3.34 \(\int F^{c (a+b x)} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx\)

Optimal. Leaf size=17 \[ x \sin (d+e x) F^{c (a+b x)} \]

[Out]

F^(c*(a + b*x))*x*Sin[d + e*x]

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Rubi [B]  time = 0.766368, antiderivative size = 327, normalized size of antiderivative = 19.24, number of steps used = 14, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {6741, 6742, 4433, 4466, 4432, 4465} \[ \frac{b^2 c^2 x \log ^2(F) \sin (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}+\frac{e^2 x \sin (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}+\frac{b c \log (F) \sin (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}-\frac{b^3 c^3 \log ^3(F) \sin (d+e x) F^{a c+b c x}}{\left (b^2 c^2 \log ^2(F)+e^2\right )^2}-\frac{b c e^2 \log (F) \sin (d+e x) F^{a c+b c x}}{\left (b^2 c^2 \log ^2(F)+e^2\right )^2}-\frac{e \cos (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}+\frac{b^2 c^2 e \log ^2(F) \cos (d+e x) F^{a c+b c x}}{\left (b^2 c^2 \log ^2(F)+e^2\right )^2}+\frac{e^3 \cos (d+e x) F^{a c+b c x}}{\left (b^2 c^2 \log ^2(F)+e^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*(e*x*Cos[d + e*x] + (1 + b*c*x*Log[F])*Sin[d + e*x]),x]

[Out]

(e^3*F^(a*c + b*c*x)*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)^2 + (b^2*c^2*e*F^(a*c + b*c*x)*Cos[d + e*x]*Log[F]
^2)/(e^2 + b^2*c^2*Log[F]^2)^2 - (e*F^(a*c + b*c*x)*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) - (b*c*e^2*F^(a*c +
b*c*x)*Log[F]*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)^2 - (b^3*c^3*F^(a*c + b*c*x)*Log[F]^3*Sin[d + e*x])/(e^2
+ b^2*c^2*Log[F]^2)^2 + (e^2*F^(a*c + b*c*x)*x*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) + (b*c*F^(a*c + b*c*x)*
Log[F]*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) + (b^2*c^2*F^(a*c + b*c*x)*x*Log[F]^2*Sin[d + e*x])/(e^2 + b^2*c
^2*Log[F]^2)

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 4433

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*C
os[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 4466

Int[Cos[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_.)*(x_))^(m_.), x_Symbol] :> Module[{u
= IntHide[F^(c*(a + b*x))*Cos[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /
; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]

Rule 4432

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*S
in[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] - Simp[(e*F^(c*(a + b*x))*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 4465

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_.)*(x_))^(m_.)*Sin[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Module[{u
= IntHide[F^(c*(a + b*x))*Sin[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /
; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]

Rubi steps

\begin{align*} \int F^{c (a+b x)} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx &=\int F^{a c+b c x} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx\\ &=\int \left (e F^{a c+b c x} x \cos (d+e x)+F^{a c+b c x} (1+b c x \log (F)) \sin (d+e x)\right ) \, dx\\ &=e \int F^{a c+b c x} x \cos (d+e x) \, dx+\int F^{a c+b c x} (1+b c x \log (F)) \sin (d+e x) \, dx\\ &=\frac{b c e F^{a c+b c x} x \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac{e^2 F^{a c+b c x} x \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-e \int \left (\frac{b c F^{a c+b c x} \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac{e F^{a c+b c x} \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}\right ) \, dx+\int \left (F^{a c+b c x} \sin (d+e x)+b c F^{a c+b c x} x \log (F) \sin (d+e x)\right ) \, dx\\ &=\frac{b c e F^{a c+b c x} x \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac{e^2 F^{a c+b c x} x \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+(b c \log (F)) \int F^{a c+b c x} x \sin (d+e x) \, dx-\frac{e^2 \int F^{a c+b c x} \sin (d+e x) \, dx}{e^2+b^2 c^2 \log ^2(F)}-\frac{(b c e \log (F)) \int F^{a c+b c x} \cos (d+e x) \, dx}{e^2+b^2 c^2 \log ^2(F)}+\int F^{a c+b c x} \sin (d+e x) \, dx\\ &=\frac{e^3 F^{a c+b c x} \cos (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac{b^2 c^2 e F^{a c+b c x} \cos (d+e x) \log ^2(F)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac{e F^{a c+b c x} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-\frac{2 b c e^2 F^{a c+b c x} \log (F) \sin (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}+\frac{e^2 F^{a c+b c x} x \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac{b c F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac{b^2 c^2 F^{a c+b c x} x \log ^2(F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-(b c \log (F)) \int \left (-\frac{e F^{a c+b c x} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac{b c F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}\right ) \, dx\\ &=\frac{e^3 F^{a c+b c x} \cos (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac{b^2 c^2 e F^{a c+b c x} \cos (d+e x) \log ^2(F)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac{e F^{a c+b c x} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-\frac{2 b c e^2 F^{a c+b c x} \log (F) \sin (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}+\frac{e^2 F^{a c+b c x} x \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac{b c F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac{b^2 c^2 F^{a c+b c x} x \log ^2(F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac{(b c e \log (F)) \int F^{a c+b c x} \cos (d+e x) \, dx}{e^2+b^2 c^2 \log ^2(F)}-\frac{\left (b^2 c^2 \log ^2(F)\right ) \int F^{a c+b c x} \sin (d+e x) \, dx}{e^2+b^2 c^2 \log ^2(F)}\\ &=\frac{e^3 F^{a c+b c x} \cos (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}+\frac{b^2 c^2 e F^{a c+b c x} \cos (d+e x) \log ^2(F)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac{e F^{a c+b c x} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-\frac{b c e^2 F^{a c+b c x} \log (F) \sin (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac{b^3 c^3 F^{a c+b c x} \log ^3(F) \sin (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}+\frac{e^2 F^{a c+b c x} x \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac{b c F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac{b^2 c^2 F^{a c+b c x} x \log ^2(F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}\\ \end{align*}

Mathematica [A]  time = 0.386972, size = 17, normalized size = 1. \[ x \sin (d+e x) F^{c (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(c*(a + b*x))*(e*x*Cos[d + e*x] + (1 + b*c*x*Log[F])*Sin[d + e*x]),x]

[Out]

F^(c*(a + b*x))*x*Sin[d + e*x]

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Maple [B]  time = 0.079, size = 682, normalized size = 40.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(b*x+a))*(e*x*cos(e*x+d)+(1+b*c*x*ln(F))*sin(e*x+d)),x)

[Out]

(1/(e^2+b^2*c^2*ln(F)^2)*e*exp(c*(b*x+a)*ln(F))*tan(1/2*d+1/2*e*x)^2-1/(e^2+b^2*c^2*ln(F)^2)*e*exp(c*(b*x+a)*l
n(F))+2*ln(F)*b*c/(e^2+b^2*c^2*ln(F)^2)*exp(c*(b*x+a)*ln(F))*tan(1/2*d+1/2*e*x))/(1+tan(1/2*d+1/2*e*x)^2)+e*((
b^2*c^2*ln(F)^2-e^2)/(e^2+b^2*c^2*ln(F)^2)^2*exp(c*(b*x+a)*ln(F))*tan(1/2*d+1/2*e*x)^2+ln(F)*b*c/(e^2+b^2*c^2*
ln(F)^2)*x*exp(c*(b*x+a)*ln(F))-(b^2*c^2*ln(F)^2-e^2)/(e^2+b^2*c^2*ln(F)^2)^2*exp(c*(b*x+a)*ln(F))+2/(e^2+b^2*
c^2*ln(F)^2)*e*x*exp(c*(b*x+a)*ln(F))*tan(1/2*d+1/2*e*x)-4*b*c*ln(F)*e/(e^2+b^2*c^2*ln(F)^2)^2*exp(c*(b*x+a)*l
n(F))*tan(1/2*d+1/2*e*x)-ln(F)*b*c/(e^2+b^2*c^2*ln(F)^2)*x*exp(c*(b*x+a)*ln(F))*tan(1/2*d+1/2*e*x)^2)/(1+tan(1
/2*d+1/2*e*x)^2)+b*c*ln(F)*(1/(e^2+b^2*c^2*ln(F)^2)*e*x*exp(c*(b*x+a)*ln(F))*tan(1/2*d+1/2*e*x)^2-1/(e^2+b^2*c
^2*ln(F)^2)*e*x*exp(c*(b*x+a)*ln(F))-2*(b^2*c^2*ln(F)^2-e^2)/(e^2+b^2*c^2*ln(F)^2)^2*exp(c*(b*x+a)*ln(F))*tan(
1/2*d+1/2*e*x)+2*b*c*ln(F)*e/(e^2+b^2*c^2*ln(F)^2)^2*exp(c*(b*x+a)*ln(F))-2*b*c*ln(F)*e/(e^2+b^2*c^2*ln(F)^2)^
2*exp(c*(b*x+a)*ln(F))*tan(1/2*d+1/2*e*x)^2+2*ln(F)*b*c/(e^2+b^2*c^2*ln(F)^2)*x*exp(c*(b*x+a)*ln(F))*tan(1/2*d
+1/2*e*x))/(1+tan(1/2*d+1/2*e*x)^2)

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Maxima [B]  time = 1.44494, size = 1866, normalized size = 109.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(e*x*cos(e*x+d)+(1+b*c*x*log(F))*sin(e*x+d)),x, algorithm="maxima")

[Out]

1/2*((F^(a*c)*b^2*c^2*log(F)^2*sin(d) + 2*F^(a*c)*b*c*e*cos(d)*log(F) - F^(a*c)*e^2*sin(d) - (F^(a*c)*b^3*c^3*
log(F)^3*sin(d) + F^(a*c)*b^2*c^2*e*cos(d)*log(F)^2 + F^(a*c)*b*c*e^2*log(F)*sin(d) + F^(a*c)*e^3*cos(d))*x)*F
^(b*c*x)*cos(e*x + 2*d) - (F^(a*c)*b^2*c^2*log(F)^2*sin(d) - 2*F^(a*c)*b*c*e*cos(d)*log(F) - F^(a*c)*e^2*sin(d
) - (F^(a*c)*b^3*c^3*log(F)^3*sin(d) - F^(a*c)*b^2*c^2*e*cos(d)*log(F)^2 + F^(a*c)*b*c*e^2*log(F)*sin(d) - F^(
a*c)*e^3*cos(d))*x)*F^(b*c*x)*cos(e*x) - (F^(a*c)*b^2*c^2*cos(d)*log(F)^2 - 2*F^(a*c)*b*c*e*log(F)*sin(d) - F^
(a*c)*e^2*cos(d) - (F^(a*c)*b^3*c^3*cos(d)*log(F)^3 - F^(a*c)*b^2*c^2*e*log(F)^2*sin(d) + F^(a*c)*b*c*e^2*cos(
d)*log(F) - F^(a*c)*e^3*sin(d))*x)*F^(b*c*x)*sin(e*x + 2*d) - (F^(a*c)*b^2*c^2*cos(d)*log(F)^2 + 2*F^(a*c)*b*c
*e*log(F)*sin(d) - F^(a*c)*e^2*cos(d) - (F^(a*c)*b^3*c^3*cos(d)*log(F)^3 + F^(a*c)*b^2*c^2*e*log(F)^2*sin(d) +
F^(a*c)*b*c*e^2*cos(d)*log(F) + F^(a*c)*e^3*sin(d))*x)*F^(b*c*x)*sin(e*x))*b*c*log(F)/(b^4*c^4*cos(d)^2*log(F
)^4 + b^4*c^4*log(F)^4*sin(d)^2 + (cos(d)^2 + sin(d)^2)*e^4 + 2*(b^2*c^2*cos(d)^2*log(F)^2 + b^2*c^2*log(F)^2*
sin(d)^2)*e^2) - 1/2*((F^(a*c)*b^2*c^2*cos(d)*log(F)^2 - 2*F^(a*c)*b*c*e*log(F)*sin(d) - F^(a*c)*e^2*cos(d) -
(F^(a*c)*b^3*c^3*cos(d)*log(F)^3 - F^(a*c)*b^2*c^2*e*log(F)^2*sin(d) + F^(a*c)*b*c*e^2*cos(d)*log(F) - F^(a*c)
*e^3*sin(d))*x)*F^(b*c*x)*cos(e*x + 2*d) + (F^(a*c)*b^2*c^2*cos(d)*log(F)^2 + 2*F^(a*c)*b*c*e*log(F)*sin(d) -
F^(a*c)*e^2*cos(d) - (F^(a*c)*b^3*c^3*cos(d)*log(F)^3 + F^(a*c)*b^2*c^2*e*log(F)^2*sin(d) + F^(a*c)*b*c*e^2*co
s(d)*log(F) + F^(a*c)*e^3*sin(d))*x)*F^(b*c*x)*cos(e*x) + (F^(a*c)*b^2*c^2*log(F)^2*sin(d) + 2*F^(a*c)*b*c*e*c
os(d)*log(F) - F^(a*c)*e^2*sin(d) - (F^(a*c)*b^3*c^3*log(F)^3*sin(d) + F^(a*c)*b^2*c^2*e*cos(d)*log(F)^2 + F^(
a*c)*b*c*e^2*log(F)*sin(d) + F^(a*c)*e^3*cos(d))*x)*F^(b*c*x)*sin(e*x + 2*d) - (F^(a*c)*b^2*c^2*log(F)^2*sin(d
) - 2*F^(a*c)*b*c*e*cos(d)*log(F) - F^(a*c)*e^2*sin(d) - (F^(a*c)*b^3*c^3*log(F)^3*sin(d) - F^(a*c)*b^2*c^2*e*
cos(d)*log(F)^2 + F^(a*c)*b*c*e^2*log(F)*sin(d) - F^(a*c)*e^3*cos(d))*x)*F^(b*c*x)*sin(e*x))*e/(b^4*c^4*cos(d)
^2*log(F)^4 + b^4*c^4*log(F)^4*sin(d)^2 + (cos(d)^2 + sin(d)^2)*e^4 + 2*(b^2*c^2*cos(d)^2*log(F)^2 + b^2*c^2*l
og(F)^2*sin(d)^2)*e^2) - 1/2*((F^(a*c)*b*c*log(F)*sin(d) + F^(a*c)*e*cos(d))*F^(b*c*x)*cos(e*x + 2*d) - (F^(a*
c)*b*c*log(F)*sin(d) - F^(a*c)*e*cos(d))*F^(b*c*x)*cos(e*x) - (F^(a*c)*b*c*cos(d)*log(F) - F^(a*c)*e*sin(d))*F
^(b*c*x)*sin(e*x + 2*d) - (F^(a*c)*b*c*cos(d)*log(F) + F^(a*c)*e*sin(d))*F^(b*c*x)*sin(e*x))/(b^2*c^2*cos(d)^2
*log(F)^2 + b^2*c^2*log(F)^2*sin(d)^2 + (cos(d)^2 + sin(d)^2)*e^2)

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Fricas [A]  time = 0.470724, size = 43, normalized size = 2.53 \begin{align*} F^{b c x + a c} x \sin \left (e x + d\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(e*x*cos(e*x+d)+(1+b*c*x*log(F))*sin(e*x+d)),x, algorithm="fricas")

[Out]

F^(b*c*x + a*c)*x*sin(e*x + d)

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Sympy [A]  time = 31.3272, size = 19, normalized size = 1.12 \begin{align*} F^{a c} F^{b c x} x \sin{\left (d + e x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a))*(e*x*cos(e*x+d)+(1+b*c*x*ln(F))*sin(e*x+d)),x)

[Out]

F**(a*c)*F**(b*c*x)*x*sin(d + e*x)

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Giac [C]  time = 1.41614, size = 5310, normalized size = 312.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(e*x*cos(e*x+d)+(1+b*c*x*log(F))*sin(e*x+d)),x, algorithm="giac")

[Out]

1/2*(2*((pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)) + 2*b*c*e*log(abs(F)))*(pi*b*c*x*sgn(F) - pi*b
*c*x + 2*x*e)/((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2 - 2*pi*b*c*e*sgn(F) + 2*pi*b*c*e
- 2*e^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)) + 2*b*c*e*log(abs(F)))^2) + (pi^2*b^2*c
^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2 - 2*pi*b*c*e*sgn(F) + 2*pi*b*c*e - 2*e^2)*(b*c*x*log(abs(F)
) - 1)/((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2 - 2*pi*b*c*e*sgn(F) + 2*pi*b*c*e - 2*e^2
)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)) + 2*b*c*e*log(abs(F)))^2))*cos(1/2*pi*b*c*x*sg
n(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c + x*e + d) - ((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2
*c^2*log(abs(F))^2 - 2*pi*b*c*e*sgn(F) + 2*pi*b*c*e - 2*e^2)*(pi*b*c*x*sgn(F) - pi*b*c*x + 2*x*e)/((pi^2*b^2*c
^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2 - 2*pi*b*c*e*sgn(F) + 2*pi*b*c*e - 2*e^2)^2 + 4*(pi*b^2*c^2
*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)) + 2*b*c*e*log(abs(F)))^2) - 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi
*b^2*c^2*log(abs(F)) + 2*b*c*e*log(abs(F)))*(b*c*x*log(abs(F)) - 1)/((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b
^2*c^2*log(abs(F))^2 - 2*pi*b*c*e*sgn(F) + 2*pi*b*c*e - 2*e^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c
^2*log(abs(F)) + 2*b*c*e*log(abs(F)))^2))*sin(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*
a*c + x*e + d))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F)) + 1) + 1/2*(2*((pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*
c^2*log(abs(F)) - 2*b*c*e*log(abs(F)))*(pi*b*c*x*sgn(F) - pi*b*c*x - 2*x*e)/((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c
^2 + 2*b^2*c^2*log(abs(F))^2 + 2*pi*b*c*e*sgn(F) - 2*pi*b*c*e - 2*e^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) -
pi*b^2*c^2*log(abs(F)) - 2*b*c*e*log(abs(F)))^2) + (pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))
^2 + 2*pi*b*c*e*sgn(F) - 2*pi*b*c*e - 2*e^2)*(b*c*x*log(abs(F)) - 1)/((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*
b^2*c^2*log(abs(F))^2 + 2*pi*b*c*e*sgn(F) - 2*pi*b*c*e - 2*e^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*
c^2*log(abs(F)) - 2*b*c*e*log(abs(F)))^2))*cos(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi
*a*c - x*e - d) - ((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2 + 2*pi*b*c*e*sgn(F) - 2*pi*b*
c*e - 2*e^2)*(pi*b*c*x*sgn(F) - pi*b*c*x - 2*x*e)/((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))
^2 + 2*pi*b*c*e*sgn(F) - 2*pi*b*c*e - 2*e^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)) - 2
*b*c*e*log(abs(F)))^2) - 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)) - 2*b*c*e*log(abs(F)))*(b*c
*x*log(abs(F)) - 1)/((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2 + 2*pi*b*c*e*sgn(F) - 2*pi*
b*c*e - 2*e^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)) - 2*b*c*e*log(abs(F)))^2))*sin(1/
2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c - x*e - d))*e^(b*c*x*log(abs(F)) + a*c*log(a
bs(F)) + 1) - 1/2*I*((2*pi*b*c*x*sgn(F) - 2*pi*b*c*x - 4*I*b*c*x*log(abs(F)) + 4*x*e + 4*I)*e^(1/2*I*pi*b*c*x*
sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c + I*x*e + I*d)/(4*pi^2*b^2*c^2*sgn(F) + 8*I*pi*b^
2*c^2*log(abs(F))*sgn(F) - 4*pi^2*b^2*c^2 - 8*I*pi*b^2*c^2*log(abs(F)) + 8*b^2*c^2*log(abs(F))^2 - 8*pi*b*c*e*
sgn(F) + 8*pi*b*c*e + 16*I*b*c*e*log(abs(F)) - 8*e^2) + (2*pi*b*c*x*sgn(F) - 2*pi*b*c*x + 4*I*b*c*x*log(abs(F)
) + 4*x*e - 4*I)*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c - I*x*e - I*d
)/(4*pi^2*b^2*c^2*sgn(F) - 8*I*pi*b^2*c^2*log(abs(F))*sgn(F) - 4*pi^2*b^2*c^2 + 8*I*pi*b^2*c^2*log(abs(F)) + 8
*b^2*c^2*log(abs(F))^2 - 8*pi*b*c*e*sgn(F) + 8*pi*b*c*e - 16*I*b*c*e*log(abs(F)) - 8*e^2))*e^(b*c*x*log(abs(F)
) + a*c*log(abs(F)) + 1) - 1/2*I*((2*pi*b*c*x*sgn(F) - 2*pi*b*c*x - 4*I*b*c*x*log(abs(F)) - 4*x*e + 4*I)*e^(1/
2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c - I*x*e - I*d)/(4*pi^2*b^2*c^2*sgn(F
) + 8*I*pi*b^2*c^2*log(abs(F))*sgn(F) - 4*pi^2*b^2*c^2 - 8*I*pi*b^2*c^2*log(abs(F)) + 8*b^2*c^2*log(abs(F))^2
+ 8*pi*b*c*e*sgn(F) - 8*pi*b*c*e - 16*I*b*c*e*log(abs(F)) - 8*e^2) + (2*pi*b*c*x*sgn(F) - 2*pi*b*c*x + 4*I*b*c
*x*log(abs(F)) - 4*x*e - 4*I)*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c
+ I*x*e + I*d)/(4*pi^2*b^2*c^2*sgn(F) - 8*I*pi*b^2*c^2*log(abs(F))*sgn(F) - 4*pi^2*b^2*c^2 + 8*I*pi*b^2*c^2*lo
g(abs(F)) + 8*b^2*c^2*log(abs(F))^2 + 8*pi*b*c*e*sgn(F) - 8*pi*b*c*e + 16*I*b*c*e*log(abs(F)) - 8*e^2))*e^(b*c
*x*log(abs(F)) + a*c*log(abs(F)) + 1) - 1/2*((2*pi*b^2*c^2*x*log(F)*sgn(F) - 2*pi*b^2*c^2*x*log(F) - 4*I*b^2*c
^2*x*log(F)*log(abs(F)) + 4*b*c*x*e*log(F) + 2*pi*b*c*sgn(F) - 2*pi*b*c + 4*I*b*c*log(F) - 4*I*b*c*log(abs(F))
+ 4*e)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c + I*x*e + I*d)/(4*pi^2*
b^2*c^2*sgn(F) + 8*I*pi*b^2*c^2*log(abs(F))*sgn(F) - 4*pi^2*b^2*c^2 - 8*I*pi*b^2*c^2*log(abs(F)) + 8*b^2*c^2*l
og(abs(F))^2 - 8*pi*b*c*e*sgn(F) + 8*pi*b*c*e + 16*I*b*c*e*log(abs(F)) - 8*e^2) - (2*pi*b^2*c^2*x*log(F)*sgn(F
) - 2*pi*b^2*c^2*x*log(F) + 4*I*b^2*c^2*x*log(F)*log(abs(F)) + 4*b*c*x*e*log(F) + 2*pi*b*c*sgn(F) - 2*pi*b*c -
4*I*b*c*log(F) + 4*I*b*c*log(abs(F)) + 4*e)*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F)
+ 1/2*I*pi*a*c - I*x*e - I*d)/(4*pi^2*b^2*c^2*sgn(F) - 8*I*pi*b^2*c^2*log(abs(F))*sgn(F) - 4*pi^2*b^2*c^2 + 8*
I*pi*b^2*c^2*log(abs(F)) + 8*b^2*c^2*log(abs(F))^2 - 8*pi*b*c*e*sgn(F) + 8*pi*b*c*e - 16*I*b*c*e*log(abs(F)) -
8*e^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 1/2*I*((2*I*pi*b^2*c^2*x*log(F)*sgn(F) - 2*I*pi*b^2*c^2*x*l
og(F) + 4*b^2*c^2*x*log(F)*log(abs(F)) + 4*I*b*c*x*e*log(F) + 2*I*pi*b*c*sgn(F) - 2*I*pi*b*c - 4*b*c*log(F) +
4*b*c*log(abs(F)) + 4*I*e)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c + I*
x*e + I*d)/(4*pi^2*b^2*c^2*sgn(F) + 8*I*pi*b^2*c^2*log(abs(F))*sgn(F) - 4*pi^2*b^2*c^2 - 8*I*pi*b^2*c^2*log(ab
s(F)) + 8*b^2*c^2*log(abs(F))^2 - 8*pi*b*c*e*sgn(F) + 8*pi*b*c*e + 16*I*b*c*e*log(abs(F)) - 8*e^2) - (-2*I*pi*
b^2*c^2*x*log(F)*sgn(F) + 2*I*pi*b^2*c^2*x*log(F) + 4*b^2*c^2*x*log(F)*log(abs(F)) - 4*I*b*c*x*e*log(F) - 2*I*
pi*b*c*sgn(F) + 2*I*pi*b*c - 4*b*c*log(F) + 4*b*c*log(abs(F)) - 4*I*e)*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*
c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c - I*x*e - I*d)/(4*pi^2*b^2*c^2*sgn(F) - 8*I*pi*b^2*c^2*log(abs(F))*sg
n(F) - 4*pi^2*b^2*c^2 + 8*I*pi*b^2*c^2*log(abs(F)) + 8*b^2*c^2*log(abs(F))^2 - 8*pi*b*c*e*sgn(F) + 8*pi*b*c*e
- 16*I*b*c*e*log(abs(F)) - 8*e^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + 1/2*((2*pi*b^2*c^2*x*log(F)*sgn(F
) - 2*pi*b^2*c^2*x*log(F) - 4*I*b^2*c^2*x*log(F)*log(abs(F)) - 4*b*c*x*e*log(F) + 2*pi*b*c*sgn(F) - 2*pi*b*c +
4*I*b*c*log(F) - 4*I*b*c*log(abs(F)) - 4*e)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) -
1/2*I*pi*a*c - I*x*e - I*d)/(4*pi^2*b^2*c^2*sgn(F) + 8*I*pi*b^2*c^2*log(abs(F))*sgn(F) - 4*pi^2*b^2*c^2 - 8*I
*pi*b^2*c^2*log(abs(F)) + 8*b^2*c^2*log(abs(F))^2 + 8*pi*b*c*e*sgn(F) - 8*pi*b*c*e - 16*I*b*c*e*log(abs(F)) -
8*e^2) - (2*pi*b^2*c^2*x*log(F)*sgn(F) - 2*pi*b^2*c^2*x*log(F) + 4*I*b^2*c^2*x*log(F)*log(abs(F)) - 4*b*c*x*e*
log(F) + 2*pi*b*c*sgn(F) - 2*pi*b*c - 4*I*b*c*log(F) + 4*I*b*c*log(abs(F)) - 4*e)*e^(-1/2*I*pi*b*c*x*sgn(F) +
1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c + I*x*e + I*d)/(4*pi^2*b^2*c^2*sgn(F) - 8*I*pi*b^2*c^2*log
(abs(F))*sgn(F) - 4*pi^2*b^2*c^2 + 8*I*pi*b^2*c^2*log(abs(F)) + 8*b^2*c^2*log(abs(F))^2 + 8*pi*b*c*e*sgn(F) -
8*pi*b*c*e + 16*I*b*c*e*log(abs(F)) - 8*e^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 1/2*I*((-2*I*pi*b^2*c^
2*x*log(F)*sgn(F) + 2*I*pi*b^2*c^2*x*log(F) - 4*b^2*c^2*x*log(F)*log(abs(F)) + 4*I*b*c*x*e*log(F) - 2*I*pi*b*c
*sgn(F) + 2*I*pi*b*c + 4*b*c*log(F) - 4*b*c*log(abs(F)) + 4*I*e)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1
/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c - I*x*e - I*d)/(4*pi^2*b^2*c^2*sgn(F) + 8*I*pi*b^2*c^2*log(abs(F))*sgn(F) -
4*pi^2*b^2*c^2 - 8*I*pi*b^2*c^2*log(abs(F)) + 8*b^2*c^2*log(abs(F))^2 + 8*pi*b*c*e*sgn(F) - 8*pi*b*c*e - 16*I*
b*c*e*log(abs(F)) - 8*e^2) - (2*I*pi*b^2*c^2*x*log(F)*sgn(F) - 2*I*pi*b^2*c^2*x*log(F) - 4*b^2*c^2*x*log(F)*lo
g(abs(F)) - 4*I*b*c*x*e*log(F) + 2*I*pi*b*c*sgn(F) - 2*I*pi*b*c + 4*b*c*log(F) - 4*b*c*log(abs(F)) - 4*I*e)*e^
(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c + I*x*e + I*d)/(4*pi^2*b^2*c^2*s
gn(F) - 8*I*pi*b^2*c^2*log(abs(F))*sgn(F) - 4*pi^2*b^2*c^2 + 8*I*pi*b^2*c^2*log(abs(F)) + 8*b^2*c^2*log(abs(F)
)^2 + 8*pi*b*c*e*sgn(F) - 8*pi*b*c*e + 16*I*b*c*e*log(abs(F)) - 8*e^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))
)