\(\int \frac {\csc (c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\) [229]

Optimal result

Integrand size = 22, antiderivative size = 617 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {b \cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]

[Out]

-arctanh(cos(d*x+c))/a^3/d-1/8*b*cos(d*x+c)*(2-cos(d*x+c)^2)/a/(a-b)/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)^2
-1/4*b*cos(d*x+c)*(2-cos(d*x+c)^2)/a^2/(a-b)/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)-1/32*b*cos(d*x+c)*(11*a+b
-(5*a+b)*cos(d*x+c)^2)/a^2/(a-b)^2/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)-1/64*b^(1/4)*arctan(b^(1/4)*cos(d*x
+c)/(a^(1/2)-b^(1/2))^(1/2))*(5*a^(1/2)-2*b^(1/2))/a^(5/2)/d/(a^(1/2)-b^(1/2))^(5/2)-1/8*b^(1/4)*arctan(b^(1/4
)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^(5/2)/d/(a^(1/2)-b^(1/2))^(3/2)+1/8*b^(1/4)*arctanh(b^(1/4)*cos(d*x+c)
/(a^(1/2)+b^(1/2))^(1/2))/a^(5/2)/d/(a^(1/2)+b^(1/2))^(3/2)+1/64*b^(1/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b
^(1/2))^(1/2))*(5*a^(1/2)+2*b^(1/2))/a^(5/2)/d/(a^(1/2)+b^(1/2))^(5/2)-1/2*b^(1/4)*arctan(b^(1/4)*cos(d*x+c)/(
a^(1/2)-b^(1/2))^(1/2))/a^3/d/(a^(1/2)-b^(1/2))^(1/2)+1/2*b^(1/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))
^(1/2))/a^3/d/(a^(1/2)+b^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {3294, 1252, 213, 1192, 1180, 211, 214} \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \left (5 \sqrt {a}-2 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\sqrt [4]{b} \left (5 \sqrt {a}+2 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {b \cos (c+d x) \left (-\left ((5 a+b) \cos ^2(c+d x)\right )+11 a+b\right )}{32 a^2 d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]

[In]

Int[Csc[c + d*x]/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

-1/64*((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(a^(5/2)*(Sqrt[
a] - Sqrt[b])^(5/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a]
- Sqrt[b])^(3/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] - S
qrt[b]]*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])
/(8*a^(5/2)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(
2*a^3*Sqrt[Sqrt[a] + Sqrt[b]]*d) + ((5*Sqrt[a] + 2*Sqrt[b])*b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a
] + Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*a*(a - b)*d
*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*a^2*(a - b)*d*(
a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)) - (b*Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^2))/(32*a
^2*(a - b)^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1252

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d
+ e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((Intege
rQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])

Rule 3294

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {1}{a^3 \left (-1+x^2\right )}+\frac {b-b x^2}{a \left (a-b+2 b x^2-b x^4\right )^3}+\frac {b-b x^2}{a^2 \left (a-b+2 b x^2-b x^4\right )^2}+\frac {b-b x^2}{a^3 \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cos (c+d x)\right )}{a d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {-4 a b^2+2 a b^2 x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{8 a^3 (a-b) b d}+\frac {\text {Subst}\left (\int \frac {-12 a b^2+10 a b^2 x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{16 a^2 (a-b) b d}+\frac {b \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^3 d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^3 d} \\ & = -\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {b \cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {4 a (13 a-b) b^3-4 a b^3 (5 a+b) x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{128 a^3 (a-b)^2 b^2 d}+\frac {b \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right ) d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right ) d} \\ & = -\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {b \cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac {\left (\left (5 \sqrt {a}-2 \sqrt {b}\right ) b\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^2 d}+\frac {\left (\left (5 \sqrt {a}+2 \sqrt {b}\right ) b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^2 d} \\ & = -\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {b \cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 10.95 (sec) , antiderivative size = 920, normalized size of antiderivative = 1.49 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {\frac {32 a b \cos (c+d x) (-41 a+23 b+(13 a-7 b) \cos (2 (c+d x)))}{(a-b)^2 (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}+\frac {512 a^2 b (-5 \cos (c+d x)+\cos (3 (c+d x)))}{(a-b) (-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}-256 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+256 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {i b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-90 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+142 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-64 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+45 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-71 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+32 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+398 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-506 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+192 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-199 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+253 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-96 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-398 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+506 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-192 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+199 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-253 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+96 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+90 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-142 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6+64 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-45 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6+71 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6-32 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{(a-b)^2}}{256 a^3 d} \]

[In]

Integrate[Csc[c + d*x]/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

((32*a*b*Cos[c + d*x]*(-41*a + 23*b + (13*a - 7*b)*Cos[2*(c + d*x)]))/((a - b)^2*(8*a - 3*b + 4*b*Cos[2*(c + d
*x)] - b*Cos[4*(c + d*x)])) + (512*a^2*b*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/((a - b)*(-8*a + 3*b - 4*b*Cos[
2*(c + d*x)] + b*Cos[4*(c + d*x)])^2) - 256*Log[Cos[(c + d*x)/2]] + 256*Log[Sin[(c + d*x)/2]] - (I*b*RootSum[b
- 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-90*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] +
142*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - 64*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + (45*I)*a
^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (71*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (32*I)*b^2*Log[1 - 2*Cos
[c + d*x]*#1 + #1^2] + 398*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 506*a*b*ArcTan[Sin[c + d*x]/(Co
s[c + d*x] - #1)]*#1^2 + 192*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (199*I)*a^2*Log[1 - 2*Cos[c +
d*x]*#1 + #1^2]*#1^2 + (253*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (96*I)*b^2*Log[1 - 2*Cos[c + d*x]
*#1 + #1^2]*#1^2 - 398*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 506*a*b*ArcTan[Sin[c + d*x]/(Cos[c
+ d*x] - #1)]*#1^4 - 192*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (199*I)*a^2*Log[1 - 2*Cos[c + d*x
]*#1 + #1^2]*#1^4 - (253*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (96*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1
+ #1^2]*#1^4 + 90*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 142*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x
] - #1)]*#1^6 + 64*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - (45*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 +
#1^2]*#1^6 + (71*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 - (32*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*
#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b)^2)/(256*a^3*d)

Maple [A] (verified)

Time = 7.50 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.62

method	result	size
derivativedivides	\(\frac {\frac {b \left (\frac {-\frac {a b \left (13 a -7 b \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (53 a -29 b \right ) a b \left (\cos ^{5}\left (d x +c \right )\right )}{32 a^{2}-64 a b +32 b^{2}}+\frac {a \left (17 a^{2}-78 a b +37 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{32 a^{2}-64 a b +32 b^{2}}-\frac {5 \left (7 a -3 b \right ) a \cos \left (d x +c \right )}{32 \left (a -b \right )}}{{\left (a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )\right )}^{2}}+\frac {b \left (\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}-16 a^{2} b +10 a \,b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}+16 a^{2} b -10 a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{32 a^{2}-64 a b +32 b^{2}}\right )}{a^{3}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a^{3}}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{3}}}{d}\)	\(384\)
default	\(\frac {\frac {b \left (\frac {-\frac {a b \left (13 a -7 b \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (53 a -29 b \right ) a b \left (\cos ^{5}\left (d x +c \right )\right )}{32 a^{2}-64 a b +32 b^{2}}+\frac {a \left (17 a^{2}-78 a b +37 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{32 a^{2}-64 a b +32 b^{2}}-\frac {5 \left (7 a -3 b \right ) a \cos \left (d x +c \right )}{32 \left (a -b \right )}}{{\left (a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )\right )}^{2}}+\frac {b \left (\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}-16 a^{2} b +10 a \,b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}+16 a^{2} b -10 a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{32 a^{2}-64 a b +32 b^{2}}\right )}{a^{3}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a^{3}}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{3}}}{d}\)	\(384\)
risch	\(\text {Expression too large to display}\)	\(1530\)

[In]

int(csc(d*x+c)/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(b/a^3*((-1/32*a*b*(13*a-7*b)/(a^2-2*a*b+b^2)*cos(d*x+c)^7+1/32*(53*a-29*b)*a*b/(a^2-2*a*b+b^2)*cos(d*x+c)
^5+1/32*a*(17*a^2-78*a*b+37*b^2)/(a^2-2*a*b+b^2)*cos(d*x+c)^3-5/32*(7*a-3*b)*a/(a-b)*cos(d*x+c))/(a-b+2*b*cos(
d*x+c)^2-b*cos(d*x+c)^4)^2+1/32/(a^2-2*a*b+b^2)*b*(1/2*(-45*a^2*(a*b)^(1/2)+71*a*b*(a*b)^(1/2)-32*b^2*(a*b)^(1
/2)-16*a^2*b+10*a*b^2)/(a*b)^(1/2)/b/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))-
1/2*(-45*a^2*(a*b)^(1/2)+71*a*b*(a*b)^(1/2)-32*b^2*(a*b)^(1/2)+16*a^2*b-10*a*b^2)/(a*b)^(1/2)/b/(((a*b)^(1/2)+
b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))))-1/2/a^3*ln(1+cos(d*x+c))+1/2/a^3*ln(cos(d*x+c)-1
))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5020 vs. \(2 (481) = 962\).

Time = 2.20 (sec) , antiderivative size = 5020, normalized size of antiderivative = 8.14 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)/(a-b*sin(d*x+c)**4)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\csc \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]

[In]

integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/16*(8*(13*a^2*b^4 - 7*a*b^5)*cos(2*d*x + 2*c)*cos(d*x + c) - 8*(121*a^2*b^4 - 67*a*b^5)*sin(3*d*x + 3*c)*sin
(2*d*x + 2*c) + 8*(13*a^2*b^4 - 7*a*b^5)*sin(2*d*x + 2*c)*sin(d*x + c) - ((13*a^2*b^4 - 7*a*b^5)*cos(15*d*x +
15*c) - (121*a^2*b^4 - 67*a*b^5)*cos(13*d*x + 13*c) - (272*a^3*b^3 - 461*a^2*b^4 + 159*a*b^5)*cos(11*d*x + 11*
c) + (1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5)*cos(9*d*x + 9*c) + (1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5)*cos(
7*d*x + 7*c) - (272*a^3*b^3 - 461*a^2*b^4 + 159*a*b^5)*cos(5*d*x + 5*c) - (121*a^2*b^4 - 67*a*b^5)*cos(3*d*x +
3*c) + (13*a^2*b^4 - 7*a*b^5)*cos(d*x + c))*cos(16*d*x + 16*c) - (13*a^2*b^4 - 7*a*b^5 - 8*(13*a^2*b^4 - 7*a*
b^5)*cos(14*d*x + 14*c) - 4*(104*a^3*b^3 - 147*a^2*b^4 + 49*a*b^5)*cos(12*d*x + 12*c) + 8*(208*a^3*b^3 - 203*a
^2*b^4 + 49*a*b^5)*cos(10*d*x + 10*c) + 2*(1664*a^4*b^2 - 2144*a^3*b^3 + 1127*a^2*b^4 - 245*a*b^5)*cos(8*d*x +
8*c) + 8*(208*a^3*b^3 - 203*a^2*b^4 + 49*a*b^5)*cos(6*d*x + 6*c) - 4*(104*a^3*b^3 - 147*a^2*b^4 + 49*a*b^5)*c
os(4*d*x + 4*c) - 8*(13*a^2*b^4 - 7*a*b^5)*cos(2*d*x + 2*c))*cos(15*d*x + 15*c) - 8*((121*a^2*b^4 - 67*a*b^5)*
cos(13*d*x + 13*c) + (272*a^3*b^3 - 461*a^2*b^4 + 159*a*b^5)*cos(11*d*x + 11*c) - (1424*a^3*b^3 - 1121*a^2*b^4
+ 99*a*b^5)*cos(9*d*x + 9*c) - (1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5)*cos(7*d*x + 7*c) + (272*a^3*b^3 - 461
*a^2*b^4 + 159*a*b^5)*cos(5*d*x + 5*c) + (121*a^2*b^4 - 67*a*b^5)*cos(3*d*x + 3*c) - (13*a^2*b^4 - 7*a*b^5)*co
s(d*x + c))*cos(14*d*x + 14*c) + (121*a^2*b^4 - 67*a*b^5 - 4*(968*a^3*b^3 - 1383*a^2*b^4 + 469*a*b^5)*cos(12*d
*x + 12*c) + 8*(1936*a^3*b^3 - 1919*a^2*b^4 + 469*a*b^5)*cos(10*d*x + 10*c) + 2*(15488*a^4*b^2 - 20192*a^3*b^3
+ 10667*a^2*b^4 - 2345*a*b^5)*cos(8*d*x + 8*c) + 8*(1936*a^3*b^3 - 1919*a^2*b^4 + 469*a*b^5)*cos(6*d*x + 6*c)
- 4*(968*a^3*b^3 - 1383*a^2*b^4 + 469*a*b^5)*cos(4*d*x + 4*c) - 8*(121*a^2*b^4 - 67*a*b^5)*cos(2*d*x + 2*c))*
cos(13*d*x + 13*c) - 4*((2176*a^4*b^2 - 5592*a^3*b^3 + 4499*a^2*b^4 - 1113*a*b^5)*cos(11*d*x + 11*c) - (11392*
a^4*b^2 - 18936*a^3*b^3 + 8639*a^2*b^4 - 693*a*b^5)*cos(9*d*x + 9*c) - (11392*a^4*b^2 - 18936*a^3*b^3 + 8639*a
^2*b^4 - 693*a*b^5)*cos(7*d*x + 7*c) + (2176*a^4*b^2 - 5592*a^3*b^3 + 4499*a^2*b^4 - 1113*a*b^5)*cos(5*d*x + 5
*c) + (968*a^3*b^3 - 1383*a^2*b^4 + 469*a*b^5)*cos(3*d*x + 3*c) - (104*a^3*b^3 - 147*a^2*b^4 + 49*a*b^5)*cos(d
*x + c))*cos(12*d*x + 12*c) + (272*a^3*b^3 - 461*a^2*b^4 + 159*a*b^5 + 8*(4352*a^4*b^2 - 9280*a^3*b^3 + 5771*a
^2*b^4 - 1113*a*b^5)*cos(10*d*x + 10*c) + 2*(34816*a^5*b - 85120*a^4*b^2 + 74128*a^3*b^3 - 31399*a^2*b^4 + 556
5*a*b^5)*cos(8*d*x + 8*c) + 8*(4352*a^4*b^2 - 9280*a^3*b^3 + 5771*a^2*b^4 - 1113*a*b^5)*cos(6*d*x + 6*c) - 4*(
2176*a^4*b^2 - 5592*a^3*b^3 + 4499*a^2*b^4 - 1113*a*b^5)*cos(4*d*x + 4*c) - 8*(272*a^3*b^3 - 461*a^2*b^4 + 159
*a*b^5)*cos(2*d*x + 2*c))*cos(11*d*x + 11*c) - 8*((22784*a^4*b^2 - 27904*a^3*b^3 + 9431*a^2*b^4 - 693*a*b^5)*c
os(9*d*x + 9*c) + (22784*a^4*b^2 - 27904*a^3*b^3 + 9431*a^2*b^4 - 693*a*b^5)*cos(7*d*x + 7*c) - (4352*a^4*b^2
- 9280*a^3*b^3 + 5771*a^2*b^4 - 1113*a*b^5)*cos(5*d*x + 5*c) - (1936*a^3*b^3 - 1919*a^2*b^4 + 469*a*b^5)*cos(3
*d*x + 3*c) + (208*a^3*b^3 - 203*a^2*b^4 + 49*a*b^5)*cos(d*x + c))*cos(10*d*x + 10*c) - (1424*a^3*b^3 - 1121*a
^2*b^4 + 99*a*b^5 + 2*(182272*a^5*b - 280192*a^4*b^2 + 170128*a^3*b^3 - 48739*a^2*b^4 + 3465*a*b^5)*cos(8*d*x
+ 8*c) + 8*(22784*a^4*b^2 - 27904*a^3*b^3 + 9431*a^2*b^4 - 693*a*b^5)*cos(6*d*x + 6*c) - 4*(11392*a^4*b^2 - 18
936*a^3*b^3 + 8639*a^2*b^4 - 693*a*b^5)*cos(4*d*x + 4*c) - 8*(1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5)*cos(2*d*
x + 2*c))*cos(9*d*x + 9*c) - 2*((182272*a^5*b - 280192*a^4*b^2 + 170128*a^3*b^3 - 48739*a^2*b^4 + 3465*a*b^5)*
cos(7*d*x + 7*c) - (34816*a^5*b - 85120*a^4*b^2 + 74128*a^3*b^3 - 31399*a^2*b^4 + 5565*a*b^5)*cos(5*d*x + 5*c)
- (15488*a^4*b^2 - 20192*a^3*b^3 + 10667*a^2*b^4 - 2345*a*b^5)*cos(3*d*x + 3*c) + (1664*a^4*b^2 - 2144*a^3*b^
3 + 1127*a^2*b^4 - 245*a*b^5)*cos(d*x + c))*cos(8*d*x + 8*c) - (1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5 + 8*(22
784*a^4*b^2 - 27904*a^3*b^3 + 9431*a^2*b^4 - 693*a*b^5)*cos(6*d*x + 6*c) - 4*(11392*a^4*b^2 - 18936*a^3*b^3 +
8639*a^2*b^4 - 693*a*b^5)*cos(4*d*x + 4*c) - 8*(1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5)*cos(2*d*x + 2*c))*cos(
7*d*x + 7*c) + 8*((4352*a^4*b^2 - 9280*a^3*b^3 + 5771*a^2*b^4 - 1113*a*b^5)*cos(5*d*x + 5*c) + (1936*a^3*b^3 -
1919*a^2*b^4 + 469*a*b^5)*cos(3*d*x + 3*c) - (208*a^3*b^3 - 203*a^2*b^4 + 49*a*b^5)*cos(d*x + c))*cos(6*d*x +
6*c) + (272*a^3*b^3 - 461*a^2*b^4 + 159*a*b^5 - 4*(2176*a^4*b^2 - 5592*a^3*b^3 + 4499*a^2*b^4 - 1113*a*b^5)*c
os(4*d*x + 4*c) - 8*(272*a^3*b^3 - 461*a^2*b^4 + 159*a*b^5)*cos(2*d*x + 2*c))*cos(5*d*x + 5*c) - 4*((968*a^3*b
^3 - 1383*a^2*b^4 + 469*a*b^5)*cos(3*d*x + 3*c) - (104*a^3*b^3 - 147*a^2*b^4 + 49*a*b^5)*cos(d*x + c))*cos(4*d
*x + 4*c) + (121*a^2*b^4 - 67*a*b^5 - 8*(121*a^2*b^4 - 67*a*b^5)*cos(2*d*x + 2*c))*cos(3*d*x + 3*c) - (13*a^2*
b^4 - 7*a*b^5)*cos(d*x + c) - 16*((a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(16*d*x + 16*c)^2 + 64*(a^5*b^4 - 2*a^4
*b^5 + a^3*b^6)*d*cos(14*d*x + 14*c)^2 + 16*(64*a^7*b^2 - 240*a^6*b^3 + 337*a^5*b^4 - 210*a^4*b^5 + 49*a^3*b^6
)*d*cos(12*d*x + 12*c)^2 + 64*(256*a^7*b^2 - 736*a^6*b^3 + 753*a^5*b^4 - 322*a^4*b^5 + 49*a^3*b^6)*d*cos(10*d*
x + 10*c)^2 + 4*(16384*a^9 - 57344*a^8*b + 83712*a^7*b^2 - 67648*a^6*b^3 + 32841*a^5*b^4 - 9170*a^4*b^5 + 1225
*a^3*b^6)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^7*b^2 - 736*a^6*b^3 + 753*a^5*b^4 - 322*a^4*b^5 + 49*a^3*b^6)*d*cos
(6*d*x + 6*c)^2 + 16*(64*a^7*b^2 - 240*a^6*b^3 + 337*a^5*b^4 - 210*a^4*b^5 + 49*a^3*b^6)*d*cos(4*d*x + 4*c)^2
+ 64*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(2*d*x + 2*c)^2 + (a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*sin(16*d*x + 16*
c)^2 + 64*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*sin(14*d*x + 14*c)^2 + 16*(64*a^7*b^2 - 240*a^6*b^3 + 337*a^5*b^4
- 210*a^4*b^5 + 49*a^3*b^6)*d*sin(12*d*x + 12*c)^2 + 64*(256*a^7*b^2 - 736*a^6*b^3 + 753*a^5*b^4 - 322*a^4*b^5
+ 49*a^3*b^6)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^9 - 57344*a^8*b + 83712*a^7*b^2 - 67648*a^6*b^3 + 32841*a^5
*b^4 - 9170*a^4*b^5 + 1225*a^3*b^6)*d*sin(8*d*x + 8*c)^2 + 64*(256*a^7*b^2 - 736*a^6*b^3 + 753*a^5*b^4 - 322*a
^4*b^5 + 49*a^3*b^6)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^7*b^2 - 240*a^6*b^3 + 337*a^5*b^4 - 210*a^4*b^5 + 49*a^3*
b^6)*d*sin(4*d*x + 4*c)^2 + 64*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*sin(4*d*x + 4*c)*sin(2*d*x
+ 2*c) + 64*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*sin(2*d*x + 2*c)^2 - 16*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(2*
d*x + 2*c) + (a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d - 2*(8*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(14*d*x + 14*c) + 4
*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*
b^5 - 7*a^3*b^6)*d*cos(10*d*x + 10*c) - 2*(128*a^7*b^2 - 352*a^6*b^3 + 355*a^5*b^4 - 166*a^4*b^5 + 35*a^3*b^6)
*d*cos(8*d*x + 8*c) - 8*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^6*b^3 -
23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*cos(4*d*x + 4*c) + 8*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(2*d*x + 2*c
) - (a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d)*cos(16*d*x + 16*c) + 16*(4*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3
*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*cos(10*d*x + 10*c) - 2*(12
8*a^7*b^2 - 352*a^6*b^3 + 355*a^5*b^4 - 166*a^4*b^5 + 35*a^3*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^6*b^3 - 39*a^5*
b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*cos(4
*d*x + 4*c) + 8*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(2*d*x + 2*c) - (a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d)*cos(14
*d*x + 14*c) - 8*(8*(128*a^7*b^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*cos(10*d*x + 10*c)
+ 2*(1024*a^8*b - 3712*a^7*b^2 + 5304*a^6*b^3 - 3813*a^5*b^4 + 1442*a^4*b^5 - 245*a^3*b^6)*d*cos(8*d*x + 8*c)
+ 8*(128*a^7*b^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*cos(6*d*x + 6*c) - 4*(64*a^7*b^2 -
240*a^6*b^3 + 337*a^5*b^4 - 210*a^4*b^5 + 49*a^3*b^6)*d*cos(4*d*x + 4*c) - 8*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*
b^5 - 7*a^3*b^6)*d*cos(2*d*x + 2*c) + (8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d)*cos(12*d*x + 12*c)
+ 16*(2*(2048*a^8*b - 6528*a^7*b^2 + 8144*a^6*b^3 - 5141*a^5*b^4 + 1722*a^4*b^5 - 245*a^3*b^6)*d*cos(8*d*x + 8
*c) + 8*(256*a^7*b^2 - 736*a^6*b^3 + 753*a^5*b^4 - 322*a^4*b^5 + 49*a^3*b^6)*d*cos(6*d*x + 6*c) - 4*(128*a^7*b
^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*cos(4*d*x + 4*c) - 8*(16*a^6*b^3 - 39*a^5*b^4 + 3
0*a^4*b^5 - 7*a^3*b^6)*d*cos(2*d*x + 2*c) + (16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d)*cos(10*d*x +
10*c) + 4*(8*(2048*a^8*b - 6528*a^7*b^2 + 8144*a^6*b^3 - 5141*a^5*b^4 + 1722*a^4*b^5 - 245*a^3*b^6)*d*cos(6*d
*x + 6*c) - 4*(1024*a^8*b - 3712*a^7*b^2 + 5304*a^6*b^3 - 3813*a^5*b^4 + 1442*a^4*b^5 - 245*a^3*b^6)*d*cos(4*d
*x + 4*c) - 8*(128*a^7*b^2 - 352*a^6*b^3 + 355*a^5*b^4 - 166*a^4*b^5 + 35*a^3*b^6)*d*cos(2*d*x + 2*c) + (128*a
^7*b^2 - 352*a^6*b^3 + 355*a^5*b^4 - 166*a^4*b^5 + 35*a^3*b^6)*d)*cos(8*d*x + 8*c) - 16*(4*(128*a^7*b^2 - 424*
a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*cos(4*d*x + 4*c) + 8*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5
- 7*a^3*b^6)*d*cos(2*d*x + 2*c) - (16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d)*cos(6*d*x + 6*c) + 8*
(8*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*cos(2*d*x + 2*c) - (8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5
- 7*a^3*b^6)*d)*cos(4*d*x + 4*c) - 4*(4*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*sin(14*d*x + 14*c) + 2*(8*a^6*b^3 -
23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*sin(12*d*x + 12*c) - 4*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b
^6)*d*sin(10*d*x + 10*c) - (128*a^7*b^2 - 352*a^6*b^3 + 355*a^5*b^4 - 166*a^4*b^5 + 35*a^3*b^6)*d*sin(8*d*x +
8*c) - 4*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*sin(6*d*x + 6*c) + 2*(8*a^6*b^3 - 23*a^5*b^4 + 2
2*a^4*b^5 - 7*a^3*b^6)*d*sin(4*d*x + 4*c) + 4*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*sin(2*d*x + 2*c))*sin(16*d*x +
16*c) + 32*(2*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*sin(12*d*x + 12*c) - 4*(16*a^6*b^3 - 39*a^5
*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*sin(10*d*x + 10*c) - (128*a^7*b^2 - 352*a^6*b^3 + 355*a^5*b^4 - 166*a^4*b^5 +
35*a^3*b^6)*d*sin(8*d*x + 8*c) - 4*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*sin(6*d*x + 6*c) + 2*
(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*sin(4*d*x + 4*c) + 4*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*sin
(2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*(128*a^7*b^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)
*d*sin(10*d*x + 10*c) + (1024*a^8*b - 3712*a^7*b^2 + 5304*a^6*b^3 - 3813*a^5*b^4 + 1442*a^4*b^5 - 245*a^3*b^6)
*d*sin(8*d*x + 8*c) + 4*(128*a^7*b^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*sin(6*d*x + 6*c
) - 2*(64*a^7*b^2 - 240*a^6*b^3 + 337*a^5*b^4 - 210*a^4*b^5 + 49*a^3*b^6)*d*sin(4*d*x + 4*c) - 4*(8*a^6*b^3 -
23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 32*((2048*a^8*b - 6528*a^7*b^2 +
8144*a^6*b^3 - 5141*a^5*b^4 + 1722*a^4*b^5 - 245*a^3*b^6)*d*sin(8*d*x + 8*c) + 4*(256*a^7*b^2 - 736*a^6*b^3 +
753*a^5*b^4 - 322*a^4*b^5 + 49*a^3*b^6)*d*sin(6*d*x + 6*c) - 2*(128*a^7*b^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266
*a^4*b^5 + 49*a^3*b^6)*d*sin(4*d*x + 4*c) - 4*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*sin(2*d*x +
2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^8*b - 6528*a^7*b^2 + 8144*a^6*b^3 - 5141*a^5*b^4 + 1722*a^4*b^5 - 24
5*a^3*b^6)*d*sin(6*d*x + 6*c) - (1024*a^8*b - 3712*a^7*b^2 + 5304*a^6*b^3 - 3813*a^5*b^4 + 1442*a^4*b^5 - 245*
a^3*b^6)*d*sin(4*d*x + 4*c) - 2*(128*a^7*b^2 - 352*a^6*b^3 + 355*a^5*b^4 - 166*a^4*b^5 + 35*a^3*b^6)*d*sin(2*d
*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^7*b^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*sin(4
*d*x + 4*c) + 2*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integ
rate(-1/16*(4*(199*a^2*b^2 - 253*a*b^3 + 96*b^4)*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) - 4*(45*a^2*b^2 - 71*a*b^3
+ 32*b^4)*cos(d*x + c)*sin(2*d*x + 2*c) + 4*(45*a^2*b^2 - 71*a*b^3 + 32*b^4)*cos(2*d*x + 2*c)*sin(d*x + c) + (
(45*a^2*b^2 - 71*a*b^3 + 32*b^4)*sin(7*d*x + 7*c) - (199*a^2*b^2 - 253*a*b^3 + 96*b^4)*sin(5*d*x + 5*c) + (199
*a^2*b^2 - 253*a*b^3 + 96*b^4)*sin(3*d*x + 3*c) - (45*a^2*b^2 - 71*a*b^3 + 32*b^4)*sin(d*x + c))*cos(8*d*x + 8
*c) + 2*(2*(45*a^2*b^2 - 71*a*b^3 + 32*b^4)*sin(6*d*x + 6*c) + (360*a^3*b - 703*a^2*b^2 + 469*a*b^3 - 96*b^4)*
sin(4*d*x + 4*c) + 2*(45*a^2*b^2 - 71*a*b^3 + 32*b^4)*sin(2*d*x + 2*c))*cos(7*d*x + 7*c) + 4*((199*a^2*b^2 - 2
53*a*b^3 + 96*b^4)*sin(5*d*x + 5*c) - (199*a^2*b^2 - 253*a*b^3 + 96*b^4)*sin(3*d*x + 3*c) + (45*a^2*b^2 - 71*a
*b^3 + 32*b^4)*sin(d*x + c))*cos(6*d*x + 6*c) - 2*((1592*a^3*b - 2621*a^2*b^2 + 1527*a*b^3 - 288*b^4)*sin(4*d*
x + 4*c) + 2*(199*a^2*b^2 - 253*a*b^3 + 96*b^4)*sin(2*d*x + 2*c))*cos(5*d*x + 5*c) - 2*((1592*a^3*b - 2621*a^2
*b^2 + 1527*a*b^3 - 288*b^4)*sin(3*d*x + 3*c) - (360*a^3*b - 703*a^2*b^2 + 469*a*b^3 - 96*b^4)*sin(d*x + c))*c
os(4*d*x + 4*c) - ((45*a^2*b^2 - 71*a*b^3 + 32*b^4)*cos(7*d*x + 7*c) - (199*a^2*b^2 - 253*a*b^3 + 96*b^4)*cos(
5*d*x + 5*c) + (199*a^2*b^2 - 253*a*b^3 + 96*b^4)*cos(3*d*x + 3*c) - (45*a^2*b^2 - 71*a*b^3 + 32*b^4)*cos(d*x
+ c))*sin(8*d*x + 8*c) + (45*a^2*b^2 - 71*a*b^3 + 32*b^4 - 4*(45*a^2*b^2 - 71*a*b^3 + 32*b^4)*cos(6*d*x + 6*c)
- 2*(360*a^3*b - 703*a^2*b^2 + 469*a*b^3 - 96*b^4)*cos(4*d*x + 4*c) - 4*(45*a^2*b^2 - 71*a*b^3 + 32*b^4)*cos(
2*d*x + 2*c))*sin(7*d*x + 7*c) - 4*((199*a^2*b^2 - 253*a*b^3 + 96*b^4)*cos(5*d*x + 5*c) - (199*a^2*b^2 - 253*a
*b^3 + 96*b^4)*cos(3*d*x + 3*c) + (45*a^2*b^2 - 71*a*b^3 + 32*b^4)*cos(d*x + c))*sin(6*d*x + 6*c) - (199*a^2*b
^2 - 253*a*b^3 + 96*b^4 - 2*(1592*a^3*b - 2621*a^2*b^2 + 1527*a*b^3 - 288*b^4)*cos(4*d*x + 4*c) - 4*(199*a^2*b
^2 - 253*a*b^3 + 96*b^4)*cos(2*d*x + 2*c))*sin(5*d*x + 5*c) + 2*((1592*a^3*b - 2621*a^2*b^2 + 1527*a*b^3 - 288
*b^4)*cos(3*d*x + 3*c) - (360*a^3*b - 703*a^2*b^2 + 469*a*b^3 - 96*b^4)*cos(d*x + c))*sin(4*d*x + 4*c) + (199*
a^2*b^2 - 253*a*b^3 + 96*b^4 - 4*(199*a^2*b^2 - 253*a*b^3 + 96*b^4)*cos(2*d*x + 2*c))*sin(3*d*x + 3*c) - (45*a
^2*b^2 - 71*a*b^3 + 32*b^4)*sin(d*x + c))/(a^5*b^2 - 2*a^4*b^3 + a^3*b^4 + (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*cos
(8*d*x + 8*c)^2 + 16*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*cos(6*d*x + 6*c)^2 + 4*(64*a^7 - 176*a^6*b + 169*a^5*b^2
- 66*a^4*b^3 + 9*a^3*b^4)*cos(4*d*x + 4*c)^2 + 16*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*cos(2*d*x + 2*c)^2 + (a^5*b^
2 - 2*a^4*b^3 + a^3*b^4)*sin(8*d*x + 8*c)^2 + 16*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*sin(6*d*x + 6*c)^2 + 4*(64*a^
7 - 176*a^6*b + 169*a^5*b^2 - 66*a^4*b^3 + 9*a^3*b^4)*sin(4*d*x + 4*c)^2 + 16*(8*a^6*b - 19*a^5*b^2 + 14*a^4*b
^3 - 3*a^3*b^4)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*sin(2*d*x + 2*c)^2 + 2*
(a^5*b^2 - 2*a^4*b^3 + a^3*b^4 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*cos(6*d*x + 6*c) - 2*(8*a^6*b - 19*a^5*b^2
+ 14*a^4*b^3 - 3*a^3*b^4)*cos(4*d*x + 4*c) - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*cos(2*d*x + 2*c))*cos(8*d*x + 8
*c) - 8*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4 - 2*(8*a^6*b - 19*a^5*b^2 + 14*a^4*b^3 - 3*a^3*b^4)*cos(4*d*x + 4*c) -
4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^6*b - 19*a^5*b^2 + 14*a^4*b^3 -
3*a^3*b^4 - 4*(8*a^6*b - 19*a^5*b^2 + 14*a^4*b^3 - 3*a^3*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a^5*b^2
- 2*a^4*b^3 + a^3*b^4)*cos(2*d*x + 2*c) - 4*(2*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*sin(6*d*x + 6*c) + (8*a^6*b - 1
9*a^5*b^2 + 14*a^4*b^3 - 3*a^3*b^4)*sin(4*d*x + 4*c) + 2*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*sin(2*d*x + 2*c))*sin
(8*d*x + 8*c) + 16*((8*a^6*b - 19*a^5*b^2 + 14*a^4*b^3 - 3*a^3*b^4)*sin(4*d*x + 4*c) + 2*(a^5*b^2 - 2*a^4*b^3
+ a^3*b^4)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) - 8*(a^2*b^4 - 2*a*b^5 + b^6 + (a^2*b^4 - 2*a*b^5 + b^6)*co
s(16*d*x + 16*c)^2 + 64*(a^2*b^4 - 2*a*b^5 + b^6)*cos(14*d*x + 14*c)^2 + 16*(64*a^4*b^2 - 240*a^3*b^3 + 337*a^
2*b^4 - 210*a*b^5 + 49*b^6)*cos(12*d*x + 12*c)^2 + 64*(256*a^4*b^2 - 736*a^3*b^3 + 753*a^2*b^4 - 322*a*b^5 + 4
9*b^6)*cos(10*d*x + 10*c)^2 + 4*(16384*a^6 - 57344*a^5*b + 83712*a^4*b^2 - 67648*a^3*b^3 + 32841*a^2*b^4 - 917
0*a*b^5 + 1225*b^6)*cos(8*d*x + 8*c)^2 + 64*(256*a^4*b^2 - 736*a^3*b^3 + 753*a^2*b^4 - 322*a*b^5 + 49*b^6)*cos
(6*d*x + 6*c)^2 + 16*(64*a^4*b^2 - 240*a^3*b^3 + 337*a^2*b^4 - 210*a*b^5 + 49*b^6)*cos(4*d*x + 4*c)^2 + 64*(a^
2*b^4 - 2*a*b^5 + b^6)*cos(2*d*x + 2*c)^2 + (a^2*b^4 - 2*a*b^5 + b^6)*sin(16*d*x + 16*c)^2 + 64*(a^2*b^4 - 2*a
*b^5 + b^6)*sin(14*d*x + 14*c)^2 + 16*(64*a^4*b^2 - 240*a^3*b^3 + 337*a^2*b^4 - 210*a*b^5 + 49*b^6)*sin(12*d*x
+ 12*c)^2 + 64*(256*a^4*b^2 - 736*a^3*b^3 + 753*a^2*b^4 - 322*a*b^5 + 49*b^6)*sin(10*d*x + 10*c)^2 + 4*(16384
*a^6 - 57344*a^5*b + 83712*a^4*b^2 - 67648*a^3*b^3 + 32841*a^2*b^4 - 9170*a*b^5 + 1225*b^6)*sin(8*d*x + 8*c)^2
+ 64*(256*a^4*b^2 - 736*a^3*b^3 + 753*a^2*b^4 - 322*a*b^5 + 49*b^6)*sin(6*d*x + 6*c)^2 + 16*(64*a^4*b^2 - 240
*a^3*b^3 + 337*a^2*b^4 - 210*a*b^5 + 49*b^6)*sin(4*d*x + 4*c)^2 + 64*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^
6)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^2*b^4 - 2*a*b^5 + b^6)*sin(2*d*x + 2*c)^2 + 2*(a^2*b^4 - 2*a*b^5
+ b^6 - 8*(a^2*b^4 - 2*a*b^5 + b^6)*cos(14*d*x + 14*c) - 4*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*cos(12*
d*x + 12*c) + 8*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*cos(10*d*x + 10*c) + 2*(128*a^4*b^2 - 352*a^3*b^3
+ 355*a^2*b^4 - 166*a*b^5 + 35*b^6)*cos(8*d*x + 8*c) + 8*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*cos(6*d
*x + 6*c) - 4*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*cos(4*d*x + 4*c) - 8*(a^2*b^4 - 2*a*b^5 + b^6)*cos(2
*d*x + 2*c))*cos(16*d*x + 16*c) - 16*(a^2*b^4 - 2*a*b^5 + b^6 - 4*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*
cos(12*d*x + 12*c) + 8*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*cos(10*d*x + 10*c) + 2*(128*a^4*b^2 - 352*
a^3*b^3 + 355*a^2*b^4 - 166*a*b^5 + 35*b^6)*cos(8*d*x + 8*c) + 8*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*
cos(6*d*x + 6*c) - 4*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*cos(4*d*x + 4*c) - 8*(a^2*b^4 - 2*a*b^5 + b^6
)*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) - 8*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6 + 8*(128*a^4*b^2 - 424*a
^3*b^3 + 513*a^2*b^4 - 266*a*b^5 + 49*b^6)*cos(10*d*x + 10*c) + 2*(1024*a^5*b - 3712*a^4*b^2 + 5304*a^3*b^3 -
3813*a^2*b^4 + 1442*a*b^5 - 245*b^6)*cos(8*d*x + 8*c) + 8*(128*a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 - 266*a*b^5
+ 49*b^6)*cos(6*d*x + 6*c) - 4*(64*a^4*b^2 - 240*a^3*b^3 + 337*a^2*b^4 - 210*a*b^5 + 49*b^6)*cos(4*d*x + 4*c)
- 8*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 16*(16*a^3*b^3 - 39*a^
2*b^4 + 30*a*b^5 - 7*b^6 + 2*(2048*a^5*b - 6528*a^4*b^2 + 8144*a^3*b^3 - 5141*a^2*b^4 + 1722*a*b^5 - 245*b^6)*
cos(8*d*x + 8*c) + 8*(256*a^4*b^2 - 736*a^3*b^3 + 753*a^2*b^4 - 322*a*b^5 + 49*b^6)*cos(6*d*x + 6*c) - 4*(128*
a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 - 266*a*b^5 + 49*b^6)*cos(4*d*x + 4*c) - 8*(16*a^3*b^3 - 39*a^2*b^4 + 30*a
*b^5 - 7*b^6)*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 4*(128*a^4*b^2 - 352*a^3*b^3 + 355*a^2*b^4 - 166*a*b^5 +
35*b^6 + 8*(2048*a^5*b - 6528*a^4*b^2 + 8144*a^3*b^3 - 5141*a^2*b^4 + 1722*a*b^5 - 245*b^6)*cos(6*d*x + 6*c) -
4*(1024*a^5*b - 3712*a^4*b^2 + 5304*a^3*b^3 - 3813*a^2*b^4 + 1442*a*b^5 - 245*b^6)*cos(4*d*x + 4*c) - 8*(128*
a^4*b^2 - 352*a^3*b^3 + 355*a^2*b^4 - 166*a*b^5 + 35*b^6)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 16*(16*a^3*b^3
- 39*a^2*b^4 + 30*a*b^5 - 7*b^6 - 4*(128*a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 - 266*a*b^5 + 49*b^6)*cos(4*d*x +
4*c) - 8*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 8*(8*a^3*b^3 - 23*
a^2*b^4 + 22*a*b^5 - 7*b^6 - 8*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c)
- 16*(a^2*b^4 - 2*a*b^5 + b^6)*cos(2*d*x + 2*c) - 4*(4*(a^2*b^4 - 2*a*b^5 + b^6)*sin(14*d*x + 14*c) + 2*(8*a^3
*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*sin(12*d*x + 12*c) - 4*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*sin(
10*d*x + 10*c) - (128*a^4*b^2 - 352*a^3*b^3 + 355*a^2*b^4 - 166*a*b^5 + 35*b^6)*sin(8*d*x + 8*c) - 4*(16*a^3*b
^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*sin(6*d*x + 6*c) + 2*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*sin(4*d*x
+ 4*c) + 4*(a^2*b^4 - 2*a*b^5 + b^6)*sin(2*d*x + 2*c))*sin(16*d*x + 16*c) + 32*(2*(8*a^3*b^3 - 23*a^2*b^4 + 2
2*a*b^5 - 7*b^6)*sin(12*d*x + 12*c) - 4*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*sin(10*d*x + 10*c) - (128
*a^4*b^2 - 352*a^3*b^3 + 355*a^2*b^4 - 166*a*b^5 + 35*b^6)*sin(8*d*x + 8*c) - 4*(16*a^3*b^3 - 39*a^2*b^4 + 30*
a*b^5 - 7*b^6)*sin(6*d*x + 6*c) + 2*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*sin(4*d*x + 4*c) + 4*(a^2*b^4
- 2*a*b^5 + b^6)*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*(128*a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 - 266*a
*b^5 + 49*b^6)*sin(10*d*x + 10*c) + (1024*a^5*b - 3712*a^4*b^2 + 5304*a^3*b^3 - 3813*a^2*b^4 + 1442*a*b^5 - 24
5*b^6)*sin(8*d*x + 8*c) + 4*(128*a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 - 266*a*b^5 + 49*b^6)*sin(6*d*x + 6*c) -
2*(64*a^4*b^2 - 240*a^3*b^3 + 337*a^2*b^4 - 210*a*b^5 + 49*b^6)*sin(4*d*x + 4*c) - 4*(8*a^3*b^3 - 23*a^2*b^4 +
22*a*b^5 - 7*b^6)*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 32*((2048*a^5*b - 6528*a^4*b^2 + 8144*a^3*b^3 - 5141
*a^2*b^4 + 1722*a*b^5 - 245*b^6)*sin(8*d*x + 8*c) + 4*(256*a^4*b^2 - 736*a^3*b^3 + 753*a^2*b^4 - 322*a*b^5 + 4
9*b^6)*sin(6*d*x + 6*c) - 2*(128*a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 - 266*a*b^5 + 49*b^6)*sin(4*d*x + 4*c) -
4*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^5*b - 6528
*a^4*b^2 + 8144*a^3*b^3 - 5141*a^2*b^4 + 1722*a*b^5 - 245*b^6)*sin(6*d*x + 6*c) - (1024*a^5*b - 3712*a^4*b^2 +
5304*a^3*b^3 - 3813*a^2*b^4 + 1442*a*b^5 - 245*b^6)*sin(4*d*x + 4*c) - 2*(128*a^4*b^2 - 352*a^3*b^3 + 355*a^2
*b^4 - 166*a*b^5 + 35*b^6)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 -
266*a*b^5 + 49*b^6)*sin(4*d*x + 4*c) + 2*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*sin(2*d*x + 2*c))*sin(6
*d*x + 6*c))*log(cos(d*x)^2 + 2*cos(d*x)*cos(c) + cos(c)^2 + sin(d*x)^2 - 2*sin(d*x)*sin(c) + sin(c)^2) + 8*(a
^2*b^4 - 2*a*b^5 + b^6 + (a^2*b^4 - 2*a*b^5 + b^6)*cos(16*d*x + 16*c)^2 + 64*(a^2*b^4 - 2*a*b^5 + b^6)*cos(14*
d*x + 14*c)^2 + 16*(64*a^4*b^2 - 240*a^3*b^3 + 337*a^2*b^4 - 210*a*b^5 + 49*b^6)*cos(12*d*x + 12*c)^2 + 64*(25
6*a^4*b^2 - 736*a^3*b^3 + 753*a^2*b^4 - 322*a*b^5 + 49*b^6)*cos(10*d*x + 10*c)^2 + 4*(16384*a^6 - 57344*a^5*b
+ 83712*a^4*b^2 - 67648*a^3*b^3 + 32841*a^2*b^4 - 9170*a*b^5 + 1225*b^6)*cos(8*d*x + 8*c)^2 + 64*(256*a^4*b^2
- 736*a^3*b^3 + 753*a^2*b^4 - 322*a*b^5 + 49*b^6)*cos(6*d*x + 6*c)^2 + 16*(64*a^4*b^2 - 240*a^3*b^3 + 337*a^2*
b^4 - 210*a*b^5 + 49*b^6)*cos(4*d*x + 4*c)^2 + 64*(a^2*b^4 - 2*a*b^5 + b^6)*cos(2*d*x + 2*c)^2 + (a^2*b^4 - 2*
a*b^5 + b^6)*sin(16*d*x + 16*c)^2 + 64*(a^2*b^4 - 2*a*b^5 + b^6)*sin(14*d*x + 14*c)^2 + 16*(64*a^4*b^2 - 240*a
^3*b^3 + 337*a^2*b^4 - 210*a*b^5 + 49*b^6)*sin(12*d*x + 12*c)^2 + 64*(256*a^4*b^2 - 736*a^3*b^3 + 753*a^2*b^4
- 322*a*b^5 + 49*b^6)*sin(10*d*x + 10*c)^2 + 4*(16384*a^6 - 57344*a^5*b + 83712*a^4*b^2 - 67648*a^3*b^3 + 3284
1*a^2*b^4 - 9170*a*b^5 + 1225*b^6)*sin(8*d*x + 8*c)^2 + 64*(256*a^4*b^2 - 736*a^3*b^3 + 753*a^2*b^4 - 322*a*b^
5 + 49*b^6)*sin(6*d*x + 6*c)^2 + 16*(64*a^4*b^2 - 240*a^3*b^3 + 337*a^2*b^4 - 210*a*b^5 + 49*b^6)*sin(4*d*x +
4*c)^2 + 64*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^2*b^4 - 2*a*
b^5 + b^6)*sin(2*d*x + 2*c)^2 + 2*(a^2*b^4 - 2*a*b^5 + b^6 - 8*(a^2*b^4 - 2*a*b^5 + b^6)*cos(14*d*x + 14*c) -
4*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*cos(12*d*x + 12*c) + 8*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b
^6)*cos(10*d*x + 10*c) + 2*(128*a^4*b^2 - 352*a^3*b^3 + 355*a^2*b^4 - 166*a*b^5 + 35*b^6)*cos(8*d*x + 8*c) + 8
*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*cos(6*d*x + 6*c) - 4*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)
*cos(4*d*x + 4*c) - 8*(a^2*b^4 - 2*a*b^5 + b^6)*cos(2*d*x + 2*c))*cos(16*d*x + 16*c) - 16*(a^2*b^4 - 2*a*b^5 +
b^6 - 4*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*cos(12*d*x + 12*c) + 8*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^
5 - 7*b^6)*cos(10*d*x + 10*c) + 2*(128*a^4*b^2 - 352*a^3*b^3 + 355*a^2*b^4 - 166*a*b^5 + 35*b^6)*cos(8*d*x + 8
*c) + 8*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*cos(6*d*x + 6*c) - 4*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 -
7*b^6)*cos(4*d*x + 4*c) - 8*(a^2*b^4 - 2*a*b^5 + b^6)*cos(2*d*x + 2*c))*cos(14*d*x + 14*c) - 8*(8*a^3*b^3 - 2
3*a^2*b^4 + 22*a*b^5 - 7*b^6 + 8*(128*a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 - 266*a*b^5 + 49*b^6)*cos(10*d*x + 1
0*c) + 2*(1024*a^5*b - 3712*a^4*b^2 + 5304*a^3*b^3 - 3813*a^2*b^4 + 1442*a*b^5 - 245*b^6)*cos(8*d*x + 8*c) + 8
*(128*a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 - 266*a*b^5 + 49*b^6)*cos(6*d*x + 6*c) - 4*(64*a^4*b^2 - 240*a^3*b^3
+ 337*a^2*b^4 - 210*a*b^5 + 49*b^6)*cos(4*d*x + 4*c) - 8*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*cos(2*d*
x + 2*c))*cos(12*d*x + 12*c) + 16*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6 + 2*(2048*a^5*b - 6528*a^4*b^2 +
8144*a^3*b^3 - 5141*a^2*b^4 + 1722*a*b^5 - 245*b^6)*cos(8*d*x + 8*c) + 8*(256*a^4*b^2 - 736*a^3*b^3 + 753*a^2
*b^4 - 322*a*b^5 + 49*b^6)*cos(6*d*x + 6*c) - 4*(128*a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 - 266*a*b^5 + 49*b^6)
*cos(4*d*x + 4*c) - 8*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 4*(1
28*a^4*b^2 - 352*a^3*b^3 + 355*a^2*b^4 - 166*a*b^5 + 35*b^6 + 8*(2048*a^5*b - 6528*a^4*b^2 + 8144*a^3*b^3 - 51
41*a^2*b^4 + 1722*a*b^5 - 245*b^6)*cos(6*d*x + 6*c) - 4*(1024*a^5*b - 3712*a^4*b^2 + 5304*a^3*b^3 - 3813*a^2*b
^4 + 1442*a*b^5 - 245*b^6)*cos(4*d*x + 4*c) - 8*(128*a^4*b^2 - 352*a^3*b^3 + 355*a^2*b^4 - 166*a*b^5 + 35*b^6)
*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 16*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6 - 4*(128*a^4*b^2 - 424*a^
3*b^3 + 513*a^2*b^4 - 266*a*b^5 + 49*b^6)*cos(4*d*x + 4*c) - 8*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*co
s(2*d*x + 2*c))*cos(6*d*x + 6*c) - 8*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6 - 8*(8*a^3*b^3 - 23*a^2*b^4 +
22*a*b^5 - 7*b^6)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 16*(a^2*b^4 - 2*a*b^5 + b^6)*cos(2*d*x + 2*c) - 4*(4*(a
^2*b^4 - 2*a*b^5 + b^6)*sin(14*d*x + 14*c) + 2*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*sin(12*d*x + 12*c)
- 4*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*sin(10*d*x + 10*c) - (128*a^4*b^2 - 352*a^3*b^3 + 355*a^2*b^4
- 166*a*b^5 + 35*b^6)*sin(8*d*x + 8*c) - 4*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*sin(6*d*x + 6*c) + 2*
(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*sin(4*d*x + 4*c) + 4*(a^2*b^4 - 2*a*b^5 + b^6)*sin(2*d*x + 2*c))*s
in(16*d*x + 16*c) + 32*(2*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*sin(12*d*x + 12*c) - 4*(16*a^3*b^3 - 39*
a^2*b^4 + 30*a*b^5 - 7*b^6)*sin(10*d*x + 10*c) - (128*a^4*b^2 - 352*a^3*b^3 + 355*a^2*b^4 - 166*a*b^5 + 35*b^6
)*sin(8*d*x + 8*c) - 4*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*sin(6*d*x + 6*c) + 2*(8*a^3*b^3 - 23*a^2*b
^4 + 22*a*b^5 - 7*b^6)*sin(4*d*x + 4*c) + 4*(a^2*b^4 - 2*a*b^5 + b^6)*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) - 1
6*(4*(128*a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 - 266*a*b^5 + 49*b^6)*sin(10*d*x + 10*c) + (1024*a^5*b - 3712*a^
4*b^2 + 5304*a^3*b^3 - 3813*a^2*b^4 + 1442*a*b^5 - 245*b^6)*sin(8*d*x + 8*c) + 4*(128*a^4*b^2 - 424*a^3*b^3 +
513*a^2*b^4 - 266*a*b^5 + 49*b^6)*sin(6*d*x + 6*c) - 2*(64*a^4*b^2 - 240*a^3*b^3 + 337*a^2*b^4 - 210*a*b^5 + 4
9*b^6)*sin(4*d*x + 4*c) - 4*(8*a^3*b^3 - 23*a^2*b^4 + 22*a*b^5 - 7*b^6)*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) +
32*((2048*a^5*b - 6528*a^4*b^2 + 8144*a^3*b^3 - 5141*a^2*b^4 + 1722*a*b^5 - 245*b^6)*sin(8*d*x + 8*c) + 4*(25
6*a^4*b^2 - 736*a^3*b^3 + 753*a^2*b^4 - 322*a*b^5 + 49*b^6)*sin(6*d*x + 6*c) - 2*(128*a^4*b^2 - 424*a^3*b^3 +
513*a^2*b^4 - 266*a*b^5 + 49*b^6)*sin(4*d*x + 4*c) - 4*(16*a^3*b^3 - 39*a^2*b^4 + 30*a*b^5 - 7*b^6)*sin(2*d*x
+ 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^5*b - 6528*a^4*b^2 + 8144*a^3*b^3 - 5141*a^2*b^4 + 1722*a*b^5 - 245
*b^6)*sin(6*d*x + 6*c) - (1024*a^5*b - 3712*a^4*b^2 + 5304*a^3*b^3 - 3813*a^2*b^4 + 1442*a*b^5 - 245*b^6)*sin(
4*d*x + 4*c) - 2*(128*a^4*b^2 - 352*a^3*b^3 + 355*a^2*b^4 - 166*a*b^5 + 35*b^6)*sin(2*d*x + 2*c))*sin(8*d*x +
8*c) - 64*((128*a^4*b^2 - 424*a^3*b^3 + 513*a^2*b^4 - 266*a*b^5 + 49*b^6)*sin(4*d*x + 4*c) + 2*(16*a^3*b^3 - 3
9*a^2*b^4 + 30*a*b^5 - 7*b^6)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(cos(d*x)^2 - 2*cos(d*x)*cos(c) + cos(c)^
2 + sin(d*x)^2 + 2*sin(d*x)*sin(c) + sin(c)^2) - ((13*a^2*b^4 - 7*a*b^5)*sin(15*d*x + 15*c) - (121*a^2*b^4 - 6
7*a*b^5)*sin(13*d*x + 13*c) - (272*a^3*b^3 - 461*a^2*b^4 + 159*a*b^5)*sin(11*d*x + 11*c) + (1424*a^3*b^3 - 112
1*a^2*b^4 + 99*a*b^5)*sin(9*d*x + 9*c) + (1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5)*sin(7*d*x + 7*c) - (272*a^3*
b^3 - 461*a^2*b^4 + 159*a*b^5)*sin(5*d*x + 5*c) - (121*a^2*b^4 - 67*a*b^5)*sin(3*d*x + 3*c) + (13*a^2*b^4 - 7*
a*b^5)*sin(d*x + c))*sin(16*d*x + 16*c) + 2*(4*(13*a^2*b^4 - 7*a*b^5)*sin(14*d*x + 14*c) + 2*(104*a^3*b^3 - 14
7*a^2*b^4 + 49*a*b^5)*sin(12*d*x + 12*c) - 4*(208*a^3*b^3 - 203*a^2*b^4 + 49*a*b^5)*sin(10*d*x + 10*c) - (1664
*a^4*b^2 - 2144*a^3*b^3 + 1127*a^2*b^4 - 245*a*b^5)*sin(8*d*x + 8*c) - 4*(208*a^3*b^3 - 203*a^2*b^4 + 49*a*b^5
)*sin(6*d*x + 6*c) + 2*(104*a^3*b^3 - 147*a^2*b^4 + 49*a*b^5)*sin(4*d*x + 4*c) + 4*(13*a^2*b^4 - 7*a*b^5)*sin(
2*d*x + 2*c))*sin(15*d*x + 15*c) - 8*((121*a^2*b^4 - 67*a*b^5)*sin(13*d*x + 13*c) + (272*a^3*b^3 - 461*a^2*b^4
+ 159*a*b^5)*sin(11*d*x + 11*c) - (1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5)*sin(9*d*x + 9*c) - (1424*a^3*b^3 -
1121*a^2*b^4 + 99*a*b^5)*sin(7*d*x + 7*c) + (272*a^3*b^3 - 461*a^2*b^4 + 159*a*b^5)*sin(5*d*x + 5*c) + (121*a
^2*b^4 - 67*a*b^5)*sin(3*d*x + 3*c) - (13*a^2*b^4 - 7*a*b^5)*sin(d*x + c))*sin(14*d*x + 14*c) - 2*(2*(968*a^3*
b^3 - 1383*a^2*b^4 + 469*a*b^5)*sin(12*d*x + 12*c) - 4*(1936*a^3*b^3 - 1919*a^2*b^4 + 469*a*b^5)*sin(10*d*x +
10*c) - (15488*a^4*b^2 - 20192*a^3*b^3 + 10667*a^2*b^4 - 2345*a*b^5)*sin(8*d*x + 8*c) - 4*(1936*a^3*b^3 - 1919
*a^2*b^4 + 469*a*b^5)*sin(6*d*x + 6*c) + 2*(968*a^3*b^3 - 1383*a^2*b^4 + 469*a*b^5)*sin(4*d*x + 4*c) + 4*(121*
a^2*b^4 - 67*a*b^5)*sin(2*d*x + 2*c))*sin(13*d*x + 13*c) - 4*((2176*a^4*b^2 - 5592*a^3*b^3 + 4499*a^2*b^4 - 11
13*a*b^5)*sin(11*d*x + 11*c) - (11392*a^4*b^2 - 18936*a^3*b^3 + 8639*a^2*b^4 - 693*a*b^5)*sin(9*d*x + 9*c) - (
11392*a^4*b^2 - 18936*a^3*b^3 + 8639*a^2*b^4 - 693*a*b^5)*sin(7*d*x + 7*c) + (2176*a^4*b^2 - 5592*a^3*b^3 + 44
99*a^2*b^4 - 1113*a*b^5)*sin(5*d*x + 5*c) + (968*a^3*b^3 - 1383*a^2*b^4 + 469*a*b^5)*sin(3*d*x + 3*c) - (104*a
^3*b^3 - 147*a^2*b^4 + 49*a*b^5)*sin(d*x + c))*sin(12*d*x + 12*c) + 2*(4*(4352*a^4*b^2 - 9280*a^3*b^3 + 5771*a
^2*b^4 - 1113*a*b^5)*sin(10*d*x + 10*c) + (34816*a^5*b - 85120*a^4*b^2 + 74128*a^3*b^3 - 31399*a^2*b^4 + 5565*
a*b^5)*sin(8*d*x + 8*c) + 4*(4352*a^4*b^2 - 9280*a^3*b^3 + 5771*a^2*b^4 - 1113*a*b^5)*sin(6*d*x + 6*c) - 2*(21
76*a^4*b^2 - 5592*a^3*b^3 + 4499*a^2*b^4 - 1113*a*b^5)*sin(4*d*x + 4*c) - 4*(272*a^3*b^3 - 461*a^2*b^4 + 159*a
*b^5)*sin(2*d*x + 2*c))*sin(11*d*x + 11*c) - 8*((22784*a^4*b^2 - 27904*a^3*b^3 + 9431*a^2*b^4 - 693*a*b^5)*sin
(9*d*x + 9*c) + (22784*a^4*b^2 - 27904*a^3*b^3 + 9431*a^2*b^4 - 693*a*b^5)*sin(7*d*x + 7*c) - (4352*a^4*b^2 -
9280*a^3*b^3 + 5771*a^2*b^4 - 1113*a*b^5)*sin(5*d*x + 5*c) - (1936*a^3*b^3 - 1919*a^2*b^4 + 469*a*b^5)*sin(3*d
*x + 3*c) + (208*a^3*b^3 - 203*a^2*b^4 + 49*a*b^5)*sin(d*x + c))*sin(10*d*x + 10*c) - 2*((182272*a^5*b - 28019
2*a^4*b^2 + 170128*a^3*b^3 - 48739*a^2*b^4 + 3465*a*b^5)*sin(8*d*x + 8*c) + 4*(22784*a^4*b^2 - 27904*a^3*b^3 +
9431*a^2*b^4 - 693*a*b^5)*sin(6*d*x + 6*c) - 2*(11392*a^4*b^2 - 18936*a^3*b^3 + 8639*a^2*b^4 - 693*a*b^5)*sin
(4*d*x + 4*c) - 4*(1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5)*sin(2*d*x + 2*c))*sin(9*d*x + 9*c) - 2*((182272*a^5
*b - 280192*a^4*b^2 + 170128*a^3*b^3 - 48739*a^2*b^4 + 3465*a*b^5)*sin(7*d*x + 7*c) - (34816*a^5*b - 85120*a^4
*b^2 + 74128*a^3*b^3 - 31399*a^2*b^4 + 5565*a*b^5)*sin(5*d*x + 5*c) - (15488*a^4*b^2 - 20192*a^3*b^3 + 10667*a
^2*b^4 - 2345*a*b^5)*sin(3*d*x + 3*c) + (1664*a^4*b^2 - 2144*a^3*b^3 + 1127*a^2*b^4 - 245*a*b^5)*sin(d*x + c))
*sin(8*d*x + 8*c) - 4*(2*(22784*a^4*b^2 - 27904*a^3*b^3 + 9431*a^2*b^4 - 693*a*b^5)*sin(6*d*x + 6*c) - (11392*
a^4*b^2 - 18936*a^3*b^3 + 8639*a^2*b^4 - 693*a*b^5)*sin(4*d*x + 4*c) - 2*(1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b
^5)*sin(2*d*x + 2*c))*sin(7*d*x + 7*c) + 8*((4352*a^4*b^2 - 9280*a^3*b^3 + 5771*a^2*b^4 - 1113*a*b^5)*sin(5*d*
x + 5*c) + (1936*a^3*b^3 - 1919*a^2*b^4 + 469*a*b^5)*sin(3*d*x + 3*c) - (208*a^3*b^3 - 203*a^2*b^4 + 49*a*b^5)
*sin(d*x + c))*sin(6*d*x + 6*c) - 4*((2176*a^4*b^2 - 5592*a^3*b^3 + 4499*a^2*b^4 - 1113*a*b^5)*sin(4*d*x + 4*c
) + 2*(272*a^3*b^3 - 461*a^2*b^4 + 159*a*b^5)*sin(2*d*x + 2*c))*sin(5*d*x + 5*c) - 4*((968*a^3*b^3 - 1383*a^2*
b^4 + 469*a*b^5)*sin(3*d*x + 3*c) - (104*a^3*b^3 - 147*a^2*b^4 + 49*a*b^5)*sin(d*x + c))*sin(4*d*x + 4*c))/((a
^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(16*d*x + 16*c)^2 + 64*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(14*d*x + 14*c)
^2 + 16*(64*a^7*b^2 - 240*a^6*b^3 + 337*a^5*b^4 - 210*a^4*b^5 + 49*a^3*b^6)*d*cos(12*d*x + 12*c)^2 + 64*(256*a
^7*b^2 - 736*a^6*b^3 + 753*a^5*b^4 - 322*a^4*b^5 + 49*a^3*b^6)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^9 - 57344*a
^8*b + 83712*a^7*b^2 - 67648*a^6*b^3 + 32841*a^5*b^4 - 9170*a^4*b^5 + 1225*a^3*b^6)*d*cos(8*d*x + 8*c)^2 + 64*
(256*a^7*b^2 - 736*a^6*b^3 + 753*a^5*b^4 - 322*a^4*b^5 + 49*a^3*b^6)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^7*b^2 - 2
40*a^6*b^3 + 337*a^5*b^4 - 210*a^4*b^5 + 49*a^3*b^6)*d*cos(4*d*x + 4*c)^2 + 64*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)
*d*cos(2*d*x + 2*c)^2 + (a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*sin(16*d*x + 16*c)^2 + 64*(a^5*b^4 - 2*a^4*b^5 + a^3
*b^6)*d*sin(14*d*x + 14*c)^2 + 16*(64*a^7*b^2 - 240*a^6*b^3 + 337*a^5*b^4 - 210*a^4*b^5 + 49*a^3*b^6)*d*sin(12
*d*x + 12*c)^2 + 64*(256*a^7*b^2 - 736*a^6*b^3 + 753*a^5*b^4 - 322*a^4*b^5 + 49*a^3*b^6)*d*sin(10*d*x + 10*c)^
2 + 4*(16384*a^9 - 57344*a^8*b + 83712*a^7*b^2 - 67648*a^6*b^3 + 32841*a^5*b^4 - 9170*a^4*b^5 + 1225*a^3*b^6)*
d*sin(8*d*x + 8*c)^2 + 64*(256*a^7*b^2 - 736*a^6*b^3 + 753*a^5*b^4 - 322*a^4*b^5 + 49*a^3*b^6)*d*sin(6*d*x + 6
*c)^2 + 16*(64*a^7*b^2 - 240*a^6*b^3 + 337*a^5*b^4 - 210*a^4*b^5 + 49*a^3*b^6)*d*sin(4*d*x + 4*c)^2 + 64*(8*a^
6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^5*b^4 - 2*a^4*b^5 + a
^3*b^6)*d*sin(2*d*x + 2*c)^2 - 16*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(2*d*x + 2*c) + (a^5*b^4 - 2*a^4*b^5 +
a^3*b^6)*d - 2*(8*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(14*d*x + 14*c) + 4*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^
5 - 7*a^3*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*cos(10*d*x + 10*c
) - 2*(128*a^7*b^2 - 352*a^6*b^3 + 355*a^5*b^4 - 166*a^4*b^5 + 35*a^3*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^6*b^3
- 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6
)*d*cos(4*d*x + 4*c) + 8*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(2*d*x + 2*c) - (a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*
d)*cos(16*d*x + 16*c) + 16*(4*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a
^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*cos(10*d*x + 10*c) - 2*(128*a^7*b^2 - 352*a^6*b^3 + 355*a^5*b^
4 - 166*a^4*b^5 + 35*a^3*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*cos(
6*d*x + 6*c) + 4*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*cos(4*d*x + 4*c) + 8*(a^5*b^4 - 2*a^4*b^5
+ a^3*b^6)*d*cos(2*d*x + 2*c) - (a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d)*cos(14*d*x + 14*c) - 8*(8*(128*a^7*b^2 - 4
24*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*cos(10*d*x + 10*c) + 2*(1024*a^8*b - 3712*a^7*b^2 + 530
4*a^6*b^3 - 3813*a^5*b^4 + 1442*a^4*b^5 - 245*a^3*b^6)*d*cos(8*d*x + 8*c) + 8*(128*a^7*b^2 - 424*a^6*b^3 + 513
*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*cos(6*d*x + 6*c) - 4*(64*a^7*b^2 - 240*a^6*b^3 + 337*a^5*b^4 - 210*a^4*
b^5 + 49*a^3*b^6)*d*cos(4*d*x + 4*c) - 8*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*cos(2*d*x + 2*c)
+ (8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d)*cos(12*d*x + 12*c) + 16*(2*(2048*a^8*b - 6528*a^7*b^2 +
8144*a^6*b^3 - 5141*a^5*b^4 + 1722*a^4*b^5 - 245*a^3*b^6)*d*cos(8*d*x + 8*c) + 8*(256*a^7*b^2 - 736*a^6*b^3 +
753*a^5*b^4 - 322*a^4*b^5 + 49*a^3*b^6)*d*cos(6*d*x + 6*c) - 4*(128*a^7*b^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266
*a^4*b^5 + 49*a^3*b^6)*d*cos(4*d*x + 4*c) - 8*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*cos(2*d*x +
2*c) + (16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d)*cos(10*d*x + 10*c) + 4*(8*(2048*a^8*b - 6528*a^7
*b^2 + 8144*a^6*b^3 - 5141*a^5*b^4 + 1722*a^4*b^5 - 245*a^3*b^6)*d*cos(6*d*x + 6*c) - 4*(1024*a^8*b - 3712*a^7
*b^2 + 5304*a^6*b^3 - 3813*a^5*b^4 + 1442*a^4*b^5 - 245*a^3*b^6)*d*cos(4*d*x + 4*c) - 8*(128*a^7*b^2 - 352*a^6
*b^3 + 355*a^5*b^4 - 166*a^4*b^5 + 35*a^3*b^6)*d*cos(2*d*x + 2*c) + (128*a^7*b^2 - 352*a^6*b^3 + 355*a^5*b^4 -
166*a^4*b^5 + 35*a^3*b^6)*d)*cos(8*d*x + 8*c) - 16*(4*(128*a^7*b^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5
+ 49*a^3*b^6)*d*cos(4*d*x + 4*c) + 8*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*cos(2*d*x + 2*c) - (
16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d)*cos(6*d*x + 6*c) + 8*(8*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*
b^5 - 7*a^3*b^6)*d*cos(2*d*x + 2*c) - (8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d)*cos(4*d*x + 4*c) -
4*(4*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*sin(14*d*x + 14*c) + 2*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6
)*d*sin(12*d*x + 12*c) - 4*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*sin(10*d*x + 10*c) - (128*a^7*
b^2 - 352*a^6*b^3 + 355*a^5*b^4 - 166*a^4*b^5 + 35*a^3*b^6)*d*sin(8*d*x + 8*c) - 4*(16*a^6*b^3 - 39*a^5*b^4 +
30*a^4*b^5 - 7*a^3*b^6)*d*sin(6*d*x + 6*c) + 2*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*sin(4*d*x +
4*c) + 4*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*sin(2*d*x + 2*c))*sin(16*d*x + 16*c) + 32*(2*(8*a^6*b^3 - 23*a^5*b
^4 + 22*a^4*b^5 - 7*a^3*b^6)*d*sin(12*d*x + 12*c) - 4*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*sin
(10*d*x + 10*c) - (128*a^7*b^2 - 352*a^6*b^3 + 355*a^5*b^4 - 166*a^4*b^5 + 35*a^3*b^6)*d*sin(8*d*x + 8*c) - 4*
(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*sin(6*d*x + 6*c) + 2*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5
- 7*a^3*b^6)*d*sin(4*d*x + 4*c) + 4*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) -
16*(4*(128*a^7*b^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*sin(10*d*x + 10*c) + (1024*a^8*b
- 3712*a^7*b^2 + 5304*a^6*b^3 - 3813*a^5*b^4 + 1442*a^4*b^5 - 245*a^3*b^6)*d*sin(8*d*x + 8*c) + 4*(128*a^7*b^2
- 424*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*sin(6*d*x + 6*c) - 2*(64*a^7*b^2 - 240*a^6*b^3 + 33
7*a^5*b^4 - 210*a^4*b^5 + 49*a^3*b^6)*d*sin(4*d*x + 4*c) - 4*(8*a^6*b^3 - 23*a^5*b^4 + 22*a^4*b^5 - 7*a^3*b^6)
*d*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 32*((2048*a^8*b - 6528*a^7*b^2 + 8144*a^6*b^3 - 5141*a^5*b^4 + 1722*
a^4*b^5 - 245*a^3*b^6)*d*sin(8*d*x + 8*c) + 4*(256*a^7*b^2 - 736*a^6*b^3 + 753*a^5*b^4 - 322*a^4*b^5 + 49*a^3*
b^6)*d*sin(6*d*x + 6*c) - 2*(128*a^7*b^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*sin(4*d*x +
4*c) - 4*(16*a^6*b^3 - 39*a^5*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2
048*a^8*b - 6528*a^7*b^2 + 8144*a^6*b^3 - 5141*a^5*b^4 + 1722*a^4*b^5 - 245*a^3*b^6)*d*sin(6*d*x + 6*c) - (102
4*a^8*b - 3712*a^7*b^2 + 5304*a^6*b^3 - 3813*a^5*b^4 + 1442*a^4*b^5 - 245*a^3*b^6)*d*sin(4*d*x + 4*c) - 2*(128
*a^7*b^2 - 352*a^6*b^3 + 355*a^5*b^4 - 166*a^4*b^5 + 35*a^3*b^6)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((1
28*a^7*b^2 - 424*a^6*b^3 + 513*a^5*b^4 - 266*a^4*b^5 + 49*a^3*b^6)*d*sin(4*d*x + 4*c) + 2*(16*a^6*b^3 - 39*a^5
*b^4 + 30*a^4*b^5 - 7*a^3*b^6)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))

Giac [F]

\[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\csc \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]

[In]

integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 20.97 (sec) , antiderivative size = 12247, normalized size of antiderivative = 19.85 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

int(1/(sin(c + d*x)*(a - b*sin(c + d*x)^4)^3),x)

[Out]

- ((5*cos(c + d*x)*(7*a*b - 3*b^2))/(32*a^2*(a - b)) - (cos(c + d*x)^3*(17*a^2*b - 78*a*b^2 + 37*b^3))/(32*a^2
*(a - b)^2) - (cos(c + d*x)^5*(53*a*b^2 - 29*b^3))/(32*a^2*(a - b)^2) + (b*cos(c + d*x)^7*(13*a*b - 7*b^2))/(3
2*a^2*(a - b)^2))/(d*(a^2 - 2*a*b + b^2 + cos(c + d*x)^2*(4*a*b - 4*b^2) - cos(c + d*x)^4*(2*a*b - 6*b^2) - 4*
b^2*cos(c + d*x)^6 + b^2*cos(c + d*x)^8)) - (atan((((((((192*a^11*b^9 - 990*a^12*b^8 + 2050*a^13*b^7 - 2154*a^
14*b^6 + 1158*a^15*b^5 - 256*a^16*b^4)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) - (cos(c + d
*x)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 1476395008*a^16*b^
5 - 268435456*a^17*b^4))/(2097152*a^3*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) + (cos(c
+ d*x)*(75497472*a^6*b^9 - 337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(1
048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) - (12*a^5*b^9 - (4311*a^6*b^8)/64 + (307
961*a^7*b^7)/2048 - (1290253*a^8*b^6)/8192 + (546059*a^9*b^5)/8192)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^
3 + 6*a^12*b^2)))*1i)/(2*a^3) - (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*
b^6 + 8247825*a^4*b^5)*1i)/(1048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/a^3 - ((((((192*a^1
1*b^9 - 990*a^12*b^8 + 2050*a^13*b^7 - 2154*a^14*b^6 + 1158*a^15*b^5 - 256*a^16*b^4)/(2*(a^14 - 4*a^13*b + a^1
0*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) + (cos(c + d*x)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*
b^7 - 3221225472*a^15*b^6 + 1476395008*a^16*b^5 - 268435456*a^17*b^4))/(2097152*a^3*(a^12 - 4*a^11*b + a^8*b^4
- 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) - (cos(c + d*x)*(75497472*a^6*b^9 - 337215488*a^7*b^8 + 592748544*a^8*b^7
- 489406464*a^9*b^6 + 163684352*a^10*b^5))/(1048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2
*a^3) - (12*a^5*b^9 - (4311*a^6*b^8)/64 + (307961*a^7*b^7)/2048 - (1290253*a^8*b^6)/8192 + (546059*a^9*b^5)/81
92)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*1i)/(2*a^3) + (cos(c + d*x)*(3145728*b^9 - 144
17920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6 + 8247825*a^4*b^5)*1i)/(1048576*(a^12 - 4*a^11*b + a^8*b^4 -
4*a^9*b^3 + 6*a^10*b^2)))/a^3)/((((((192*a^11*b^9 - 990*a^12*b^8 + 2050*a^13*b^7 - 2154*a^14*b^6 + 1158*a^15*
b^5 - 256*a^16*b^4)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) - (cos(c + d*x)*(402653184*a^12
*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 1476395008*a^16*b^5 - 268435456*a^17*
b^4))/(2097152*a^3*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) + (cos(c + d*x)*(75497472*a^
6*b^9 - 337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(1048576*(a^12 - 4*a^
11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) - (12*a^5*b^9 - (4311*a^6*b^8)/64 + (307961*a^7*b^7)/2048 -
(1290253*a^8*b^6)/8192 + (546059*a^9*b^5)/8192)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))/(
2*a^3) - (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6 + 8247825*a^4*b^5))
/(1048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/a^3 + (((((192*a^11*b^9 - 990*a^12*b^8 + 2050
*a^13*b^7 - 2154*a^14*b^6 + 1158*a^15*b^5 - 256*a^16*b^4)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12
*b^2)) + (cos(c + d*x)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 +
1476395008*a^16*b^5 - 268435456*a^17*b^4))/(2097152*a^3*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2))
)/(2*a^3) - (cos(c + d*x)*(75497472*a^6*b^9 - 337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 1636
84352*a^10*b^5))/(1048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) - (12*a^5*b^9 - (4311
*a^6*b^8)/64 + (307961*a^7*b^7)/2048 - (1290253*a^8*b^6)/8192 + (546059*a^9*b^5)/8192)/(2*(a^14 - 4*a^13*b + a
^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))/(2*a^3) + (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7
- 23076232*a^3*b^6 + 8247825*a^4*b^5))/(1048576*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/a^3 + (
(90009*a*b^7)/32768 - (81*b^8)/128 - (271845*a^2*b^6)/65536 + (1184625*a^3*b^5)/524288)/(a^14 - 4*a^13*b + a^1
0*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*1i)/(a^3*d) - (atan(((((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 214958
0800*a^13*b^7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b + a^
10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) - (cos(c + d*x)*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*
a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(
a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 1
0*a^15*b^2)))^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 14
76395008*a^16*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(202
5*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 93
06*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^
16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) + (cos(c + d*x)*(75497472*a^6*b^9 - 3
37215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(524288*(a^12 - 4*a^11*b + a^8
*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*
b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 44
29*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2
) - (12582912*a^5*b^9 - 70631424*a^6*b^8 + 157676032*a^7*b^7 - 165152384*a^8*b^6 + 69895552*a^9*b^5)/(1048576*
(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) -
3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a
^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b
^3 - 10*a^15*b^2)))^(1/2) - (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6
+ 8247825*a^4*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2)
+ 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b
^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^
5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2)*1i - (((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 21495
80800*a^13*b^7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b + a
^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) + (cos(c + d*x)*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465
*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*
(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 -
10*a^15*b^2)))^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 1
476395008*a^16*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(20
25*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9
306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a
^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) - (cos(c + d*x)*(75497472*a^6*b^9 -
337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(524288*(a^12 - 4*a^11*b + a^
8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6
*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4
429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/
2) - (12582912*a^5*b^9 - 70631424*a^6*b^8 + 157676032*a^7*b^7 - 165152384*a^8*b^6 + 69895552*a^9*b^5)/(1048576
*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2)
- 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*
a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*
b^3 - 10*a^15*b^2)))^(1/2) + (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6
+ 8247825*a^4*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2)
+ 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*
b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b
^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2)*1i)/((((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 2149
580800*a^13*b^7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b +
a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) - (cos(c + d*x)*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 346
5*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b
*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 -
10*a^15*b^2)))^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 +
1476395008*a^16*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2
025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 +
9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*
a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) + (cos(c + d*x)*(75497472*a^6*b^9 -
337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(524288*(a^12 - 4*a^11*b + a
^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^
6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) +
4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1
/2) - (12582912*a^5*b^9 - 70631424*a^6*b^8 + 157676032*a^7*b^7 - 165152384*a^8*b^6 + 69895552*a^9*b^5)/(104857
6*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2)
- 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694
*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14
*b^3 - 10*a^15*b^2)))^(1/2) - (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^
6 + 8247825*a^4*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2
) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a
*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*
b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) + (((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 214958
0800*a^13*b^7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b + a^
10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) + (cos(c + d*x)*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*
a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(
a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 1
0*a^15*b^2)))^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 14
76395008*a^16*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(202
5*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 93
06*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^
16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) - (cos(c + d*x)*(75497472*a^6*b^9 - 3
37215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(524288*(a^12 - 4*a^11*b + a^8
*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*
b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 44
29*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2
) - (12582912*a^5*b^9 - 70631424*a^6*b^8 + 157676032*a^7*b^7 - 165152384*a^8*b^6 + 69895552*a^9*b^5)/(1048576*
(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) -
3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a
^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b
^3 - 10*a^15*b^2)))^(1/2) + (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6
+ 8247825*a^4*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*(-(2025*a^4*(a^13*b)^(1/2)
+ 384*b^4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b
^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^
5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) + (1440144*a*b^7 - 331776*b^8 - 2174760*a^2*b^6 + 1184625*
a^3*b^5)/(524288*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2))))*(-(2025*a^4*(a^13*b)^(1/2) + 384*b^
4*(a^13*b)^(1/2) - 3465*a^10*b - 1024*a^6*b^5 + 5084*a^7*b^4 - 10045*a^8*b^3 + 9306*a^9*b^2 - 2000*a*b^3*(a^13
*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^
13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2)*2i)/d - (atan(((((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 214
9580800*a^13*b^7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b +
a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) - (cos(c + d*x)*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 346
5*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b
*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 -
10*a^15*b^2)))^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 +
1476395008*a^16*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*((20
25*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9
306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a
^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) + (cos(c + d*x)*(75497472*a^6*b^9 -
337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(524288*(a^12 - 4*a^11*b + a^
8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1024*a^6*
b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 44
29*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2
) - (12582912*a^5*b^9 - 70631424*a^6*b^8 + 157676032*a^7*b^7 - 165152384*a^8*b^6 + 69895552*a^9*b^5)/(1048576*
(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) +
3465*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^
3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^
3 - 10*a^15*b^2)))^(1/2) - (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6 +
8247825*a^4*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*((2025*a^4*(a^13*b)^(1/2) +
384*b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3
*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5
- 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2)*1i - (((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 2149580
800*a^13*b^7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b + a^1
0*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) + (cos(c + d*x)*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^
10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^
13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*
a^15*b^2)))^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 1476
395008*a^16*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*((2025*a
^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*
a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*
b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) - (cos(c + d*x)*(75497472*a^6*b^9 - 3372
15488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^
4 - 4*a^9*b^3 + 6*a^10*b^2)))*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1024*a^6*b^5
- 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a
^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) -
(12582912*a^5*b^9 - 70631424*a^6*b^8 + 157676032*a^7*b^7 - 165152384*a^8*b^6 + 69895552*a^9*b^5)/(1048576*(a^1
4 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465
*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*
(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 -
10*a^15*b^2)))^(1/2) + (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6 + 824
7825*a^4*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*((2025*a^4*(a^13*b)^(1/2) + 384*
b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^
13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*
a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2)*1i)/((((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 2149580800*
a^13*b^7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b + a^10*b^
4 - 4*a^11*b^3 + 6*a^12*b^2)) - (cos(c + d*x)*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^10*b
+ 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b
)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15
*b^2)))^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 14763950
08*a^16*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*((2025*a^4*(
a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*
b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b -
a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) + (cos(c + d*x)*(75497472*a^6*b^9 - 33721548
8*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 -
4*a^9*b^3 + 6*a^10*b^2)))*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1024*a^6*b^5 - 50
84*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b
^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) - (125
82912*a^5*b^9 - 70631424*a^6*b^8 + 157676032*a^7*b^7 - 165152384*a^8*b^6 + 69895552*a^9*b^5)/(1048576*(a^14 -
4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^1
0*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^1
3*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a
^15*b^2)))^(1/2) - (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6 + 8247825
*a^4*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*
(a^13*b)^(1/2) + 3465*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b
)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13
*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) + (((((201326592*a^11*b^9 - 1038090240*a^12*b^8 + 2149580800*a^13*b^
7 - 2258632704*a^14*b^6 + 1214251008*a^15*b^5 - 268435456*a^16*b^4)/(1048576*(a^14 - 4*a^13*b + a^10*b^4 - 4*a
^11*b^3 + 6*a^12*b^2)) + (cos(c + d*x)*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1024
*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2)
+ 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))
^(1/2)*(402653184*a^12*b^9 - 1879048192*a^13*b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 1476395008*a^16
*b^5 - 268435456*a^17*b^4))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*((2025*a^4*(a^13*b)
^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2
000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 +
a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) - (cos(c + d*x)*(75497472*a^6*b^9 - 337215488*a^7*b
^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b
^3 + 6*a^10*b^2)))*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1024*a^6*b^5 - 5084*a^7*
b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^1
3*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2)))^(1/2) - (12582912*a
^5*b^9 - 70631424*a^6*b^8 + 157676032*a^7*b^7 - 165152384*a^8*b^6 + 69895552*a^9*b^5)/(1048576*(a^14 - 4*a^13*
b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)))*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) + 3465*a^10*b + 1
024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a^3*b*(a^13*b)^(1
/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b^3 - 10*a^15*b^2
)))^(1/2) + (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 26453264*a^2*b^7 - 23076232*a^3*b^6 + 8247825*a^4*b^
5))/(524288*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b
)^(1/2) + 3465*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2)
- 4694*a^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 +
10*a^14*b^3 - 10*a^15*b^2)))^(1/2) + (1440144*a*b^7 - 331776*b^8 - 2174760*a^2*b^6 + 1184625*a^3*b^5)/(524288*
(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2))))*((2025*a^4*(a^13*b)^(1/2) + 384*b^4*(a^13*b)^(1/2) +
3465*a^10*b + 1024*a^6*b^5 - 5084*a^7*b^4 + 10045*a^8*b^3 - 9306*a^9*b^2 - 2000*a*b^3*(a^13*b)^(1/2) - 4694*a
^3*b*(a^13*b)^(1/2) + 4429*a^2*b^2*(a^13*b)^(1/2))/(16384*(5*a^16*b - a^17 + a^12*b^5 - 5*a^13*b^4 + 10*a^14*b
^3 - 10*a^15*b^2)))^(1/2)*2i)/d