{3} \left ( dx+c \right ) ^{3}-{f}^{2}ed \left ( dx+c \right ) ^{3}-{\frac{3\,{c}^{2}{f}^{3} \left ( dx+c \right ) ^{2}}{2}}+3\,cde{f}^{2} \left ( dx+c \right ) ^{2}-{\frac{3\,{d}^{2}{e}^{2}f \left ( dx+c \right ) ^{2}}{2}}+{f}^{3}{c}^{3} \left ( dx+c \right ) -3\,{c}^{2}{f}^{2}ed \left ( dx+c \right ) +3\,cf{e}^{2}{d}^{2} \left ( dx+c \right ) -{d}^{3}{e}^{3} \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d^4/a*(I*d^3*e^3*cosh(d*x+c)+3*I*c^2*f^2*e*d*cosh(d*x+c)+I*f^3*((d*x+c)^3*cosh(d*x+c)-3*(d*x+c)^2*sinh(d*x+
c)+6*(d*x+c)*cosh(d*x+c)-6*sinh(d*x+c))-6*I*f^2*c*e*d*((d*x+c)*cosh(d*x+c)-sinh(d*x+c))+3*I*f^2*e*d*((d*x+c)^2
*cosh(d*x+c)-2*(d*x+c)*sinh(d*x+c)+2*cosh(d*x+c))+3*I*f*e^2*d^2*((d*x+c)*cosh(d*x+c)-sinh(d*x+c))-3*I*c*f^3*((
d*x+c)^2*cosh(d*x+c)-2*(d*x+c)*sinh(d*x+c)+2*cosh(d*x+c))-I*f^3*c^3*cosh(d*x+c)+3*I*c^2*f^3*((d*x+c)*cosh(d*x+
c)-sinh(d*x+c))-3*I*c*f*e^2*d^2*cosh(d*x+c)-1/4*f^3*(d*x+c)^4+c*f^3*(d*x+c)^3-f^2*e*d*(d*x+c)^3-3/2*c^2*f^3*(d
*x+c)^2+3*c*d*e*f^2*(d*x+c)^2-3/2*d^2*e^2*f*(d*x+c)^2+f^3*c^3*(d*x+c)-3*c^2*f^2*e*d*(d*x+c)+3*c*f*e^2*d^2*(d*x
+c)-d^3*e^3*(d*x+c))

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Maxima [B]  time = 2.02196, size = 504, normalized size = 4.67 \begin{align*} \frac{3}{4} \, e^{2} f{\left (\frac{4 \, x e^{\left (d x + c\right )}}{a d e^{\left (d x + c\right )} - i \, a d} + \frac{-2 i \, d^{2} x^{2} e^{c} - 2 i \, d x e^{c} -{\left (2 i \, d x e^{\left (3 \, c\right )} - 2 i \, e^{\left (3 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 2 \,{\left (d^{2} x^{2} e^{\left (2 \, c\right )} - 3 \, d x e^{\left (2 \, c\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - 2 \,{\left (d x + 1\right )} e^{\left (-d x\right )} - 2 i \, e^{c}}{a d^{2} e^{\left (d x + 2 \, c\right )} - i \, a d^{2} e^{c}}\right )} + \frac{1}{4} \, e^{3}{\left (\frac{4 \,{\left (d x + c\right )}}{a d} - \frac{2 i \, e^{\left (d x + c\right )}}{a d} - \frac{2 i \, e^{\left (-d x - c\right )}}{a d}\right )} + \frac{{\left (4 \, d^{3} x^{3} e^{c} -{\left (6 i \, d^{2} x^{2} e^{\left (2 \, c\right )} - 12 i \, d x e^{\left (2 \, c\right )} + 12 i \, e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} -{\left (6 i \, d^{2} x^{2} + 12 i \, d x + 12 i\right )} e^{\left (-d x\right )}\right )} e f^{2} e^{\left (-c\right )}}{4 \, a d^{3}} + \frac{{\left (d^{4} x^{4} e^{c} -{\left (2 i \, d^{3} x^{3} e^{\left (2 \, c\right )} - 6 i \, d^{2} x^{2} e^{\left (2 \, c\right )} + 12 i \, d x e^{\left (2 \, c\right )} - 12 i \, e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} -{\left (2 i \, d^{3} x^{3} + 6 i \, d^{2} x^{2} + 12 i \, d x + 12 i\right )} e^{\left (-d x\right )}\right )} f^{3} e^{\left (-c\right )}}{4 \, 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)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.21788, size = 605, normalized size = 5.6 \begin{align*} \frac{{\left (-2 i \, d^{3} f^{3} x^{3} - 2 i \, d^{3} e^{3} - 6 i \, d^{2} e^{2} f - 12 i \, d e f^{2} - 12 i \, f^{3} +{\left (-6 i \, d^{3} e f^{2} - 6 i \, d^{2} f^{3}\right )} x^{2} +{\left (-6 i \, d^{3} e^{2} f - 12 i \, d^{2} e f^{2} - 12 i \, d f^{3}\right )} x +{\left (-2 i \, d^{3} f^{3} x^{3} - 2 i \, d^{3} e^{3} + 6 i \, d^{2} e^{2} f - 12 i \, d e f^{2} + 12 i \, f^{3} +{\left (-6 i \, d^{3} e f^{2} + 6 i \, d^{2} f^{3}\right )} x^{2} +{\left (-6 i \, d^{3} e^{2} f + 12 i \, d^{2} e f^{2} - 12 i \, d f^{3}\right )} x\right )} e^{\left (2 \, d x + 2 \, c\right )} +{\left (d^{4} f^{3} x^{4} + 4 \, d^{4} e f^{2} x^{3} + 6 \, d^{4} e^{2} f x^{2} + 4 \, d^{4} e^{3} x\right )} e^{\left (d x + c\right )}\right )} e^{\left (-d x - c\right )}}{4 \, 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)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.66624, size = 610, normalized size = 5.65 \begin{align*} \begin{cases} \frac{\left (\left (- 2 i a^{7} d^{19} e^{3} e^{3 c} - 6 i a^{7} d^{19} e^{2} f x e^{3 c} - 6 i a^{7} d^{19} e f^{2} x^{2} e^{3 c} - 2 i a^{7} d^{19} f^{3} x^{3} e^{3 c} - 6 i a^{7} d^{18} e^{2} f e^{3 c} - 12 i a^{7} d^{18} e f^{2} x e^{3 c} - 6 i a^{7} d^{18} f^{3} x^{2} e^{3 c} - 12 i a^{7} d^{17} e f^{2} e^{3 c} - 12 i a^{7} d^{17} f^{3} x e^{3 c} - 12 i a^{7} d^{16} f^{3} e^{3 c}\right ) e^{- d x} + \left (- 2 i a^{7} d^{19} e^{3} e^{5 c} - 6 i a^{7} d^{19} e^{2} f x e^{5 c} - 6 i a^{7} d^{19} e f^{2} x^{2} e^{5 c} - 2 i a^{7} d^{19} f^{3} x^{3} e^{5 c} + 6 i a^{7} d^{18} e^{2} f e^{5 c} + 12 i a^{7} d^{18} e f^{2} x e^{5 c} + 6 i a^{7} d^{18} f^{3} x^{2} e^{5 c} - 12 i a^{7} d^{17} e f^{2} e^{5 c} - 12 i a^{7} d^{17} f^{3} x e^{5 c} + 12 i a^{7} d^{16} f^{3} e^{5 c}\right ) e^{d x}\right ) e^{- 4 c}}{4 a^{8} d^{20}} & \text{for}\: 4 a^{8} d^{20} e^{4 c} \neq 0 \\- \frac{x^{4} \left (i f^{3} e^{2 c} - i f^{3}\right ) e^{- c}}{8 a} - \frac{x^{3} \left (i e f^{2} e^{2 c} - i e f^{2}\right ) e^{- c}}{2 a} - \frac{x^{2} \left (3 i e^{2} f e^{2 c} - 3 i e^{2} f\right ) e^{- c}}{4 a} - \frac{x \left (i e^{3} e^{2 c} - i e^{3}\right ) e^{- c}}{2 a} & \text{otherwise} \end{cases} + \frac{e^{3} x}{a} + \frac{3 e^{2} f x^{2}}{2 a} + \frac{e f^{2} x^{3}}{a} + \frac{f^{3} x^{4}}{4 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((((-2*I*a**7*d**19*e**3*exp(3*c) - 6*I*a**7*d**19*e**2*f*x*exp(3*c) - 6*I*a**7*d**19*e*f**2*x**2*exp
(3*c) - 2*I*a**7*d**19*f**3*x**3*exp(3*c) - 6*I*a**7*d**18*e**2*f*exp(3*c) - 12*I*a**7*d**18*e*f**2*x*exp(3*c)
- 6*I*a**7*d**18*f**3*x**2*exp(3*c) - 12*I*a**7*d**17*e*f**2*exp(3*c) - 12*I*a**7*d**17*f**3*x*exp(3*c) - 12*
I*a**7*d**16*f**3*exp(3*c))*exp(-d*x) + (-2*I*a**7*d**19*e**3*exp(5*c) - 6*I*a**7*d**19*e**2*f*x*exp(5*c) - 6*
I*a**7*d**19*e*f**2*x**2*exp(5*c) - 2*I*a**7*d**19*f**3*x**3*exp(5*c) + 6*I*a**7*d**18*e**2*f*exp(5*c) + 12*I*
a**7*d**18*e*f**2*x*exp(5*c) + 6*I*a**7*d**18*f**3*x**2*exp(5*c) - 12*I*a**7*d**17*e*f**2*exp(5*c) - 12*I*a**7
*d**17*f**3*x*exp(5*c) + 12*I*a**7*d**16*f**3*exp(5*c))*exp(d*x))*exp(-4*c)/(4*a**8*d**20), Ne(4*a**8*d**20*ex
p(4*c), 0)), (-x**4*(I*f**3*exp(2*c) - I*f**3)*exp(-c)/(8*a) - x**3*(I*e*f**2*exp(2*c) - I*e*f**2)*exp(-c)/(2*
a) - x**2*(3*I*e**2*f*exp(2*c) - 3*I*e**2*f)*exp(-c)/(4*a) - x*(I*e**3*exp(2*c) - I*e**3)*exp(-c)/(2*a), True)
) + e**3*x/a + 3*e**2*f*x**2/(2*a) + e*f**2*x**3/a + f**3*x**4/(4*a)

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Giac [B]  time = 1.34559, size = 1084, normalized size = 10.04 \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)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

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