\(\int F^{c (a+b x)} \csc ^4(d+e x) \, dx\) [8]

Optimal result

Integrand size = 18, antiderivative size = 141 \[ \int F^{c (a+b x)} \csc ^4(d+e x) \, dx=-\frac {F^{c (a+b x)} \cot (d+e x) \csc ^2(d+e x)}{3 e}-\frac {b c F^{c (a+b x)} \csc ^2(d+e x) \log (F)}{6 e^2}+\frac {2 e^{2 i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {i b c \log (F)}{2 e},2-\frac {i b c \log (F)}{2 e},e^{2 i (d+e x)}\right ) (2 i e-b c \log (F))}{3 e^2} \]

[Out]

-1/3*F^(c*(b*x+a))*cot(e*x+d)*csc(e*x+d)^2/e-1/6*b*c*F^(c*(b*x+a))*csc(e*x+d)^2*ln(F)/e^2+2/3*exp(2*I*(e*x+d))
*F^(c*(b*x+a))*hypergeom([2, 1-1/2*I*b*c*ln(F)/e],[2-1/2*I*b*c*ln(F)/e],exp(2*I*(e*x+d)))*(2*I*e-b*c*ln(F))/e^
2

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4534, 4538} \[ \int F^{c (a+b x)} \csc ^4(d+e x) \, dx=\frac {2 e^{2 i (d+e x)} F^{c (a+b x)} (-b c \log (F)+2 i e) \operatorname {Hypergeometric2F1}\left (2,1-\frac {i b c \log (F)}{2 e},2-\frac {i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )}{3 e^2}-\frac {b c \log (F) \csc ^2(d+e x) F^{c (a+b x)}}{6 e^2}-\frac {\cot (d+e x) \csc ^2(d+e x) F^{c (a+b x)}}{3 e} \]

[In]

Int[F^(c*(a + b*x))*Csc[d + e*x]^4,x]

[Out]

-1/3*(F^(c*(a + b*x))*Cot[d + e*x]*Csc[d + e*x]^2)/e - (b*c*F^(c*(a + b*x))*Csc[d + e*x]^2*Log[F])/(6*e^2) + (
2*E^((2*I)*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 - ((I/2)*b*c*Log[F])/e, 2 - ((I/2)*b*c*Log[F])/e,
E^((2*I)*(d + e*x))]*((2*I)*e - b*c*Log[F]))/(3*e^2)

Rule 4534

Int[Csc[(d_.) + (e_.)*(x_)]^(n_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(-b)*c*Log[F]*F^(c*(a +
b*x))*(Csc[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2))), x] + (Dist[(e^2*(n - 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n - 1)
*(n - 2)), Int[F^(c*(a + b*x))*Csc[d + e*x]^(n - 2), x], x] - Simp[F^(c*(a + b*x))*Csc[d + e*x]^(n - 1)*(Cos[d
+ e*x]/(e*(n - 1))), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && GtQ[n
, 1] && NeQ[n, 2]

Rule 4538

Int[Csc[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(-2*I)^n*E^(I*n*(d + e*
x))*(F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 - I*b*c*(Log[F]/(2*e)), 1 + n/2 - I*b*c*(L
og[F]/(2*e)), E^(2*I*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = -\frac {F^{c (a+b x)} \cot (d+e x) \csc ^2(d+e x)}{3 e}-\frac {b c F^{c (a+b x)} \csc ^2(d+e x) \log (F)}{6 e^2}+\frac {1}{6} \left (4+\frac {b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \csc ^2(d+e x) \, dx \\ & = -\frac {F^{c (a+b x)} \cot (d+e x) \csc ^2(d+e x)}{3 e}-\frac {b c F^{c (a+b x)} \csc ^2(d+e x) \log (F)}{6 e^2}+\frac {2 e^{2 i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {i b c \log (F)}{2 e},2-\frac {i b c \log (F)}{2 e},e^{2 i (d+e x)}\right ) (2 i e-b c \log (F))}{3 e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.18 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.23 \[ \int F^{c (a+b x)} \csc ^4(d+e x) \, dx=\frac {F^{c (a+b x)} \left (-e \csc ^2(d+e x) (2 e \cot (d)+b c \log (F))-\frac {2 i \left (1+\left (-1+e^{2 i d}\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c \log (F)}{2 e},1-\frac {i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )\right ) \left (4 e^2+b^2 c^2 \log ^2(F)\right )}{-1+e^{2 i d}}+2 e^2 \csc (d) \csc ^3(d+e x) \sin (e x)+\csc (d) \csc (d+e x) \left (4 e^2+b^2 c^2 \log ^2(F)\right ) \sin (e x)\right )}{6 e^3} \]

[In]

Integrate[F^(c*(a + b*x))*Csc[d + e*x]^4,x]

[Out]

(F^(c*(a + b*x))*(-(e*Csc[d + e*x]^2*(2*e*Cot[d] + b*c*Log[F])) - ((2*I)*(1 + (-1 + E^((2*I)*d))*Hypergeometri
c2F1[1, ((-1/2*I)*b*c*Log[F])/e, 1 - ((I/2)*b*c*Log[F])/e, E^((2*I)*(d + e*x))])*(4*e^2 + b^2*c^2*Log[F]^2))/(
-1 + E^((2*I)*d)) + 2*e^2*Csc[d]*Csc[d + e*x]^3*Sin[e*x] + Csc[d]*Csc[d + e*x]*(4*e^2 + b^2*c^2*Log[F]^2)*Sin[
e*x]))/(6*e^3)

Maple [F]

\[\int F^{c \left (x b +a \right )} \csc \left (e x +d \right )^{4}d x\]

[In]

int(F^(c*(b*x+a))*csc(e*x+d)^4,x)

[Out]

int(F^(c*(b*x+a))*csc(e*x+d)^4,x)

Fricas [F]

\[ \int F^{c (a+b x)} \csc ^4(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{4} \,d x } \]

[In]

integrate(F^(c*(b*x+a))*csc(e*x+d)^4,x, algorithm="fricas")

[Out]

integral(F^(b*c*x + a*c)*csc(e*x + d)^4, x)

Sympy [F]

\[ \int F^{c (a+b x)} \csc ^4(d+e x) \, dx=\int F^{c \left (a + b x\right )} \csc ^{4}{\left (d + e x \right )}\, dx \]

[In]

integrate(F**(c*(b*x+a))*csc(e*x+d)**4,x)

[Out]

Integral(F**(c*(a + b*x))*csc(d + e*x)**4, x)

Maxima [F]

\[ \int F^{c (a+b x)} \csc ^4(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{4} \,d x } \]

[In]

integrate(F^(c*(b*x+a))*csc(e*x+d)^4,x, algorithm="maxima")

[Out]

16*(6*(F^(a*c)*b^5*c^5*log(F)^5 + 100*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 2304*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*co
s(4*e*x + 4*d)^2 + 320*(F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*cos(2*e*x + 2*d)^2
+ 6*(F^(a*c)*b^5*c^5*log(F)^5 + 100*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 2304*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*sin
(4*e*x + 4*d)^2 + 320*(F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*sin(2*e*x + 2*d)^2
+ 560*(F^(a*c)*b^3*c^3*e^2*log(F)^3 - 32*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*cos(2*e*x + 2*d) - 40*(F^(a*c)*b^4*
c^4*e*log(F)^4 - 104*F^(a*c)*b^2*c^2*e^3*log(F)^2)*F^(b*c*x)*sin(2*e*x + 2*d) - 160*(F^(a*c)*b^3*c^3*e^2*log(F
)^3 - 20*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x) + ((F^(a*c)*b^5*c^5*log(F)^5 + 100*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 2
304*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*cos(4*e*x + 4*d) - 80*(F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4
*log(F))*F^(b*c*x)*cos(2*e*x + 2*d) - 4*(F^(a*c)*b^4*c^4*e*log(F)^4 + 100*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 2304*
F^(a*c)*e^5)*F^(b*c*x)*sin(4*e*x + 4*d) - 8*(F^(a*c)*b^4*c^4*e*log(F)^4 + 40*F^(a*c)*b^2*c^2*e^3*log(F)^2 - 15
36*F^(a*c)*e^5)*F^(b*c*x)*sin(2*e*x + 2*d) - 160*(F^(a*c)*b^3*c^3*e^2*log(F)^3 - 20*F^(a*c)*b*c*e^4*log(F))*F^
(b*c*x))*cos(8*e*x + 8*d) - 4*((F^(a*c)*b^5*c^5*log(F)^5 + 100*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 2304*F^(a*c)*b*c
*e^4*log(F))*F^(b*c*x)*cos(4*e*x + 4*d) - 80*(F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F))*F^(b*c
*x)*cos(2*e*x + 2*d) - 4*(F^(a*c)*b^4*c^4*e*log(F)^4 + 100*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 2304*F^(a*c)*e^5)*F^
(b*c*x)*sin(4*e*x + 4*d) - 8*(F^(a*c)*b^4*c^4*e*log(F)^4 + 40*F^(a*c)*b^2*c^2*e^3*log(F)^2 - 1536*F^(a*c)*e^5)
*F^(b*c*x)*sin(2*e*x + 2*d) - 160*(F^(a*c)*b^3*c^3*e^2*log(F)^3 - 20*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x))*cos(6*
e*x + 6*d) - (4*(F^(a*c)*b^5*c^5*log(F)^5 + 220*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 9984*F^(a*c)*b*c*e^4*log(F))*F^
(b*c*x)*cos(2*e*x + 2*d) + 64*(F^(a*c)*b^4*c^4*e*log(F)^4 + 55*F^(a*c)*b^2*c^2*e^3*log(F)^2 - 576*F^(a*c)*e^5)
*F^(b*c*x)*sin(2*e*x + 2*d) - (F^(a*c)*b^5*c^5*log(F)^5 - 860*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 21504*F^(a*c)*b*c
*e^4*log(F))*F^(b*c*x))*cos(4*e*x + 4*d) + 16*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5
+ 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F) + (F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a
*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(8*e*x + 8*d)^
2 + 16*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 +
36864*F^(a*c)*b*c*e^8*log(F))*cos(6*e*x + 6*d)^2 + 36*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*
log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(4*e*x + 4*d)^2 + 16*(F^(a*c)*
b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*
c*e^8*log(F))*cos(2*e*x + 2*d)^2 + (F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(
a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(8*e*x + 8*d)^2 + 16*(F^(a*c)*b^7*c^7*e^2*log(F)^
7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(6
*e*x + 6*d)^2 + 36*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6
*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(4*e*x + 4*d)^2 - 48*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*
b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(4*e*x + 4*d)*sin(
2*e*x + 2*d) + 16*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*
log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(2*e*x + 2*d)^2 + 2*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^
5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F) - 4*(F^(a*c)*b^7*c^7*e^2
*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F
))*cos(6*e*x + 6*d) + 6*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^
3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(4*e*x + 4*d) - 4*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c
)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(2*e*x + 2*d))*c
os(8*e*x + 8*d) - 8*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^
6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F) + 6*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5
+ 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(4*e*x + 4*d) - 4*(F^(a*c)*b^7*c^7*e^2
*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F
))*cos(2*e*x + 2*d))*cos(6*e*x + 6*d) + 12*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 +
3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F) - 4*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*
c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(2*e*x + 2*d))*
cos(4*e*x + 4*d) - 8*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e
^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*cos(2*e*x + 2*d) - 4*(2*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c
)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(6*e*x + 6*d) -
3*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864
*F^(a*c)*b*c*e^8*log(F))*sin(4*e*x + 4*d) + 2*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5
+ 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(2*e*x + 2*d))*sin(8*e*x + 8*d) - 16*(
3*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5 + 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864
*F^(a*c)*b*c*e^8*log(F))*sin(4*e*x + 4*d) - 2*(F^(a*c)*b^7*c^7*e^2*log(F)^7 + 116*F^(a*c)*b^5*c^5*e^4*log(F)^5
+ 3904*F^(a*c)*b^3*c^3*e^6*log(F)^3 + 36864*F^(a*c)*b*c*e^8*log(F))*sin(2*e*x + 2*d))*sin(6*e*x + 6*d))*integ
rate(-((b^3*c^3*log(F)^3 - 104*b*c*e^2*log(F))*F^(b*c*x)*cos(10*e*x + 10*d) - 5*(b^3*c^3*log(F)^3 - 104*b*c*e^
2*log(F))*F^(b*c*x)*cos(8*e*x + 8*d) + 10*(b^3*c^3*log(F)^3 - 104*b*c*e^2*log(F))*F^(b*c*x)*cos(6*e*x + 6*d) -
10*(b^3*c^3*log(F)^3 - 104*b*c*e^2*log(F))*F^(b*c*x)*cos(4*e*x + 4*d) + 5*(b^3*c^3*log(F)^3 - 104*b*c*e^2*log
(F))*F^(b*c*x)*cos(2*e*x + 2*d) + 6*(3*b^2*c^2*e*log(F)^2 - 32*e^3)*F^(b*c*x)*sin(10*e*x + 10*d) - 30*(3*b^2*c
^2*e*log(F)^2 - 32*e^3)*F^(b*c*x)*sin(8*e*x + 8*d) + 60*(3*b^2*c^2*e*log(F)^2 - 32*e^3)*F^(b*c*x)*sin(6*e*x +
6*d) - 60*(3*b^2*c^2*e*log(F)^2 - 32*e^3)*F^(b*c*x)*sin(4*e*x + 4*d) + 30*(3*b^2*c^2*e*log(F)^2 - 32*e^3)*F^(b
*c*x)*sin(2*e*x + 2*d) - (b^3*c^3*log(F)^3 - 104*b*c*e^2*log(F))*F^(b*c*x))/(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^
2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 + (b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c
^2*e^4*log(F)^2 + 36864*e^6)*cos(10*e*x + 10*d)^2 + 25*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2
*c^2*e^4*log(F)^2 + 36864*e^6)*cos(8*e*x + 8*d)^2 + 100*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^
2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(6*e*x + 6*d)^2 + 100*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b
^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(4*e*x + 4*d)^2 + 25*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b
^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(2*e*x + 2*d)^2 + (b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*
c^2*e^4*log(F)^2 + 36864*e^6)*sin(10*e*x + 10*d)^2 + 25*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^
2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(8*e*x + 8*d)^2 + 100*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b
^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(6*e*x + 6*d)^2 + 100*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*
b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d)^2 - 100*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904
*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) + 25*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*
log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(2*e*x + 2*d)^2 - 2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*l
og(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 + 5*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^
2*e^4*log(F)^2 + 36864*e^6)*cos(8*e*x + 8*d) - 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*
e^4*log(F)^2 + 36864*e^6)*cos(6*e*x + 6*d) + 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^
4*log(F)^2 + 36864*e^6)*cos(4*e*x + 4*d) - 5*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*l
og(F)^2 + 36864*e^6)*cos(2*e*x + 2*d))*cos(10*e*x + 10*d) + 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 +
3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 - 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log
(F)^2 + 36864*e^6)*cos(6*e*x + 6*d) + 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F
)^2 + 36864*e^6)*cos(4*e*x + 4*d) - 5*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2
+ 36864*e^6)*cos(2*e*x + 2*d))*cos(8*e*x + 8*d) - 20*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*
c^2*e^4*log(F)^2 + 36864*e^6 + 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 3
6864*e^6)*cos(4*e*x + 4*d) - 5*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 3686
4*e^6)*cos(2*e*x + 2*d))*cos(6*e*x + 6*d) + 20*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4
*log(F)^2 + 36864*e^6 - 5*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6
)*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) - 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(
F)^2 + 36864*e^6)*cos(2*e*x + 2*d) - 10*((b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F
)^2 + 36864*e^6)*sin(8*e*x + 8*d) - 2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2
+ 36864*e^6)*sin(6*e*x + 6*d) + 2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 +
36864*e^6)*sin(4*e*x + 4*d) - (b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864
*e^6)*sin(2*e*x + 2*d))*sin(10*e*x + 10*d) - 50*(2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2
*e^4*log(F)^2 + 36864*e^6)*sin(6*e*x + 6*d) - 2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^
4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d) + (b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log
(F)^2 + 36864*e^6)*sin(2*e*x + 2*d))*sin(8*e*x + 8*d) - 100*(2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 +
3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d) - (b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*
b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(2*e*x + 2*d))*sin(6*e*x + 6*d)), x) - 8*(F^(a*c)*b^8*c^8*e*log(F)^8 + 11
6*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2 + (F^(
a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)
*b^2*c^2*e^7*log(F)^2)*cos(8*e*x + 8*d)^2 + 16*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6
+ 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*cos(6*e*x + 6*d)^2 + 36*(F^(a*c)*b^8
*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2
*e^7*log(F)^2)*cos(4*e*x + 4*d)^2 + 16*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F
^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*cos(2*e*x + 2*d)^2 + (F^(a*c)*b^8*c^8*e*log(
F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)
^2)*sin(8*e*x + 8*d)^2 + 16*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*
c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*sin(6*e*x + 6*d)^2 + 36*(F^(a*c)*b^8*c^8*e*log(F)^8 + 1
16*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*sin(
4*e*x + 4*d)^2 - 48*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*
log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) + 16*(F^(a*c)*b^8*c^8*e*log(F
)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^
2)*sin(2*e*x + 2*d)^2 + 2*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^
4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2 - 4*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*
log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*cos(6*e*x + 6*d) + 6*(F^(a*
c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b
^2*c^2*e^7*log(F)^2)*cos(4*e*x + 4*d) - 4*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 390
4*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*cos(2*e*x + 2*d))*cos(8*e*x + 8*d) - 8*(F
^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*
c)*b^2*c^2*e^7*log(F)^2 + 6*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*
c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*cos(4*e*x + 4*d) - 4*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*
F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*cos(2*e
*x + 2*d))*cos(6*e*x + 6*d) + 12*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)
*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2 - 4*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c
^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*cos(2*e*x + 2*d))*co
s(4*e*x + 4*d) - 8*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*l
og(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*cos(2*e*x + 2*d) - 4*(2*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c
)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*sin(6*e*x + 6
*d) - 3*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 3
6864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*sin(4*e*x + 4*d) + 2*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*
log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*sin(2*e*x + 2*d))*sin(8*e*x
+ 8*d) - 16*(3*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(
F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*sin(4*e*x + 4*d) - 2*(F^(a*c)*b^8*c^8*e*log(F)^8 + 116*F^(a*c)*b^6*
c^6*e^3*log(F)^6 + 3904*F^(a*c)*b^4*c^4*e^5*log(F)^4 + 36864*F^(a*c)*b^2*c^2*e^7*log(F)^2)*sin(2*e*x + 2*d))*s
in(6*e*x + 6*d))*integrate((6*(3*b^2*c^2*e*log(F)^2 - 32*e^3)*F^(b*c*x)*cos(10*e*x + 10*d) - 30*(3*b^2*c^2*e*l
og(F)^2 - 32*e^3)*F^(b*c*x)*cos(8*e*x + 8*d) + 60*(3*b^2*c^2*e*log(F)^2 - 32*e^3)*F^(b*c*x)*cos(6*e*x + 6*d) -
60*(3*b^2*c^2*e*log(F)^2 - 32*e^3)*F^(b*c*x)*cos(4*e*x + 4*d) + 30*(3*b^2*c^2*e*log(F)^2 - 32*e^3)*F^(b*c*x)*
cos(2*e*x + 2*d) - (b^3*c^3*log(F)^3 - 104*b*c*e^2*log(F))*F^(b*c*x)*sin(10*e*x + 10*d) + 5*(b^3*c^3*log(F)^3
- 104*b*c*e^2*log(F))*F^(b*c*x)*sin(8*e*x + 8*d) - 10*(b^3*c^3*log(F)^3 - 104*b*c*e^2*log(F))*F^(b*c*x)*sin(6*
e*x + 6*d) + 10*(b^3*c^3*log(F)^3 - 104*b*c*e^2*log(F))*F^(b*c*x)*sin(4*e*x + 4*d) - 5*(b^3*c^3*log(F)^3 - 104
*b*c*e^2*log(F))*F^(b*c*x)*sin(2*e*x + 2*d) - 6*(3*b^2*c^2*e*log(F)^2 - 32*e^3)*F^(b*c*x))/(b^6*c^6*log(F)^6 +
116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 + (b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)
^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(10*e*x + 10*d)^2 + 25*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(
F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(8*e*x + 8*d)^2 + 100*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log
(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(6*e*x + 6*d)^2 + 100*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*lo
g(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(4*e*x + 4*d)^2 + 25*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*lo
g(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(2*e*x + 2*d)^2 + (b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F
)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(10*e*x + 10*d)^2 + 25*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log
(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(8*e*x + 8*d)^2 + 100*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*lo
g(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(6*e*x + 6*d)^2 + 100*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*l
og(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d)^2 - 100*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*
log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) + 25*(b^6*c^6*log(F)^6 + 1
16*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(2*e*x + 2*d)^2 - 2*(b^6*c^6*log(F)^6 + 11
6*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 + 5*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^
4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(8*e*x + 8*d) - 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4
+ 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(6*e*x + 6*d) + 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 +
3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(4*e*x + 4*d) - 5*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 390
4*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(2*e*x + 2*d))*cos(10*e*x + 10*d) + 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*
e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 - 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*
b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(6*e*x + 6*d) + 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^
2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(4*e*x + 4*d) - 5*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c
^2*e^4*log(F)^2 + 36864*e^6)*cos(2*e*x + 2*d))*cos(8*e*x + 8*d) - 20*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F
)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 + 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e
^4*log(F)^2 + 36864*e^6)*cos(4*e*x + 4*d) - 5*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*
log(F)^2 + 36864*e^6)*cos(2*e*x + 2*d))*cos(6*e*x + 6*d) + 20*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3
904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 - 5*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F
)^2 + 36864*e^6)*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) - 10*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b
^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(2*e*x + 2*d) - 10*((b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^
2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(8*e*x + 8*d) - 2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c
^2*e^4*log(F)^2 + 36864*e^6)*sin(6*e*x + 6*d) + 2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*
e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d) - (b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*l
og(F)^2 + 36864*e^6)*sin(2*e*x + 2*d))*sin(10*e*x + 10*d) - 50*(2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4
+ 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(6*e*x + 6*d) - 2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 +
3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d) + (b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*
b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(2*e*x + 2*d))*sin(8*e*x + 8*d) - 100*(2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*
e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d) - (b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*l
og(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(2*e*x + 2*d))*sin(6*e*x + 6*d)), x) + (4*(F^(a*c)*b^4*c^4
*e*log(F)^4 + 100*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 2304*F^(a*c)*e^5)*F^(b*c*x)*cos(4*e*x + 4*d) + 8*(F^(a*c)*b^4
*c^4*e*log(F)^4 + 40*F^(a*c)*b^2*c^2*e^3*log(F)^2 - 1536*F^(a*c)*e^5)*F^(b*c*x)*cos(2*e*x + 2*d) + (F^(a*c)*b^
5*c^5*log(F)^5 + 100*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 2304*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*sin(4*e*x + 4*d) -
80*(F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*sin(2*e*x + 2*d) + 8*(F^(a*c)*b^4*c^4*
e*log(F)^4 - 140*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 384*F^(a*c)*e^5)*F^(b*c*x))*sin(8*e*x + 8*d) - 4*(4*(F^(a*c)*b
^4*c^4*e*log(F)^4 + 100*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 2304*F^(a*c)*e^5)*F^(b*c*x)*cos(4*e*x + 4*d) + 8*(F^(a*
c)*b^4*c^4*e*log(F)^4 + 40*F^(a*c)*b^2*c^2*e^3*log(F)^2 - 1536*F^(a*c)*e^5)*F^(b*c*x)*cos(2*e*x + 2*d) + (F^(a
*c)*b^5*c^5*log(F)^5 + 100*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 2304*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*sin(4*e*x + 4
*d) - 80*(F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*sin(2*e*x + 2*d) + 8*(F^(a*c)*b^
4*c^4*e*log(F)^4 - 140*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 384*F^(a*c)*e^5)*F^(b*c*x))*sin(6*e*x + 6*d) + 4*(16*(F^
(a*c)*b^4*c^4*e*log(F)^4 + 55*F^(a*c)*b^2*c^2*e^3*log(F)^2 - 576*F^(a*c)*e^5)*F^(b*c*x)*cos(2*e*x + 2*d) - (F^
(a*c)*b^5*c^5*log(F)^5 + 220*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 9984*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*sin(2*e*x +
2*d) + (11*F^(a*c)*b^4*c^4*e*log(F)^4 - 1780*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 2304*F^(a*c)*e^5)*F^(b*c*x))*sin(
4*e*x + 4*d))/(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 + (b^6*c^6*
log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(8*e*x + 8*d)^2 + 16*(b^6*c^6*
log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(6*e*x + 6*d)^2 + 36*(b^6*c^6*
log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(4*e*x + 4*d)^2 + 16*(b^6*c^6*
log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(2*e*x + 2*d)^2 + (b^6*c^6*log
(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(8*e*x + 8*d)^2 + 16*(b^6*c^6*log
(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(6*e*x + 6*d)^2 + 36*(b^6*c^6*log
(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d)^2 - 48*(b^6*c^6*log
(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) +
16*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(2*e*x + 2*d)^2 +
2*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 - 4*(b^6*c^6*log(F)^6 +
116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(6*e*x + 6*d) + 6*(b^6*c^6*log(F)^6 + 11
6*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(4*e*x + 4*d) - 4*(b^6*c^6*log(F)^6 + 116*b
^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(2*e*x + 2*d))*cos(8*e*x + 8*d) - 8*(b^6*c^6*l
og(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 + 6*(b^6*c^6*log(F)^6 + 116*b^4*c^4
*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(4*e*x + 4*d) - 4*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^
2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(2*e*x + 2*d))*cos(6*e*x + 6*d) + 12*(b^6*c^6*log(F)^6
+ 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6 - 4*(b^6*c^6*log(F)^6 + 116*b^4*c^4*e^2*log
(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) - 8*(b^6*c^6*log(F)^6 + 116*
b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*cos(2*e*x + 2*d) - 4*(2*(b^6*c^6*log(F)^6 + 116*
b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(6*e*x + 6*d) - 3*(b^6*c^6*log(F)^6 + 116*b^4
*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d) + 2*(b^6*c^6*log(F)^6 + 116*b^4*c^
4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(2*e*x + 2*d))*sin(8*e*x + 8*d) - 16*(3*(b^6*c^6*lo
g(F)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(4*e*x + 4*d) - 2*(b^6*c^6*log(F
)^6 + 116*b^4*c^4*e^2*log(F)^4 + 3904*b^2*c^2*e^4*log(F)^2 + 36864*e^6)*sin(2*e*x + 2*d))*sin(6*e*x + 6*d))

Giac [F]

\[ \int F^{c (a+b x)} \csc ^4(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{4} \,d x } \]

[In]

integrate(F^(c*(b*x+a))*csc(e*x+d)^4,x, algorithm="giac")

[Out]

integrate(F^((b*x + a)*c)*csc(e*x + d)^4, x)

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \csc ^4(d+e x) \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\sin \left (d+e\,x\right )}^4} \,d x \]

[In]

int(F^(c*(a + b*x))/sin(d + e*x)^4,x)

[Out]

int(F^(c*(a + b*x))/sin(d + e*x)^4, x)