\(\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [266]

Optimal result

Integrand size = 31, antiderivative size = 171 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i e f x}{2 a d}-\frac {i f^2 x^2}{4 a d}-\frac {2 f (e+f x) \cosh (c+d x)}{a d^2}+\frac {2 f^2 \sinh (c+d x)}{a d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{a d}+\frac {i f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d^2}-\frac {i f^2 \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \sinh ^2(c+d x)}{2 a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {5682, 3377, 2717, 5554, 3391} \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i f^2 \sinh ^2(c+d x)}{4 a d^3}+\frac {2 f^2 \sinh (c+d x)}{a d^3}-\frac {2 f (e+f x) \cosh (c+d x)}{a d^2}+\frac {i f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x)}{2 a d}+\frac {(e+f x)^2 \sinh (c+d x)}{a d}-\frac {i e f x}{2 a d}-\frac {i f^2 x^2}{4 a d} \]

[In]

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

[Out]

((-1/2*I)*e*f*x)/(a*d) - ((I/4)*f^2*x^2)/(a*d) - (2*f*(e + f*x)*Cosh[c + d*x])/(a*d^2) + (2*f^2*Sinh[c + d*x])
/(a*d^3) + ((e + f*x)^2*Sinh[c + d*x])/(a*d) + ((I/2)*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^2) - ((I/4
)*f^2*Sinh[c + d*x]^2)/(a*d^3) - ((I/2)*(e + f*x)^2*Sinh[c + d*x]^2)/(a*d)

Rule 2717

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

Rule 3377

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 3391

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

Rule 5554

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 5682

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]

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

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.58 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {-32 d f (e+f x) \cosh (c+d x)-2 i \left (f^2+2 d^2 (e+f x)^2\right ) \cosh (2 (c+d x))+8 \left (2 \left (2 f^2+d^2 (e+f x)^2\right )+i d f (e+f x) \cosh (c+d x)\right ) \sinh (c+d x)}{16 a d^3} \]

[In]

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

[Out]

(-32*d*f*(e + f*x)*Cosh[c + d*x] - (2*I)*(f^2 + 2*d^2*(e + f*x)^2)*Cosh[2*(c + d*x)] + 8*(2*(2*f^2 + d^2*(e +
f*x)^2) + I*d*f*(e + f*x)*Cosh[c + d*x])*Sinh[c + d*x])/(16*a*d^3)

Maple [A] (verified)

Time = 16.57 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.41

method	result	size
risch	\(-\frac {i \left (2 d^{2} x^{2} f^{2}+4 d^{2} e f x +2 d^{2} e^{2}-2 x d \,f^{2}-2 d e f +f^{2}\right ) {\mathrm e}^{2 d x +2 c}}{16 d^{3} a}+\frac {\left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}-2 x d \,f^{2}-2 d e f +2 f^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{3} a}-\frac {\left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}+2 x d \,f^{2}+2 d e f +2 f^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{3} a}-\frac {i \left (2 d^{2} x^{2} f^{2}+4 d^{2} e f x +2 d^{2} e^{2}+2 x d \,f^{2}+2 d e f +f^{2}\right ) {\mathrm e}^{-2 d x -2 c}}{16 a \,d^{3}}\)	\(241\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.33 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {{\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e^{2} - 2 i \, d e f - i \, f^{2} - 2 \, {\left (2 i \, d^{2} e f + i \, d f^{2}\right )} x + {\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e^{2} + 2 i \, d e f - i \, f^{2} - 2 \, {\left (2 i \, d^{2} e f - i \, d f^{2}\right )} x\right )} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, {\left (d^{2} f^{2} x^{2} + d^{2} e^{2} - 2 \, d e f + 2 \, f^{2} + 2 \, {\left (d^{2} e f - d f^{2}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - 8 \, {\left (d^{2} f^{2} x^{2} + d^{2} e^{2} + 2 \, d e f + 2 \, f^{2} + 2 \, {\left (d^{2} e f + d f^{2}\right )} x\right )} e^{\left (d x + c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, a d^{3}} \]

[In]

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

[Out]

1/16*(-2*I*d^2*f^2*x^2 - 2*I*d^2*e^2 - 2*I*d*e*f - I*f^2 - 2*(2*I*d^2*e*f + I*d*f^2)*x + (-2*I*d^2*f^2*x^2 - 2
*I*d^2*e^2 + 2*I*d*e*f - I*f^2 - 2*(2*I*d^2*e*f - I*d*f^2)*x)*e^(4*d*x + 4*c) + 8*(d^2*f^2*x^2 + d^2*e^2 - 2*d
*e*f + 2*f^2 + 2*(d^2*e*f - d*f^2)*x)*e^(3*d*x + 3*c) - 8*(d^2*f^2*x^2 + d^2*e^2 + 2*d*e*f + 2*f^2 + 2*(d^2*e*
f + d*f^2)*x)*e^(d*x + c))*e^(-2*d*x - 2*c)/(a*d^3)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (150) = 300\).

Time = 0.44 (sec) , antiderivative size = 631, normalized size of antiderivative = 3.69 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\begin {cases} \frac {\left (\left (- 512 a^{3} d^{11} e^{2} e^{2 c} - 1024 a^{3} d^{11} e f x e^{2 c} - 512 a^{3} d^{11} f^{2} x^{2} e^{2 c} - 1024 a^{3} d^{10} e f e^{2 c} - 1024 a^{3} d^{10} f^{2} x e^{2 c} - 1024 a^{3} d^{9} f^{2} e^{2 c}\right ) e^{- d x} + \left (512 a^{3} d^{11} e^{2} e^{4 c} + 1024 a^{3} d^{11} e f x e^{4 c} + 512 a^{3} d^{11} f^{2} x^{2} e^{4 c} - 1024 a^{3} d^{10} e f e^{4 c} - 1024 a^{3} d^{10} f^{2} x e^{4 c} + 1024 a^{3} d^{9} f^{2} e^{4 c}\right ) e^{d x} + \left (- 128 i a^{3} d^{11} e^{2} e^{c} - 256 i a^{3} d^{11} e f x e^{c} - 128 i a^{3} d^{11} f^{2} x^{2} e^{c} - 128 i a^{3} d^{10} e f e^{c} - 128 i a^{3} d^{10} f^{2} x e^{c} - 64 i a^{3} d^{9} f^{2} e^{c}\right ) e^{- 2 d x} + \left (- 128 i a^{3} d^{11} e^{2} e^{5 c} - 256 i a^{3} d^{11} e f x e^{5 c} - 128 i a^{3} d^{11} f^{2} x^{2} e^{5 c} + 128 i a^{3} d^{10} e f e^{5 c} + 128 i a^{3} d^{10} f^{2} x e^{5 c} - 64 i a^{3} d^{9} f^{2} e^{5 c}\right ) e^{2 d x}\right ) e^{- 3 c}}{1024 a^{4} d^{12}} & \text {for}\: a^{4} d^{12} e^{3 c} \neq 0 \\\frac {x^{3} \left (- i f^{2} e^{4 c} + 2 f^{2} e^{3 c} + 2 f^{2} e^{c} + i f^{2}\right ) e^{- 2 c}}{12 a} + \frac {x^{2} \left (- i e f e^{4 c} + 2 e f e^{3 c} + 2 e f e^{c} + i e f\right ) e^{- 2 c}}{4 a} + \frac {x \left (- i e^{2} e^{4 c} + 2 e^{2} e^{3 c} + 2 e^{2} e^{c} + i e^{2}\right ) e^{- 2 c}}{4 a} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((((-512*a**3*d**11*e**2*exp(2*c) - 1024*a**3*d**11*e*f*x*exp(2*c) - 512*a**3*d**11*f**2*x**2*exp(2*c
) - 1024*a**3*d**10*e*f*exp(2*c) - 1024*a**3*d**10*f**2*x*exp(2*c) - 1024*a**3*d**9*f**2*exp(2*c))*exp(-d*x) +
(512*a**3*d**11*e**2*exp(4*c) + 1024*a**3*d**11*e*f*x*exp(4*c) + 512*a**3*d**11*f**2*x**2*exp(4*c) - 1024*a**
3*d**10*e*f*exp(4*c) - 1024*a**3*d**10*f**2*x*exp(4*c) + 1024*a**3*d**9*f**2*exp(4*c))*exp(d*x) + (-128*I*a**3
*d**11*e**2*exp(c) - 256*I*a**3*d**11*e*f*x*exp(c) - 128*I*a**3*d**11*f**2*x**2*exp(c) - 128*I*a**3*d**10*e*f*
exp(c) - 128*I*a**3*d**10*f**2*x*exp(c) - 64*I*a**3*d**9*f**2*exp(c))*exp(-2*d*x) + (-128*I*a**3*d**11*e**2*ex
p(5*c) - 256*I*a**3*d**11*e*f*x*exp(5*c) - 128*I*a**3*d**11*f**2*x**2*exp(5*c) + 128*I*a**3*d**10*e*f*exp(5*c)
+ 128*I*a**3*d**10*f**2*x*exp(5*c) - 64*I*a**3*d**9*f**2*exp(5*c))*exp(2*d*x))*exp(-3*c)/(1024*a**4*d**12), N
e(a**4*d**12*exp(3*c), 0)), (x**3*(-I*f**2*exp(4*c) + 2*f**2*exp(3*c) + 2*f**2*exp(c) + I*f**2)*exp(-2*c)/(12*
a) + x**2*(-I*e*f*exp(4*c) + 2*e*f*exp(3*c) + 2*e*f*exp(c) + I*e*f)*exp(-2*c)/(4*a) + x*(-I*e**2*exp(4*c) + 2*
e**2*exp(3*c) + 2*e**2*exp(c) + I*e**2)*exp(-2*c)/(4*a), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (151) = 302\).

Time = 0.30 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.98 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {{\left (2 i \, d^{2} f^{2} x^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, d^{2} f^{2} x^{2} e^{\left (3 \, d x + 3 \, c\right )} + 8 \, d^{2} f^{2} x^{2} e^{\left (d x + c\right )} + 2 i \, d^{2} f^{2} x^{2} + 4 i \, d^{2} e f x e^{\left (4 \, d x + 4 \, c\right )} - 16 \, d^{2} e f x e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d^{2} e f x e^{\left (d x + c\right )} + 4 i \, d^{2} e f x + 2 i \, d^{2} e^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 i \, d f^{2} x e^{\left (4 \, d x + 4 \, c\right )} - 8 \, d^{2} e^{2} e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d f^{2} x e^{\left (3 \, d x + 3 \, c\right )} + 8 \, d^{2} e^{2} e^{\left (d x + c\right )} + 16 \, d f^{2} x e^{\left (d x + c\right )} + 2 i \, d^{2} e^{2} + 2 i \, d f^{2} x - 2 i \, d e f e^{\left (4 \, d x + 4 \, c\right )} + 16 \, d e f e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d e f e^{\left (d x + c\right )} + 2 i \, d e f + i \, f^{2} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, f^{2} e^{\left (3 \, d x + 3 \, c\right )} + 16 \, f^{2} e^{\left (d x + c\right )} + i \, f^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, a d^{3}} \]

[In]

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

[Out]

-1/16*(2*I*d^2*f^2*x^2*e^(4*d*x + 4*c) - 8*d^2*f^2*x^2*e^(3*d*x + 3*c) + 8*d^2*f^2*x^2*e^(d*x + c) + 2*I*d^2*f
^2*x^2 + 4*I*d^2*e*f*x*e^(4*d*x + 4*c) - 16*d^2*e*f*x*e^(3*d*x + 3*c) + 16*d^2*e*f*x*e^(d*x + c) + 4*I*d^2*e*f
*x + 2*I*d^2*e^2*e^(4*d*x + 4*c) - 2*I*d*f^2*x*e^(4*d*x + 4*c) - 8*d^2*e^2*e^(3*d*x + 3*c) + 16*d*f^2*x*e^(3*d
*x + 3*c) + 8*d^2*e^2*e^(d*x + c) + 16*d*f^2*x*e^(d*x + c) + 2*I*d^2*e^2 + 2*I*d*f^2*x - 2*I*d*e*f*e^(4*d*x +
4*c) + 16*d*e*f*e^(3*d*x + 3*c) + 16*d*e*f*e^(d*x + c) + 2*I*d*e*f + I*f^2*e^(4*d*x + 4*c) - 16*f^2*e^(3*d*x +
3*c) + 16*f^2*e^(d*x + c) + I*f^2)*e^(-2*d*x - 2*c)/(a*d^3)

Mupad [B] (verification not implemented)

Time = 1.55 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.58 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx={\mathrm {e}}^{c+d\,x}\,\left (\frac {d^2\,e^2-2\,d\,e\,f+2\,f^2}{2\,a\,d^3}+\frac {f^2\,x^2}{2\,a\,d}-\frac {f\,x\,\left (f-d\,e\right )}{a\,d^2}\right )-{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (\frac {\left (2\,d^2\,e^2+2\,d\,e\,f+f^2\right )\,1{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,1{}\mathrm {i}}{8\,a\,d}+\frac {f\,x\,\left (f+2\,d\,e\right )\,1{}\mathrm {i}}{8\,a\,d^2}\right )-{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (\frac {\left (2\,d^2\,e^2-2\,d\,e\,f+f^2\right )\,1{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,1{}\mathrm {i}}{8\,a\,d}-\frac {f\,x\,\left (f-2\,d\,e\right )\,1{}\mathrm {i}}{8\,a\,d^2}\right )-{\mathrm {e}}^{-c-d\,x}\,\left (\frac {d^2\,e^2+2\,d\,e\,f+2\,f^2}{2\,a\,d^3}+\frac {f^2\,x^2}{2\,a\,d}+\frac {f\,x\,\left (f+d\,e\right )}{a\,d^2}\right ) \]

[In]

int((cosh(c + d*x)^3*(e + f*x)^2)/(a + a*sinh(c + d*x)*1i),x)

[Out]

exp(c + d*x)*((2*f^2 + d^2*e^2 - 2*d*e*f)/(2*a*d^3) + (f^2*x^2)/(2*a*d) - (f*x*(f - d*e))/(a*d^2)) - exp(- 2*c
- 2*d*x)*(((f^2 + 2*d^2*e^2 + 2*d*e*f)*1i)/(16*a*d^3) + (f^2*x^2*1i)/(8*a*d) + (f*x*(f + 2*d*e)*1i)/(8*a*d^2)
) - exp(2*c + 2*d*x)*(((f^2 + 2*d^2*e^2 - 2*d*e*f)*1i)/(16*a*d^3) + (f^2*x^2*1i)/(8*a*d) - (f*x*(f - 2*d*e)*1i
)/(8*a*d^2)) - exp(- c - d*x)*((2*f^2 + d^2*e^2 + 2*d*e*f)/(2*a*d^3) + (f^2*x^2)/(2*a*d) + (f*x*(f + d*e))/(a*
d^2))