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

Optimal. Leaf size=171 \[ -\frac {i f^2 \sinh ^2(c+d x)}{4 a d^3}+\frac {2 f^2 \sinh (c+d x)}{a d^3}-\frac {2 f (e+f x) \cosh (c+d x)}{a d^2}+\frac {i f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \sinh ^2(c+d x)}{2 a d}+\frac {(e+f x)^2 \sinh (c+d x)}{a d}-\frac {i e f x}{2 a d}-\frac {i f^2 x^2}{4 a d} \]

[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

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Rubi [A]  time = 0.19, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {5563, 3296, 2637, 5446, 3310} \[ -\frac {2 f (e+f x) \cosh (c+d x)}{a d^2}+\frac {i f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a d^2}-\frac {i f^2 \sinh ^2(c+d x)}{4 a d^3}+\frac {2 f^2 \sinh (c+d x)}{a d^3}-\frac {i (e+f x)^2 \sinh ^2(c+d x)}{2 a d}+\frac {(e+f x)^2 \sinh (c+d x)}{a d}-\frac {i e f x}{2 a d}-\frac {i f^2 x^2}{4 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/2)*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 2637

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

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 3310

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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.87, size = 99, normalized size = 0.58 \[ \frac {-2 i \cosh (2 (c+d x)) \left (2 d^2 (e+f x)^2+f^2\right )+8 \sinh (c+d x) \left (2 \left (d^2 (e+f x)^2+2 f^2\right )+i d f (e+f x) \cosh (c+d x)\right )-32 d f (e+f x) \cosh (c+d x)}{16 a d^3} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.49, size = 225, normalized size = 1.32 \[ \frac {{\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e^{2} - 2 i \, d e f - i \, f^{2} + {\left (-4 i \, d^{2} e f - 2 i \, d f^{2}\right )} x + {\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e^{2} + 2 i \, d e f - i \, f^{2} + {\left (-4 i \, d^{2} e f + 2 i \, d f^{2}\right )} x\right )} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, {\left (d^{2} f^{2} x^{2} + d^{2} e^{2} - 2 \, d e f + 2 \, f^{2} + 2 \, {\left (d^{2} e f - d f^{2}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - 8 \, {\left (d^{2} f^{2} x^{2} + d^{2} e^{2} + 2 \, d e f + 2 \, f^{2} + 2 \, {\left (d^{2} e f + d f^{2}\right )} x\right )} e^{\left (d x + c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, a d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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 + (-4*I*d^2*e*f - 2*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 + (-4*I*d^2*e*f + 2*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)

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giac [B]  time = 0.26, size = 566, normalized size = 3.31 \[ \frac {-2 i \, d^{2} f^{2} x^{2} e^{\left (5 \, d x + 6 \, c\right )} + 6 \, d^{2} f^{2} x^{2} e^{\left (4 \, d x + 5 \, c\right )} - 8 i \, d^{2} f^{2} x^{2} e^{\left (3 \, d x + 4 \, c\right )} - 8 \, d^{2} f^{2} x^{2} e^{\left (2 \, d x + 3 \, c\right )} + 6 i \, d^{2} f^{2} x^{2} e^{\left (d x + 2 \, c\right )} - 2 \, d^{2} f^{2} x^{2} e^{c} - 4 i \, d^{2} f x e^{\left (5 \, d x + 6 \, c + 1\right )} + 2 i \, d f^{2} x e^{\left (5 \, d x + 6 \, c\right )} + 12 \, d^{2} f x e^{\left (4 \, d x + 5 \, c + 1\right )} - 14 \, d f^{2} x e^{\left (4 \, d x + 5 \, c\right )} - 16 i \, d^{2} f x e^{\left (3 \, d x + 4 \, c + 1\right )} + 16 i \, d f^{2} x e^{\left (3 \, d x + 4 \, c\right )} - 16 \, d^{2} f x e^{\left (2 \, d x + 3 \, c + 1\right )} - 16 \, d f^{2} x e^{\left (2 \, d x + 3 \, c\right )} + 12 i \, d^{2} f x e^{\left (d x + 2 \, c + 1\right )} + 14 i \, d f^{2} x e^{\left (d x + 2 \, c\right )} - 4 \, d^{2} f x e^{\left (c + 1\right )} - 2 \, d f^{2} x e^{c} - 2 i \, d^{2} e^{\left (5 \, d x + 6 \, c + 2\right )} + 2 i \, d f e^{\left (5 \, d x + 6 \, c + 1\right )} - i \, f^{2} e^{\left (5 \, d x + 6 \, c\right )} + 6 \, d^{2} e^{\left (4 \, d x + 5 \, c + 2\right )} - 14 \, d f e^{\left (4 \, d x + 5 \, c + 1\right )} + 15 \, f^{2} e^{\left (4 \, d x + 5 \, c\right )} - 8 i \, d^{2} e^{\left (3 \, d x + 4 \, c + 2\right )} + 16 i \, d f e^{\left (3 \, d x + 4 \, c + 1\right )} - 16 i \, f^{2} e^{\left (3 \, d x + 4 \, c\right )} - 8 \, d^{2} e^{\left (2 \, d x + 3 \, c + 2\right )} - 16 \, d f e^{\left (2 \, d x + 3 \, c + 1\right )} - 16 \, f^{2} e^{\left (2 \, d x + 3 \, c\right )} + 6 i \, d^{2} e^{\left (d x + 2 \, c + 2\right )} + 14 i \, d f e^{\left (d x + 2 \, c + 1\right )} + 15 i \, f^{2} e^{\left (d x + 2 \, c\right )} - 2 \, d^{2} e^{\left (c + 2\right )} - 2 \, d f e^{\left (c + 1\right )} - f^{2} e^{c}}{16 \, {\left (a d^{3} e^{\left (3 \, d x + 4 \, c\right )} - i \, a d^{3} e^{\left (2 \, d x + 3 \, c\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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^(5*d*x + 6*c) + 6*d^2*f^2*x^2*e^(4*d*x + 5*c) - 8*I*d^2*f^2*x^2*e^(3*d*x + 4*c) - 8*d
^2*f^2*x^2*e^(2*d*x + 3*c) + 6*I*d^2*f^2*x^2*e^(d*x + 2*c) - 2*d^2*f^2*x^2*e^c - 4*I*d^2*f*x*e^(5*d*x + 6*c +
1) + 2*I*d*f^2*x*e^(5*d*x + 6*c) + 12*d^2*f*x*e^(4*d*x + 5*c + 1) - 14*d*f^2*x*e^(4*d*x + 5*c) - 16*I*d^2*f*x*
e^(3*d*x + 4*c + 1) + 16*I*d*f^2*x*e^(3*d*x + 4*c) - 16*d^2*f*x*e^(2*d*x + 3*c + 1) - 16*d*f^2*x*e^(2*d*x + 3*
c) + 12*I*d^2*f*x*e^(d*x + 2*c + 1) + 14*I*d*f^2*x*e^(d*x + 2*c) - 4*d^2*f*x*e^(c + 1) - 2*d*f^2*x*e^c - 2*I*d
^2*e^(5*d*x + 6*c + 2) + 2*I*d*f*e^(5*d*x + 6*c + 1) - I*f^2*e^(5*d*x + 6*c) + 6*d^2*e^(4*d*x + 5*c + 2) - 14*
d*f*e^(4*d*x + 5*c + 1) + 15*f^2*e^(4*d*x + 5*c) - 8*I*d^2*e^(3*d*x + 4*c + 2) + 16*I*d*f*e^(3*d*x + 4*c + 1)
- 16*I*f^2*e^(3*d*x + 4*c) - 8*d^2*e^(2*d*x + 3*c + 2) - 16*d*f*e^(2*d*x + 3*c + 1) - 16*f^2*e^(2*d*x + 3*c) +
6*I*d^2*e^(d*x + 2*c + 2) + 14*I*d*f*e^(d*x + 2*c + 1) + 15*I*f^2*e^(d*x + 2*c) - 2*d^2*e^(c + 2) - 2*d*f*e^(
c + 1) - f^2*e^c)/(a*d^3*e^(3*d*x + 4*c) - I*a*d^3*e^(2*d*x + 3*c))

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maple [A]  time = 0.24, size = 241, normalized size = 1.41 \[ -\frac {i \left (2 x^{2} f^{2} d^{2}+4 d^{2} e f x +2 d^{2} e^{2}-2 d \,f^{2} x -2 d e f +f^{2}\right ) {\mathrm e}^{2 d x +2 c}}{16 a \,d^{3}}+\frac {\left (x^{2} f^{2} d^{2}+2 d^{2} e f x +d^{2} e^{2}-2 d \,f^{2} x -2 d e f +2 f^{2}\right ) {\mathrm e}^{d x +c}}{2 a \,d^{3}}-\frac {\left (x^{2} f^{2} d^{2}+2 d^{2} e f x +d^{2} e^{2}+2 d \,f^{2} x +2 d e f +2 f^{2}\right ) {\mathrm e}^{-d x -c}}{2 a \,d^{3}}-\frac {i \left (2 x^{2} f^{2} d^{2}+4 d^{2} e f x +2 d^{2} e^{2}+2 d \,f^{2} x +2 d e f +f^{2}\right ) {\mathrm e}^{-2 d x -2 c}}{16 a \,d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)/a/d^3*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)/a/d^3*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)/a/d^3*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)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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.

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mupad [B]  time = 0.94, size = 271, normalized size = 1.58 \[ {\mathrm {e}}^{c+d\,x}\,\left (\frac {d^2\,e^2-2\,d\,e\,f+2\,f^2}{2\,a\,d^3}+\frac {f^2\,x^2}{2\,a\,d}-\frac {f\,x\,\left (f-d\,e\right )}{a\,d^2}\right )-{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (\frac {\left (2\,d^2\,e^2+2\,d\,e\,f+f^2\right )\,1{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,1{}\mathrm {i}}{8\,a\,d}+\frac {f\,x\,\left (f+2\,d\,e\right )\,1{}\mathrm {i}}{8\,a\,d^2}\right )-{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (\frac {\left (2\,d^2\,e^2-2\,d\,e\,f+f^2\right )\,1{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,1{}\mathrm {i}}{8\,a\,d}-\frac {f\,x\,\left (f-2\,d\,e\right )\,1{}\mathrm {i}}{8\,a\,d^2}\right )-{\mathrm {e}}^{-c-d\,x}\,\left (\frac {d^2\,e^2+2\,d\,e\,f+2\,f^2}{2\,a\,d^3}+\frac {f^2\,x^2}{2\,a\,d}+\frac {f\,x\,\left (f+d\,e\right )}{a\,d^2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.85, size = 632, normalized size = 3.70 \[ \begin {cases} \frac {\left (\left (- 512 a^{3} d^{11} e^{2} e^{2 c} - 1024 a^{3} d^{11} e f x e^{2 c} - 512 a^{3} d^{11} f^{2} x^{2} e^{2 c} - 1024 a^{3} d^{10} e f e^{2 c} - 1024 a^{3} d^{10} f^{2} x e^{2 c} - 1024 a^{3} d^{9} f^{2} e^{2 c}\right ) e^{- d x} + \left (512 a^{3} d^{11} e^{2} e^{4 c} + 1024 a^{3} d^{11} e f x e^{4 c} + 512 a^{3} d^{11} f^{2} x^{2} e^{4 c} - 1024 a^{3} d^{10} e f e^{4 c} - 1024 a^{3} d^{10} f^{2} x e^{4 c} + 1024 a^{3} d^{9} f^{2} e^{4 c}\right ) e^{d x} + \left (- 128 i a^{3} d^{11} e^{2} e^{c} - 256 i a^{3} d^{11} e f x e^{c} - 128 i a^{3} d^{11} f^{2} x^{2} e^{c} - 128 i a^{3} d^{10} e f e^{c} - 128 i a^{3} d^{10} f^{2} x e^{c} - 64 i a^{3} d^{9} f^{2} e^{c}\right ) e^{- 2 d x} + \left (- 128 i a^{3} d^{11} e^{2} e^{5 c} - 256 i a^{3} d^{11} e f x e^{5 c} - 128 i a^{3} d^{11} f^{2} x^{2} e^{5 c} + 128 i a^{3} d^{10} e f e^{5 c} + 128 i a^{3} d^{10} f^{2} x e^{5 c} - 64 i a^{3} d^{9} f^{2} e^{5 c}\right ) e^{2 d x}\right ) e^{- 3 c}}{1024 a^{4} d^{12}} & \text {for}\: 1024 a^{4} d^{12} e^{3 c} \neq 0 \\\frac {x^{3} \left (- i f^{2} e^{4 c} + 2 f^{2} e^{3 c} + 2 f^{2} e^{c} + i f^{2}\right ) e^{- 2 c}}{12 a} + \frac {x^{2} \left (- i e f e^{4 c} + 2 e f e^{3 c} + 2 e f e^{c} + i e f\right ) e^{- 2 c}}{4 a} + \frac {x \left (- i e^{2} e^{4 c} + 2 e^{2} e^{3 c} + 2 e^{2} e^{c} + i e^{2}\right ) e^{- 2 c}}{4 a} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[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(1024*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))

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