3.67 \(\int F^{a+b (c+d x)} x (e+f x)^2 \, dx\)

Optimal. Leaf size=242 \[ -\frac{e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac{4 e f x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{3 f^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{6 f^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac{e^2 x F^{a+b c+b d x}}{b d \log (F)}+\frac{2 e f x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac{f^2 x^3 F^{a+b c+b d x}}{b d \log (F)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.355587, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2196, 2176, 2194} \[ -\frac{e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac{4 e f x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{3 f^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{6 f^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac{e^2 x F^{a+b c+b d x}}{b d \log (F)}+\frac{2 e f x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac{f^2 x^3 F^{a+b c+b d x}}{b d \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x))*x*(e + f*x)^2,x]

[Out]

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

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

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

Mathematica [A]  time = 0.149704, size = 91, normalized size = 0.38 \[ \frac{F^{a+b (c+d x)} \left (-b^2 d^2 \log ^2(F) \left (e^2+4 e f x+3 f^2 x^2\right )+b^3 d^3 x \log ^3(F) (e+f x)^2+2 b d f \log (F) (2 e+3 f x)-6 f^2\right )}{b^4 d^4 \log ^4(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x))*x*(e + f*x)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 144, normalized size = 0.6 \begin{align*}{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}{f}^{2}{x}^{3}+2\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}ef{x}^{2}+ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}{e}^{2}x-3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{f}^{2}{x}^{2}-4\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}efx- \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{e}^{2}+6\,\ln \left ( F \right ) bd{f}^{2}x+4\,fe\ln \left ( F \right ) bd-6\,{f}^{2} \right ){F}^{bdx+bc+a}}{ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{d}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c))*x*(f*x+e)^2,x)

[Out]

(ln(F)^3*b^3*d^3*f^2*x^3+2*ln(F)^3*b^3*d^3*e*f*x^2+ln(F)^3*b^3*d^3*e^2*x-3*ln(F)^2*b^2*d^2*f^2*x^2-4*ln(F)^2*b
^2*d^2*e*f*x-ln(F)^2*b^2*d^2*e^2+6*ln(F)*b*d*f^2*x+4*f*e*ln(F)*b*d-6*f^2)*F^(b*d*x+b*c+a)/ln(F)^4/b^4/d^4

________________________________________________________________________________________

Maxima [A]  time = 1.0494, size = 265, normalized size = 1.1 \begin{align*} \frac{{\left (F^{b c + a} b d x \log \left (F\right ) - F^{b c + a}\right )} F^{b d x} e^{2}}{b^{2} d^{2} \log \left (F\right )^{2}} + \frac{2 \,{\left (F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{b c + a} b d x \log \left (F\right ) + 2 \, F^{b c + a}\right )} F^{b d x} e f}{b^{3} d^{3} \log \left (F\right )^{3}} + \frac{{\left (F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{b c + a} b d x \log \left (F\right ) - 6 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{4} d^{4} \log \left (F\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c))*x*(f*x+e)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.57332, size = 288, normalized size = 1.19 \begin{align*} \frac{{\left ({\left (b^{3} d^{3} f^{2} x^{3} + 2 \, b^{3} d^{3} e f x^{2} + b^{3} d^{3} e^{2} x\right )} \log \left (F\right )^{3} -{\left (3 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} - 6 \, f^{2} + 2 \,{\left (3 \, b d f^{2} x + 2 \, b d e f\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{4} d^{4} \log \left (F\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c))*x*(f*x+e)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.175929, size = 199, normalized size = 0.82 \begin{align*} \begin{cases} \frac{F^{a + b \left (c + d x\right )} \left (b^{3} d^{3} e^{2} x \log{\left (F \right )}^{3} + 2 b^{3} d^{3} e f x^{2} \log{\left (F \right )}^{3} + b^{3} d^{3} f^{2} x^{3} \log{\left (F \right )}^{3} - b^{2} d^{2} e^{2} \log{\left (F \right )}^{2} - 4 b^{2} d^{2} e f x \log{\left (F \right )}^{2} - 3 b^{2} d^{2} f^{2} x^{2} \log{\left (F \right )}^{2} + 4 b d e f \log{\left (F \right )} + 6 b d f^{2} x \log{\left (F \right )} - 6 f^{2}\right )}{b^{4} d^{4} \log{\left (F \right )}^{4}} & \text{for}\: b^{4} d^{4} \log{\left (F \right )}^{4} \neq 0 \\\frac{e^{2} x^{2}}{2} + \frac{2 e f x^{3}}{3} + \frac{f^{2} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c))*x*(f*x+e)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [C]  time = 1.48567, size = 6340, normalized size = 26.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c))*x*(f*x+e)^2,x, algorithm="giac")

[Out]

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