3.25 \(\int (a+b (F^{g (e+f x)})^n) (c+d x)^3 \, dx\)

Optimal. Leaf size=153 \[ \frac{a (c+d x)^4}{4 d}+\frac{6 b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac{3 b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{6 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)} \]

[Out]

(a*(c + d*x)^4)/(4*d) - (6*b*d^3*(F^(e*g + f*g*x))^n)/(f^4*g^4*n^4*Log[F]^4) + (6*b*d^2*(F^(e*g + f*g*x))^n*(c
+ d*x))/(f^3*g^3*n^3*Log[F]^3) - (3*b*d*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f^2*g^2*n^2*Log[F]^2) + (b*(F^(e*g
+ f*g*x))^n*(c + d*x)^3)/(f*g*n*Log[F])

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Rubi [A]  time = 0.238107, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2183, 2176, 2194} \[ \frac{a (c+d x)^4}{4 d}+\frac{6 b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac{3 b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{6 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(F^(g*(e + f*x)))^n)*(c + d*x)^3,x]

[Out]

(a*(c + d*x)^4)/(4*d) - (6*b*d^3*(F^(e*g + f*g*x))^n)/(f^4*g^4*n^4*Log[F]^4) + (6*b*d^2*(F^(e*g + f*g*x))^n*(c
+ d*x))/(f^3*g^3*n^3*Log[F]^3) - (3*b*d*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f^2*g^2*n^2*Log[F]^2) + (b*(F^(e*g
+ f*g*x))^n*(c + d*x)^3)/(f*g*n*Log[F])

Rule 2183

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> In
t[ExpandIntegrand[(c + d*x)^m, (a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n},
x] && IGtQ[p, 0]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^3 \, dx &=\int \left (a (c+d x)^3+b \left (F^{e g+f g x}\right )^n (c+d x)^3\right ) \, dx\\ &=\frac{a (c+d x)^4}{4 d}+b \int \left (F^{e g+f g x}\right )^n (c+d x)^3 \, dx\\ &=\frac{a (c+d x)^4}{4 d}+\frac{b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}-\frac{(3 b d) \int \left (F^{e g+f g x}\right )^n (c+d x)^2 \, dx}{f g n \log (F)}\\ &=\frac{a (c+d x)^4}{4 d}-\frac{3 b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}+\frac{b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac{\left (6 b d^2\right ) \int \left (F^{e g+f g x}\right )^n (c+d x) \, dx}{f^2 g^2 n^2 \log ^2(F)}\\ &=\frac{a (c+d x)^4}{4 d}+\frac{6 b d^2 \left (F^{e g+f g x}\right )^n (c+d x)}{f^3 g^3 n^3 \log ^3(F)}-\frac{3 b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}+\frac{b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}-\frac{\left (6 b d^3\right ) \int \left (F^{e g+f g x}\right )^n \, dx}{f^3 g^3 n^3 \log ^3(F)}\\ &=\frac{a (c+d x)^4}{4 d}-\frac{6 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac{6 b d^2 \left (F^{e g+f g x}\right )^n (c+d x)}{f^3 g^3 n^3 \log ^3(F)}-\frac{3 b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}+\frac{b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.266721, size = 130, normalized size = 0.85 \[ \frac{3}{2} a c^2 d x^2+a c^3 x+a c d^2 x^3+\frac{1}{4} a d^3 x^4+\frac{b \left (F^{g (e+f x)}\right )^n \left (6 d^2 f g n \log (F) (c+d x)-3 d f^2 g^2 n^2 \log ^2(F) (c+d x)^2+f^3 g^3 n^3 \log ^3(F) (c+d x)^3-6 d^3\right )}{f^4 g^4 n^4 \log ^4(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(F^(g*(e + f*x)))^n)*(c + d*x)^3,x]

[Out]

a*c^3*x + (3*a*c^2*d*x^2)/2 + a*c*d^2*x^3 + (a*d^3*x^4)/4 + (b*(F^(g*(e + f*x)))^n*(-6*d^3 + 6*d^2*f*g*n*(c +
d*x)*Log[F] - 3*d*f^2*g^2*n^2*(c + d*x)^2*Log[F]^2 + f^3*g^3*n^3*(c + d*x)^3*Log[F]^3))/(f^4*g^4*n^4*Log[F]^4)

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) \left ( dx+c \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^3,x)

[Out]

int((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^3,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.80348, size = 559, normalized size = 3.65 \begin{align*} \frac{{\left (a d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, a c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, a c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, a c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} - 4 \,{\left (6 \, b d^{3} -{\left (b d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, b c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, b c^{2} d f^{3} g^{3} n^{3} x + b c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 3 \,{\left (b d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, b c d^{2} f^{2} g^{2} n^{2} x + b c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \,{\left (b d^{3} f g n x + b c d^{2} f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{4 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^3,x, algorithm="fricas")

[Out]

1/4*((a*d^3*f^4*g^4*n^4*x^4 + 4*a*c*d^2*f^4*g^4*n^4*x^3 + 6*a*c^2*d*f^4*g^4*n^4*x^2 + 4*a*c^3*f^4*g^4*n^4*x)*l
og(F)^4 - 4*(6*b*d^3 - (b*d^3*f^3*g^3*n^3*x^3 + 3*b*c*d^2*f^3*g^3*n^3*x^2 + 3*b*c^2*d*f^3*g^3*n^3*x + b*c^3*f^
3*g^3*n^3)*log(F)^3 + 3*(b*d^3*f^2*g^2*n^2*x^2 + 2*b*c*d^2*f^2*g^2*n^2*x + b*c^2*d*f^2*g^2*n^2)*log(F)^2 - 6*(
b*d^3*f*g*n*x + b*c*d^2*f*g*n)*log(F))*F^(f*g*n*x + e*g*n))/(f^4*g^4*n^4*log(F)^4)

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Sympy [A]  time = 0.312302, size = 332, normalized size = 2.17 \begin{align*} a c^{3} x + \frac{3 a c^{2} d x^{2}}{2} + a c d^{2} x^{3} + \frac{a d^{3} x^{4}}{4} + \begin{cases} \frac{\left (b c^{3} f^{3} g^{3} n^{3} \log{\left (F \right )}^{3} + 3 b c^{2} d f^{3} g^{3} n^{3} x \log{\left (F \right )}^{3} - 3 b c^{2} d f^{2} g^{2} n^{2} \log{\left (F \right )}^{2} + 3 b c d^{2} f^{3} g^{3} n^{3} x^{2} \log{\left (F \right )}^{3} - 6 b c d^{2} f^{2} g^{2} n^{2} x \log{\left (F \right )}^{2} + 6 b c d^{2} f g n \log{\left (F \right )} + b d^{3} f^{3} g^{3} n^{3} x^{3} \log{\left (F \right )}^{3} - 3 b d^{3} f^{2} g^{2} n^{2} x^{2} \log{\left (F \right )}^{2} + 6 b d^{3} f g n x \log{\left (F \right )} - 6 b d^{3}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{4} g^{4} n^{4} \log{\left (F \right )}^{4}} & \text{for}\: f^{4} g^{4} n^{4} \log{\left (F \right )}^{4} \neq 0 \\b c^{3} x + \frac{3 b c^{2} d x^{2}}{2} + b c d^{2} x^{3} + \frac{b d^{3} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F**(g*(f*x+e)))**n)*(d*x+c)**3,x)

[Out]

a*c**3*x + 3*a*c**2*d*x**2/2 + a*c*d**2*x**3 + a*d**3*x**4/4 + Piecewise(((b*c**3*f**3*g**3*n**3*log(F)**3 + 3
*b*c**2*d*f**3*g**3*n**3*x*log(F)**3 - 3*b*c**2*d*f**2*g**2*n**2*log(F)**2 + 3*b*c*d**2*f**3*g**3*n**3*x**2*lo
g(F)**3 - 6*b*c*d**2*f**2*g**2*n**2*x*log(F)**2 + 6*b*c*d**2*f*g*n*log(F) + b*d**3*f**3*g**3*n**3*x**3*log(F)*
*3 - 3*b*d**3*f**2*g**2*n**2*x**2*log(F)**2 + 6*b*d**3*f*g*n*x*log(F) - 6*b*d**3)*(F**(g*(e + f*x)))**n/(f**4*
g**4*n**4*log(F)**4), Ne(f**4*g**4*n**4*log(F)**4, 0)), (b*c**3*x + 3*b*c**2*d*x**2/2 + b*c*d**2*x**3 + b*d**3
*x**4/4, True))

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Giac [C]  time = 1.49821, size = 7738, normalized size = 50.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^3,x, algorithm="giac")

[Out]

1/4*a*d^3*x^4 + a*c*d^2*x^3 + 3/2*a*c^2*d*x^2 + a*c^3*x - (((3*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))*sgn(F) -
3*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F)) + 2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^3 + 9*pi^2*b*c*d^2*f^3*g^3*n^3
*x^2*log(abs(F))*sgn(F) - 9*pi^2*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F)) + 6*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^3
+ 9*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))*sgn(F) - 9*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F)) + 6*b*c^2*d*f^3*
g^3*n^3*x*log(abs(F))^3 + 3*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F)) +
2*b*c^3*f^3*g^3*n^3*log(abs(F))^3 - 3*pi^2*b*d^3*f^2*g^2*n^2*x^2*sgn(F) + 3*pi^2*b*d^3*f^2*g^2*n^2*x^2 - 6*b*
d^3*f^2*g^2*n^2*x^2*log(abs(F))^2 - 6*pi^2*b*c*d^2*f^2*g^2*n^2*x*sgn(F) + 6*pi^2*b*c*d^2*f^2*g^2*n^2*x - 12*b*
c*d^2*f^2*g^2*n^2*x*log(abs(F))^2 - 3*pi^2*b*c^2*d*f^2*g^2*n^2*sgn(F) + 3*pi^2*b*c^2*d*f^2*g^2*n^2 - 6*b*c^2*d
*f^2*g^2*n^2*log(abs(F))^2 + 12*b*d^3*f*g*n*x*log(abs(F)) + 12*b*c*d^2*f*g*n*log(abs(F)) - 12*b*d^3)*(pi^4*f^4
*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - pi^4*f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^
2 - 2*f^4*g^4*n^4*log(abs(F))^4)/((pi^4*f^4*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - pi^4*f^
4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^2 - 2*f^4*g^4*n^4*log(abs(F))^4)^2 + 16*(pi^3*f^4*g^4*n^4*log(abs(F
))*sgn(F) - pi*f^4*g^4*n^4*log(abs(F))^3*sgn(F) - pi^3*f^4*g^4*n^4*log(abs(F)) + pi*f^4*g^4*n^4*log(abs(F))^3)
^2) - 4*(pi^3*b*d^3*f^3*g^3*n^3*x^3*sgn(F) - 3*pi*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2*sgn(F) - pi^3*b*d^3*f^3*
g^3*n^3*x^3 + 3*pi*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2 + 3*pi^3*b*c*d^2*f^3*g^3*n^3*x^2*sgn(F) - 9*pi*b*c*d^2*
f^3*g^3*n^3*x^2*log(abs(F))^2*sgn(F) - 3*pi^3*b*c*d^2*f^3*g^3*n^3*x^2 + 9*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F
))^2 + 3*pi^3*b*c^2*d*f^3*g^3*n^3*x*sgn(F) - 9*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2*sgn(F) - 3*pi^3*b*c^2*d*
f^3*g^3*n^3*x + 9*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2 + pi^3*b*c^3*f^3*g^3*n^3*sgn(F) - 3*pi*b*c^3*f^3*g^3*
n^3*log(abs(F))^2*sgn(F) - pi^3*b*c^3*f^3*g^3*n^3 + 3*pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2 + 6*pi*b*d^3*f^2*g^2*
n^2*x^2*log(abs(F))*sgn(F) - 6*pi*b*d^3*f^2*g^2*n^2*x^2*log(abs(F)) + 12*pi*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))*
sgn(F) - 12*pi*b*c*d^2*f^2*g^2*n^2*x*log(abs(F)) + 6*pi*b*c^2*d*f^2*g^2*n^2*log(abs(F))*sgn(F) - 6*pi*b*c^2*d*
f^2*g^2*n^2*log(abs(F)) - 6*pi*b*d^3*f*g*n*x*sgn(F) + 6*pi*b*d^3*f*g*n*x - 6*pi*b*c*d^2*f*g*n*sgn(F) + 6*pi*b*
c*d^2*f*g*n)*(pi^3*f^4*g^4*n^4*log(abs(F))*sgn(F) - pi*f^4*g^4*n^4*log(abs(F))^3*sgn(F) - pi^3*f^4*g^4*n^4*log
(abs(F)) + pi*f^4*g^4*n^4*log(abs(F))^3)/((pi^4*f^4*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) -
pi^4*f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^2 - 2*f^4*g^4*n^4*log(abs(F))^4)^2 + 16*(pi^3*f^4*g^4*n^4*l
og(abs(F))*sgn(F) - pi*f^4*g^4*n^4*log(abs(F))^3*sgn(F) - pi^3*f^4*g^4*n^4*log(abs(F)) + pi*f^4*g^4*n^4*log(ab
s(F))^3)^2))*cos(-1/2*pi*f*g*n*x*sgn(F) + 1/2*pi*f*g*n*x - 1/2*pi*g*n*e*sgn(F) + 1/2*pi*g*n*e) - ((pi^3*b*d^3*
f^3*g^3*n^3*x^3*sgn(F) - 3*pi*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2*sgn(F) - pi^3*b*d^3*f^3*g^3*n^3*x^3 + 3*pi*b
*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2 + 3*pi^3*b*c*d^2*f^3*g^3*n^3*x^2*sgn(F) - 9*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(
abs(F))^2*sgn(F) - 3*pi^3*b*c*d^2*f^3*g^3*n^3*x^2 + 9*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2 + 3*pi^3*b*c^2*
d*f^3*g^3*n^3*x*sgn(F) - 9*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2*sgn(F) - 3*pi^3*b*c^2*d*f^3*g^3*n^3*x + 9*pi
*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2 + pi^3*b*c^3*f^3*g^3*n^3*sgn(F) - 3*pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2*sg
n(F) - pi^3*b*c^3*f^3*g^3*n^3 + 3*pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2 + 6*pi*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))*
sgn(F) - 6*pi*b*d^3*f^2*g^2*n^2*x^2*log(abs(F)) + 12*pi*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 12*pi*b*c*d
^2*f^2*g^2*n^2*x*log(abs(F)) + 6*pi*b*c^2*d*f^2*g^2*n^2*log(abs(F))*sgn(F) - 6*pi*b*c^2*d*f^2*g^2*n^2*log(abs(
F)) - 6*pi*b*d^3*f*g*n*x*sgn(F) + 6*pi*b*d^3*f*g*n*x - 6*pi*b*c*d^2*f*g*n*sgn(F) + 6*pi*b*c*d^2*f*g*n)*(pi^4*f
^4*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - pi^4*f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F)
)^2 - 2*f^4*g^4*n^4*log(abs(F))^4)/((pi^4*f^4*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - pi^4*
f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^2 - 2*f^4*g^4*n^4*log(abs(F))^4)^2 + 16*(pi^3*f^4*g^4*n^4*log(abs
(F))*sgn(F) - pi*f^4*g^4*n^4*log(abs(F))^3*sgn(F) - pi^3*f^4*g^4*n^4*log(abs(F)) + pi*f^4*g^4*n^4*log(abs(F))^
3)^2) + 4*(3*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))*sgn(F) - 3*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F)) + 2*b*d^
3*f^3*g^3*n^3*x^3*log(abs(F))^3 + 9*pi^2*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))*sgn(F) - 9*pi^2*b*c*d^2*f^3*g^3*n
^3*x^2*log(abs(F)) + 6*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^3 + 9*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))*sgn(F)
- 9*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F)) + 6*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^3 + 3*pi^2*b*c^3*f^3*g^3*n^3
*log(abs(F))*sgn(F) - 3*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F)) + 2*b*c^3*f^3*g^3*n^3*log(abs(F))^3 - 3*pi^2*b*d^3*
f^2*g^2*n^2*x^2*sgn(F) + 3*pi^2*b*d^3*f^2*g^2*n^2*x^2 - 6*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))^2 - 6*pi^2*b*c*d^2
*f^2*g^2*n^2*x*sgn(F) + 6*pi^2*b*c*d^2*f^2*g^2*n^2*x - 12*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))^2 - 3*pi^2*b*c^2*d
*f^2*g^2*n^2*sgn(F) + 3*pi^2*b*c^2*d*f^2*g^2*n^2 - 6*b*c^2*d*f^2*g^2*n^2*log(abs(F))^2 + 12*b*d^3*f*g*n*x*log(
abs(F)) + 12*b*c*d^2*f*g*n*log(abs(F)) - 12*b*d^3)*(pi^3*f^4*g^4*n^4*log(abs(F))*sgn(F) - pi*f^4*g^4*n^4*log(a
bs(F))^3*sgn(F) - pi^3*f^4*g^4*n^4*log(abs(F)) + pi*f^4*g^4*n^4*log(abs(F))^3)/((pi^4*f^4*g^4*n^4*sgn(F) - 6*p
i^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - pi^4*f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^2 - 2*f^4*g^4*n^4*log
(abs(F))^4)^2 + 16*(pi^3*f^4*g^4*n^4*log(abs(F))*sgn(F) - pi*f^4*g^4*n^4*log(abs(F))^3*sgn(F) - pi^3*f^4*g^4*n
^4*log(abs(F)) + pi*f^4*g^4*n^4*log(abs(F))^3)^2))*sin(-1/2*pi*f*g*n*x*sgn(F) + 1/2*pi*f*g*n*x - 1/2*pi*g*n*e*
sgn(F) + 1/2*pi*g*n*e))*e^(f*g*n*x*log(abs(F)) + g*n*e*log(abs(F))) - 1/2*I*((8*pi^3*b*d^3*f^3*g^3*n^3*x^3*sgn
(F) + 24*I*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))*sgn(F) - 24*pi*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2*sgn(F) -
8*pi^3*b*d^3*f^3*g^3*n^3*x^3 - 24*I*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F)) + 24*pi*b*d^3*f^3*g^3*n^3*x^3*log(a
bs(F))^2 + 16*I*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^3 + 24*pi^3*b*c*d^2*f^3*g^3*n^3*x^2*sgn(F) + 72*I*pi^2*b*c*d
^2*f^3*g^3*n^3*x^2*log(abs(F))*sgn(F) - 72*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2*sgn(F) - 24*pi^3*b*c*d^2*f
^3*g^3*n^3*x^2 - 72*I*pi^2*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F)) + 72*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2 +
48*I*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^3 + 24*pi^3*b*c^2*d*f^3*g^3*n^3*x*sgn(F) + 72*I*pi^2*b*c^2*d*f^3*g^3
*n^3*x*log(abs(F))*sgn(F) - 72*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2*sgn(F) - 24*pi^3*b*c^2*d*f^3*g^3*n^3*x -
72*I*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F)) + 72*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2 + 48*I*b*c^2*d*f^3*g^
3*n^3*x*log(abs(F))^3 + 8*pi^3*b*c^3*f^3*g^3*n^3*sgn(F) + 24*I*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F))*sgn(F) - 24*
pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - 8*pi^3*b*c^3*f^3*g^3*n^3 - 24*I*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F))
+ 24*pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2 + 16*I*b*c^3*f^3*g^3*n^3*log(abs(F))^3 - 24*I*pi^2*b*d^3*f^2*g^2*n^2*
x^2*sgn(F) + 48*pi*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) + 24*I*pi^2*b*d^3*f^2*g^2*n^2*x^2 - 48*pi*b*d^3*f^
2*g^2*n^2*x^2*log(abs(F)) - 48*I*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))^2 - 48*I*pi^2*b*c*d^2*f^2*g^2*n^2*x*sgn(F)
+ 96*pi*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))*sgn(F) + 48*I*pi^2*b*c*d^2*f^2*g^2*n^2*x - 96*pi*b*c*d^2*f^2*g^2*n^2
*x*log(abs(F)) - 96*I*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))^2 - 24*I*pi^2*b*c^2*d*f^2*g^2*n^2*sgn(F) + 48*pi*b*c^2
*d*f^2*g^2*n^2*log(abs(F))*sgn(F) + 24*I*pi^2*b*c^2*d*f^2*g^2*n^2 - 48*pi*b*c^2*d*f^2*g^2*n^2*log(abs(F)) - 48
*I*b*c^2*d*f^2*g^2*n^2*log(abs(F))^2 - 48*pi*b*d^3*f*g*n*x*sgn(F) + 48*pi*b*d^3*f*g*n*x + 96*I*b*d^3*f*g*n*x*l
og(abs(F)) - 48*pi*b*c*d^2*f*g*n*sgn(F) + 48*pi*b*c*d^2*f*g*n + 96*I*b*c*d^2*f*g*n*log(abs(F)) - 96*I*b*d^3)*e
^(1/2*I*pi*f*g*n*x*sgn(F) - 1/2*I*pi*f*g*n*x + 1/2*I*pi*g*n*e*sgn(F) - 1/2*I*pi*g*n*e)/(8*pi^4*f^4*g^4*n^4*sgn
(F) + 32*I*pi^3*f^4*g^4*n^4*log(abs(F))*sgn(F) - 48*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - 32*I*pi*f^4*g^4*n^
4*log(abs(F))^3*sgn(F) - 8*pi^4*f^4*g^4*n^4 - 32*I*pi^3*f^4*g^4*n^4*log(abs(F)) + 48*pi^2*f^4*g^4*n^4*log(abs(
F))^2 + 32*I*pi*f^4*g^4*n^4*log(abs(F))^3 - 16*f^4*g^4*n^4*log(abs(F))^4) + (8*pi^3*b*d^3*f^3*g^3*n^3*x^3*sgn(
F) - 24*I*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))*sgn(F) - 24*pi*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2*sgn(F) - 8
*pi^3*b*d^3*f^3*g^3*n^3*x^3 + 24*I*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F)) + 24*pi*b*d^3*f^3*g^3*n^3*x^3*log(ab
s(F))^2 - 16*I*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^3 + 24*pi^3*b*c*d^2*f^3*g^3*n^3*x^2*sgn(F) - 72*I*pi^2*b*c*d^
2*f^3*g^3*n^3*x^2*log(abs(F))*sgn(F) - 72*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2*sgn(F) - 24*pi^3*b*c*d^2*f^
3*g^3*n^3*x^2 + 72*I*pi^2*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F)) + 72*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2 -
48*I*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^3 + 24*pi^3*b*c^2*d*f^3*g^3*n^3*x*sgn(F) - 72*I*pi^2*b*c^2*d*f^3*g^3*
n^3*x*log(abs(F))*sgn(F) - 72*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2*sgn(F) - 24*pi^3*b*c^2*d*f^3*g^3*n^3*x +
72*I*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F)) + 72*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2 - 48*I*b*c^2*d*f^3*g^3
*n^3*x*log(abs(F))^3 + 8*pi^3*b*c^3*f^3*g^3*n^3*sgn(F) - 24*I*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F))*sgn(F) - 24*p
i*b*c^3*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - 8*pi^3*b*c^3*f^3*g^3*n^3 + 24*I*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F))
+ 24*pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2 - 16*I*b*c^3*f^3*g^3*n^3*log(abs(F))^3 + 24*I*pi^2*b*d^3*f^2*g^2*n^2*x
^2*sgn(F) + 48*pi*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 24*I*pi^2*b*d^3*f^2*g^2*n^2*x^2 - 48*pi*b*d^3*f^2
*g^2*n^2*x^2*log(abs(F)) + 48*I*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))^2 + 48*I*pi^2*b*c*d^2*f^2*g^2*n^2*x*sgn(F) +
96*pi*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 48*I*pi^2*b*c*d^2*f^2*g^2*n^2*x - 96*pi*b*c*d^2*f^2*g^2*n^2*
x*log(abs(F)) + 96*I*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))^2 + 24*I*pi^2*b*c^2*d*f^2*g^2*n^2*sgn(F) + 48*pi*b*c^2*
d*f^2*g^2*n^2*log(abs(F))*sgn(F) - 24*I*pi^2*b*c^2*d*f^2*g^2*n^2 - 48*pi*b*c^2*d*f^2*g^2*n^2*log(abs(F)) + 48*
I*b*c^2*d*f^2*g^2*n^2*log(abs(F))^2 - 48*pi*b*d^3*f*g*n*x*sgn(F) + 48*pi*b*d^3*f*g*n*x - 96*I*b*d^3*f*g*n*x*lo
g(abs(F)) - 48*pi*b*c*d^2*f*g*n*sgn(F) + 48*pi*b*c*d^2*f*g*n - 96*I*b*c*d^2*f*g*n*log(abs(F)) + 96*I*b*d^3)*e^
(-1/2*I*pi*f*g*n*x*sgn(F) + 1/2*I*pi*f*g*n*x - 1/2*I*pi*g*n*e*sgn(F) + 1/2*I*pi*g*n*e)/(8*pi^4*f^4*g^4*n^4*sgn
(F) - 32*I*pi^3*f^4*g^4*n^4*log(abs(F))*sgn(F) - 48*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) + 32*I*pi*f^4*g^4*n^
4*log(abs(F))^3*sgn(F) - 8*pi^4*f^4*g^4*n^4 + 32*I*pi^3*f^4*g^4*n^4*log(abs(F)) + 48*pi^2*f^4*g^4*n^4*log(abs(
F))^2 - 32*I*pi*f^4*g^4*n^4*log(abs(F))^3 - 16*f^4*g^4*n^4*log(abs(F))^4))*e^(f*g*n*x*log(abs(F)) + g*n*e*log(
abs(F)))