3.322 \(\int F^{c (a+b x)} \sinh ^3(d+e x) \, dx\)
Optimal. Leaf size=202 \[ -\frac{b c \log (F) \sinh ^3(d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}+\frac{6 b c e^2 \log (F) \sinh (d+e x) F^{c (a+b x)}}{-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)+9 e^4}-\frac{6 e^3 \cosh (d+e x) F^{c (a+b x)}}{-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)+9 e^4}+\frac{3 e \sinh ^2(d+e x) \cosh (d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)} \]
[Out]
(-6*e^3*F^(c*(a + b*x))*Cosh[d + e*x])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (6*b*c*e^2*F^(c*
(a + b*x))*Log[F]*Sinh[d + e*x])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (3*e*F^(c*(a + b*x))*C
osh[d + e*x]*Sinh[d + e*x]^2)/(9*e^2 - b^2*c^2*Log[F]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x]^3)/(9*e^2
- b^2*c^2*Log[F]^2)
________________________________________________________________________________________
Rubi [A] time = 0.0817947, antiderivative size = 202, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used =
{5476, 5474} \[ -\frac{b c \log (F) \sinh ^3(d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}+\frac{6 b c e^2 \log (F) \sinh (d+e x) F^{c (a+b x)}}{-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)+9 e^4}-\frac{6 e^3 \cosh (d+e x) F^{c (a+b x)}}{-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)+9 e^4}+\frac{3 e \sinh ^2(d+e x) \cosh (d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)} \]
Antiderivative was successfully verified.
[In]
Int[F^(c*(a + b*x))*Sinh[d + e*x]^3,x]
[Out]
(-6*e^3*F^(c*(a + b*x))*Cosh[d + e*x])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (6*b*c*e^2*F^(c*
(a + b*x))*Log[F]*Sinh[d + e*x])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (3*e*F^(c*(a + b*x))*C
osh[d + e*x]*Sinh[d + e*x]^2)/(9*e^2 - b^2*c^2*Log[F]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x]^3)/(9*e^2
- b^2*c^2*Log[F]^2)
Rule 5476
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> -Simp[(b*c*Log[F]*F^(c*(a +
b*x))*Sinh[d + e*x]^n)/(e^2*n^2 - b^2*c^2*Log[F]^2), x] + (-Dist[(n*(n - 1)*e^2)/(e^2*n^2 - b^2*c^2*Log[F]^2),
Int[F^(c*(a + b*x))*Sinh[d + e*x]^(n - 2), x], x] + Simp[(e*n*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^(n
- 1))/(e^2*n^2 - b^2*c^2*Log[F]^2), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0]
&& GtQ[n, 1]
Rule 5474
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)], x_Symbol] :> -Simp[(b*c*Log[F]*F^(c*(a + b*x))
*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Cosh[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]
Rubi steps
\begin{align*} \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx &=\frac{3 e F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x)}{9 e^2-b^2 c^2 \log ^2(F)}-\frac{b c F^{c (a+b x)} \log (F) \sinh ^3(d+e x)}{9 e^2-b^2 c^2 \log ^2(F)}-\frac{\left (6 e^2\right ) \int F^{c (a+b x)} \sinh (d+e x) \, dx}{9 e^2-b^2 c^2 \log ^2(F)}\\ &=-\frac{6 e^3 F^{c (a+b x)} \cosh (d+e x)}{9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac{6 b c e^2 F^{c (a+b x)} \log (F) \sinh (d+e x)}{9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac{3 e F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x)}{9 e^2-b^2 c^2 \log ^2(F)}-\frac{b c F^{c (a+b x)} \log (F) \sinh ^3(d+e x)}{9 e^2-b^2 c^2 \log ^2(F)}\\ \end{align*}
Mathematica [A] time = 0.665594, size = 157, normalized size = 0.78 \[ \frac{F^{c (a+b x)} \left (3 \cosh (3 (d+e x)) \left (e^3-b^2 c^2 e \log ^2(F)\right )+3 \cosh (d+e x) \left (b^2 c^2 e \log ^2(F)-9 e^3\right )+2 b c \log (F) \sinh (d+e x) \left (\cosh (2 (d+e x)) \left (b^2 c^2 \log ^2(F)-e^2\right )-b^2 c^2 \log ^2(F)+13 e^2\right )\right )}{4 \left (-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)+9 e^4\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[F^(c*(a + b*x))*Sinh[d + e*x]^3,x]
[Out]
(F^(c*(a + b*x))*(3*Cosh[3*(d + e*x)]*(e^3 - b^2*c^2*e*Log[F]^2) + 3*Cosh[d + e*x]*(-9*e^3 + b^2*c^2*e*Log[F]^
2) + 2*b*c*Log[F]*(13*e^2 - b^2*c^2*Log[F]^2 + Cosh[2*(d + e*x)]*(-e^2 + b^2*c^2*Log[F]^2))*Sinh[d + e*x]))/(4
*(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4))
________________________________________________________________________________________
Maple [A] time = 0.061, size = 326, normalized size = 1.6 \begin{align*}{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}{{\rm e}^{6\,ex+6\,d}}-3\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}{{\rm e}^{4\,ex+4\,d}}-3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}e{{\rm e}^{6\,ex+6\,d}}+3\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}{{\rm e}^{2\,ex+2\,d}}+3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}e{{\rm e}^{4\,ex+4\,d}}-\ln \left ( F \right ) bc{e}^{2}{{\rm e}^{6\,ex+6\,d}}- \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}+3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}e{{\rm e}^{2\,ex+2\,d}}+27\,\ln \left ( F \right ) bc{e}^{2}{{\rm e}^{4\,ex+4\,d}}+3\,{e}^{3}{{\rm e}^{6\,ex+6\,d}}-3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}e-27\,\ln \left ( F \right ) bc{e}^{2}{{\rm e}^{2\,ex+2\,d}}-27\,{e}^{3}{{\rm e}^{4\,ex+4\,d}}+\ln \left ( F \right ) bc{e}^{2}-27\,{e}^{3}{{\rm e}^{2\,ex+2\,d}}+3\,{e}^{3} \right ){{\rm e}^{-3\,ex-3\,d}}{F}^{c \left ( bx+a \right ) }}{ \left ( 8\,bc\ln \left ( F \right ) -8\,e \right ) \left ( bc\ln \left ( F \right ) -3\,e \right ) \left ( e+bc\ln \left ( F \right ) \right ) \left ( bc\ln \left ( F \right ) +3\,e \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(c*(b*x+a))*sinh(e*x+d)^3,x)
[Out]
1/8*(ln(F)^3*b^3*c^3*exp(6*e*x+6*d)-3*ln(F)^3*b^3*c^3*exp(4*e*x+4*d)-3*ln(F)^2*b^2*c^2*e*exp(6*e*x+6*d)+3*ln(F
)^3*b^3*c^3*exp(2*e*x+2*d)+3*ln(F)^2*b^2*c^2*e*exp(4*e*x+4*d)-ln(F)*b*c*e^2*exp(6*e*x+6*d)-ln(F)^3*b^3*c^3+3*l
n(F)^2*b^2*c^2*e*exp(2*e*x+2*d)+27*ln(F)*b*c*e^2*exp(4*e*x+4*d)+3*e^3*exp(6*e*x+6*d)-3*ln(F)^2*b^2*c^2*e-27*ln
(F)*b*c*e^2*exp(2*e*x+2*d)-27*e^3*exp(4*e*x+4*d)+ln(F)*b*c*e^2-27*e^3*exp(2*e*x+2*d)+3*e^3)/(b*c*ln(F)-e)*exp(
-3*e*x-3*d)/(b*c*ln(F)-3*e)/(e+b*c*ln(F))/(b*c*ln(F)+3*e)*F^(c*(b*x+a))
________________________________________________________________________________________
Maxima [A] time = 1.19122, size = 181, normalized size = 0.9 \begin{align*} \frac{F^{a c} e^{\left (b c x \log \left (F\right ) + 3 \, e x + 3 \, d\right )}}{8 \,{\left (b c \log \left (F\right ) + 3 \, e\right )}} - \frac{3 \, F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{8 \,{\left (b c \log \left (F\right ) + e\right )}} + \frac{3 \, F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{8 \,{\left (b c e^{d} \log \left (F\right ) - e e^{d}\right )}} - \frac{F^{a c} e^{\left (b c x \log \left (F\right ) - 3 \, e x\right )}}{8 \,{\left (b c e^{\left (3 \, d\right )} \log \left (F\right ) - 3 \, e e^{\left (3 \, d\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*sinh(e*x+d)^3,x, algorithm="maxima")
[Out]
1/8*F^(a*c)*e^(b*c*x*log(F) + 3*e*x + 3*d)/(b*c*log(F) + 3*e) - 3/8*F^(a*c)*e^(b*c*x*log(F) + e*x + d)/(b*c*lo
g(F) + e) + 3/8*F^(a*c)*e^(b*c*x*log(F) - e*x)/(b*c*e^d*log(F) - e*e^d) - 1/8*F^(a*c)*e^(b*c*x*log(F) - 3*e*x)
/(b*c*e^(3*d)*log(F) - 3*e*e^(3*d))
________________________________________________________________________________________
Fricas [B] time = 2.55951, size = 5434, normalized size = 26.9 \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))*sinh(e*x+d)^3,x, algorithm="fricas")
[Out]
1/8*((3*e^3*cosh(e*x + d)^6 - 27*e^3*cosh(e*x + d)^4 + (b^3*c^3*log(F)^3 - 3*b^2*c^2*e*log(F)^2 - b*c*e^2*log(
F) + 3*e^3)*sinh(e*x + d)^6 + 6*(b^3*c^3*cosh(e*x + d)*log(F)^3 - 3*b^2*c^2*e*cosh(e*x + d)*log(F)^2 - b*c*e^2
*cosh(e*x + d)*log(F) + 3*e^3*cosh(e*x + d))*sinh(e*x + d)^5 - 27*e^3*cosh(e*x + d)^2 + 3*(15*e^3*cosh(e*x + d
)^2 + (5*b^3*c^3*cosh(e*x + d)^2 - b^3*c^3)*log(F)^3 - 9*e^3 - (15*b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*log(
F)^2 - (5*b*c*e^2*cosh(e*x + d)^2 - 9*b*c*e^2)*log(F))*sinh(e*x + d)^4 + (b^3*c^3*cosh(e*x + d)^6 - 3*b^3*c^3*
cosh(e*x + d)^4 + 3*b^3*c^3*cosh(e*x + d)^2 - b^3*c^3)*log(F)^3 + 4*(15*e^3*cosh(e*x + d)^3 - 27*e^3*cosh(e*x
+ d) + (5*b^3*c^3*cosh(e*x + d)^3 - 3*b^3*c^3*cosh(e*x + d))*log(F)^3 - 3*(5*b^2*c^2*e*cosh(e*x + d)^3 - b^2*c
^2*e*cosh(e*x + d))*log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^3 - 27*b*c*e^2*cosh(e*x + d))*log(F))*sinh(e*x + d)^3
+ 3*e^3 - 3*(b^2*c^2*e*cosh(e*x + d)^6 - b^2*c^2*e*cosh(e*x + d)^4 - b^2*c^2*e*cosh(e*x + d)^2 + b^2*c^2*e)*lo
g(F)^2 + 3*(15*e^3*cosh(e*x + d)^4 - 54*e^3*cosh(e*x + d)^2 + (5*b^3*c^3*cosh(e*x + d)^4 - 6*b^3*c^3*cosh(e*x
+ d)^2 + b^3*c^3)*log(F)^3 - 9*e^3 - (15*b^2*c^2*e*cosh(e*x + d)^4 - 6*b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*
log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^4 - 54*b*c*e^2*cosh(e*x + d)^2 + 9*b*c*e^2)*log(F))*sinh(e*x + d)^2 - (b*c
*e^2*cosh(e*x + d)^6 - 27*b*c*e^2*cosh(e*x + d)^4 + 27*b*c*e^2*cosh(e*x + d)^2 - b*c*e^2)*log(F) + 6*(3*e^3*co
sh(e*x + d)^5 - 18*e^3*cosh(e*x + d)^3 - 9*e^3*cosh(e*x + d) + (b^3*c^3*cosh(e*x + d)^5 - 2*b^3*c^3*cosh(e*x +
d)^3 + b^3*c^3*cosh(e*x + d))*log(F)^3 - (3*b^2*c^2*e*cosh(e*x + d)^5 - 2*b^2*c^2*e*cosh(e*x + d)^3 - b^2*c^2
*e*cosh(e*x + d))*log(F)^2 - (b*c*e^2*cosh(e*x + d)^5 - 18*b*c*e^2*cosh(e*x + d)^3 + 9*b*c*e^2*cosh(e*x + d))*
log(F))*sinh(e*x + d))*cosh((b*c*x + a*c)*log(F)) + (3*e^3*cosh(e*x + d)^6 - 27*e^3*cosh(e*x + d)^4 + (b^3*c^3
*log(F)^3 - 3*b^2*c^2*e*log(F)^2 - b*c*e^2*log(F) + 3*e^3)*sinh(e*x + d)^6 + 6*(b^3*c^3*cosh(e*x + d)*log(F)^3
- 3*b^2*c^2*e*cosh(e*x + d)*log(F)^2 - b*c*e^2*cosh(e*x + d)*log(F) + 3*e^3*cosh(e*x + d))*sinh(e*x + d)^5 -
27*e^3*cosh(e*x + d)^2 + 3*(15*e^3*cosh(e*x + d)^2 + (5*b^3*c^3*cosh(e*x + d)^2 - b^3*c^3)*log(F)^3 - 9*e^3 -
(15*b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^2 - 9*b*c*e^2)*log(F))*sinh(e*x
+ d)^4 + (b^3*c^3*cosh(e*x + d)^6 - 3*b^3*c^3*cosh(e*x + d)^4 + 3*b^3*c^3*cosh(e*x + d)^2 - b^3*c^3)*log(F)^3
+ 4*(15*e^3*cosh(e*x + d)^3 - 27*e^3*cosh(e*x + d) + (5*b^3*c^3*cosh(e*x + d)^3 - 3*b^3*c^3*cosh(e*x + d))*lo
g(F)^3 - 3*(5*b^2*c^2*e*cosh(e*x + d)^3 - b^2*c^2*e*cosh(e*x + d))*log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^3 - 27*
b*c*e^2*cosh(e*x + d))*log(F))*sinh(e*x + d)^3 + 3*e^3 - 3*(b^2*c^2*e*cosh(e*x + d)^6 - b^2*c^2*e*cosh(e*x + d
)^4 - b^2*c^2*e*cosh(e*x + d)^2 + b^2*c^2*e)*log(F)^2 + 3*(15*e^3*cosh(e*x + d)^4 - 54*e^3*cosh(e*x + d)^2 + (
5*b^3*c^3*cosh(e*x + d)^4 - 6*b^3*c^3*cosh(e*x + d)^2 + b^3*c^3)*log(F)^3 - 9*e^3 - (15*b^2*c^2*e*cosh(e*x + d
)^4 - 6*b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^4 - 54*b*c*e^2*cosh(e*x + d
)^2 + 9*b*c*e^2)*log(F))*sinh(e*x + d)^2 - (b*c*e^2*cosh(e*x + d)^6 - 27*b*c*e^2*cosh(e*x + d)^4 + 27*b*c*e^2*
cosh(e*x + d)^2 - b*c*e^2)*log(F) + 6*(3*e^3*cosh(e*x + d)^5 - 18*e^3*cosh(e*x + d)^3 - 9*e^3*cosh(e*x + d) +
(b^3*c^3*cosh(e*x + d)^5 - 2*b^3*c^3*cosh(e*x + d)^3 + b^3*c^3*cosh(e*x + d))*log(F)^3 - (3*b^2*c^2*e*cosh(e*x
+ d)^5 - 2*b^2*c^2*e*cosh(e*x + d)^3 - b^2*c^2*e*cosh(e*x + d))*log(F)^2 - (b*c*e^2*cosh(e*x + d)^5 - 18*b*c*
e^2*cosh(e*x + d)^3 + 9*b*c*e^2*cosh(e*x + d))*log(F))*sinh(e*x + d))*sinh((b*c*x + a*c)*log(F)))/(b^4*c^4*cos
h(e*x + d)^3*log(F)^4 - 10*b^2*c^2*e^2*cosh(e*x + d)^3*log(F)^2 + 9*e^4*cosh(e*x + d)^3 + (b^4*c^4*log(F)^4 -
10*b^2*c^2*e^2*log(F)^2 + 9*e^4)*sinh(e*x + d)^3 + 3*(b^4*c^4*cosh(e*x + d)*log(F)^4 - 10*b^2*c^2*e^2*cosh(e*x
+ d)*log(F)^2 + 9*e^4*cosh(e*x + d))*sinh(e*x + d)^2 + 3*(b^4*c^4*cosh(e*x + d)^2*log(F)^4 - 10*b^2*c^2*e^2*c
osh(e*x + d)^2*log(F)^2 + 9*e^4*cosh(e*x + d)^2)*sinh(e*x + d))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F**(c*(b*x+a))*sinh(e*x+d)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [C] time = 1.29867, size = 1673, normalized size = 8.28 \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))*sinh(e*x+d)^3,x, algorithm="giac")
[Out]
1/4*(2*(b*c*log(abs(F)) + 3*e)*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)/((pi*
b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + 3*e)^2) - (pi*b*c*sgn(F) - pi*b*c)*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)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + 3*e)^2))*e^(a*
c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)*x + 3*d) - 1/2*I*(-2*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)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) + 48*e) + 2*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)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b
*c*log(abs(F)) + 48*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)*x + 3*d) - 3/4*(2*(b*c*log(abs(F)) + e)*c
os(-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)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*
log(abs(F)) + e)^2) - (pi*b*c*sgn(F) - pi*b*c)*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)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + e
)*x + d) - 1/2*I*(6*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)/(8*I*pi*
b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) + 16*e) - 6*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)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b*c*log(abs(F)) + 16*e))*e^(a*c*log(abs(F)
) + (b*c*log(abs(F)) + e)*x + d) + 3/4*(2*(b*c*log(abs(F)) - e)*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)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) - e)^2) - (pi*b*c*sgn(F) - pi*b*c
)*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)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b
*c*log(abs(F)) - e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) - e)*x - d) - 1/2*I*(-6*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)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F))
- 16*e) + 6*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)/(-8*I*pi*b*c*s
gn(F) + 8*I*pi*b*c + 16*b*c*log(abs(F)) - 16*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) - e)*x - d) - 1/4*(2*(b
*c*log(abs(F)) - 3*e)*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)/((pi*b*c*sgn(F
) - pi*b*c)^2 + 4*(b*c*log(abs(F)) - 3*e)^2) - (pi*b*c*sgn(F) - pi*b*c)*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)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) - 3*e)^2))*e^(a*c*log(abs
(F)) + (b*c*log(abs(F)) - 3*e)*x - 3*d) - 1/2*I*(2*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)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) - 48*e) - 2*I*e^(-1/2*I*pi*b*c*x*s
gn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b*c*log(abs
(F)) - 48*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) - 3*e)*x - 3*d)