\(\int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx\) [391]

Optimal result

Integrand size = 23, antiderivative size = 299 \[ \int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {2 (-1)^{2/3} b^{2/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{2/3} \left (a^{2/3}-(-1)^{2/3} b^{2/3}\right )^{3/2} d}-\frac {2 b^{2/3} \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{2/3} \left (a^{2/3}-b^{2/3}\right )^{3/2} d}+\frac {2 \sqrt [3]{-1} b^{2/3} \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{2/3} \left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right )^{3/2} d}+\frac {\sec (c+d x) (b-a \sin (c+d x))}{\left (-a^2+b^2\right ) d} \]

[Out]

2/3*(-1)^(2/3)*b^(2/3)*arctan(((-1)^(1/3)*b^(1/3)-a^(1/3)*tan(1/2*d*x+1/2*c))/(a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/
2))/a^(2/3)/(a^(2/3)-(-1)^(2/3)*b^(2/3))^(3/2)/d-2/3*b^(2/3)*arctan((b^(1/3)+a^(1/3)*tan(1/2*d*x+1/2*c))/(a^(2
/3)-b^(2/3))^(1/2))/a^(2/3)/(a^(2/3)-b^(2/3))^(3/2)/d+2/3*(-1)^(1/3)*b^(2/3)*arctan(((-1)^(2/3)*b^(1/3)+a^(1/3
)*tan(1/2*d*x+1/2*c))/(a^(2/3)+(-1)^(1/3)*b^(2/3))^(1/2))/a^(2/3)/(a^(2/3)+(-1)^(1/3)*b^(2/3))^(3/2)/d+sec(d*x
+c)*(b-a*sin(d*x+c))/(-a^2+b^2)/d

Rubi [F]

\[ \int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx \]

[In]

Int[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]

[Out]

Defer[Int][Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.52 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.44 \[ \int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {-6 b+6 b \cos (c+d x)-i b \cos (c+d x) \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+4 i a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-12 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+6 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-4 i a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]+6 a \sin (c+d x)}{6 (a-b) (a+b) d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]

[In]

Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]

[Out]

(-6*b + 6*b*Cos[c + d*x] - I*b*Cos[c + d*x]*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6
& , (2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (4*I)*a*ArcTan[Si
n[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 2*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 12*b*ArcTan[Sin[c + d*x]/(Cos[
c + d*x] - #1)]*#1^2 + (6*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (4*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d
*x] - #1)]*#1^3 - 2*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1
^4 - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + 6*a*Sin[c + d
*x])/(6*(a - b)*(a + b)*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))

Maple [C] (verified)

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

Time = 2.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.54

method	result	size
derivativedivides	\(\frac {-\frac {2}{\left (2 a -2 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{4} b +2 \textit {\_R}^{3} a -6 \textit {\_R}^{2} b +2 \textit {\_R} a -b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 \left (a -b \right ) \left (a +b \right )}-\frac {2}{\left (2 a +2 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(162\)
default	\(\frac {-\frac {2}{\left (2 a -2 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{4} b +2 \textit {\_R}^{3} a -6 \textit {\_R}^{2} b +2 \textit {\_R} a -b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 \left (a -b \right ) \left (a +b \right )}-\frac {2}{\left (2 a +2 b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(162\)
risch	\(\text {Expression too large to display}\)	\(1344\)

[In]

int(sec(d*x+c)^2/(a+b*sin(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

1/d*(-2/(2*a-2*b)/(tan(1/2*d*x+1/2*c)+1)+1/3*b/(a-b)/(a+b)*sum((-_R^4*b+2*_R^3*a-6*_R^2*b+2*_R*a-b)/(_R^5*a+2*
_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))-2/(2*a+2*b)/(t
an(1/2*d*x+1/2*c)-1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.89 (sec) , antiderivative size = 59362, normalized size of antiderivative = 198.54 \[ \int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \]

[In]

integrate(sec(d*x+c)**2/(a+b*sin(d*x+c)**3),x)

[Out]

Integral(sec(c + d*x)**2/(a + b*sin(c + d*x)**3), x)

Maxima [F]

\[ \int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{b \sin \left (d x + c\right )^{3} + a} \,d x } \]

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm="maxima")

[Out]

-(2*b*cos(2*d*x + 2*c)*cos(d*x + c) + 2*b*cos(d*x + c) - ((a^2 - b^2)*d*cos(2*d*x + 2*c)^2 + (a^2 - b^2)*d*sin
(2*d*x + 2*c)^2 + 2*(a^2 - b^2)*d*cos(2*d*x + 2*c) + (a^2 - b^2)*d)*integrate(2*(6*a*b^2*cos(4*d*x + 4*c)^2 -
48*a*b^2*cos(3*d*x + 3*c)^2 + 6*a*b^2*cos(2*d*x + 2*c)^2 + 6*a*b^2*sin(4*d*x + 4*c)^2 - 48*a*b^2*sin(3*d*x + 3
*c)^2 - 3*b^3*cos(d*x + c)*sin(2*d*x + 2*c) + 6*a*b^2*sin(2*d*x + 2*c)^2 - b^3*sin(d*x + c) - (2*a*b^2*cos(4*d
*x + 4*c) - 2*a*b^2*cos(2*d*x + 2*c) - b^3*sin(5*d*x + 5*c) + 6*b^3*sin(3*d*x + 3*c) - b^3*sin(d*x + c))*cos(6
*d*x + 6*c) + (8*a*b^2*cos(3*d*x + 3*c) + 3*b^3*sin(4*d*x + 4*c) - 3*b^3*sin(2*d*x + 2*c))*cos(5*d*x + 5*c) -
(12*a*b^2*cos(2*d*x + 2*c) + 3*b^3*sin(d*x + c) - 2*a*b^2 + 2*(8*a^2*b - 9*b^3)*sin(3*d*x + 3*c))*cos(4*d*x +
4*c) + 2*(4*a*b^2*cos(d*x + c) - (8*a^2*b - 9*b^3)*sin(2*d*x + 2*c))*cos(3*d*x + 3*c) + (3*b^3*sin(d*x + c) -
2*a*b^2)*cos(2*d*x + 2*c) - (b^3*cos(5*d*x + 5*c) - 6*b^3*cos(3*d*x + 3*c) + b^3*cos(d*x + c) + 2*a*b^2*sin(4*
d*x + 4*c) - 2*a*b^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) - (3*b^3*cos(4*d*x + 4*c) - 3*b^3*cos(2*d*x + 2*c) - 8
*a*b^2*sin(3*d*x + 3*c) + b^3)*sin(5*d*x + 5*c) + (3*b^3*cos(d*x + c) - 12*a*b^2*sin(2*d*x + 2*c) + 2*(8*a^2*b
- 9*b^3)*cos(3*d*x + 3*c))*sin(4*d*x + 4*c) + 2*(4*a*b^2*sin(d*x + c) + 3*b^3 + (8*a^2*b - 9*b^3)*cos(2*d*x +
2*c))*sin(3*d*x + 3*c))/(a^2*b^2 - b^4 + (a^2*b^2 - b^4)*cos(6*d*x + 6*c)^2 + 9*(a^2*b^2 - b^4)*cos(4*d*x + 4
*c)^2 + 64*(a^4 - a^2*b^2)*cos(3*d*x + 3*c)^2 + 9*(a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + (a^2*b^2 - b^4)*sin(6*d
*x + 6*c)^2 + 9*(a^2*b^2 - b^4)*sin(4*d*x + 4*c)^2 + 64*(a^4 - a^2*b^2)*sin(3*d*x + 3*c)^2 - 48*(a^3*b - a*b^3
)*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) + 9*(a^2*b^2 - b^4)*sin(2*d*x + 2*c)^2 - 2*(a^2*b^2 - b^4 + 3*(a^2*b^2 - b
^4)*cos(4*d*x + 4*c) - 3*(a^2*b^2 - b^4)*cos(2*d*x + 2*c) - 8*(a^3*b - a*b^3)*sin(3*d*x + 3*c))*cos(6*d*x + 6*
c) + 6*(a^2*b^2 - b^4 - 3*(a^2*b^2 - b^4)*cos(2*d*x + 2*c) - 8*(a^3*b - a*b^3)*sin(3*d*x + 3*c))*cos(4*d*x + 4
*c) - 6*(a^2*b^2 - b^4)*cos(2*d*x + 2*c) - 2*(8*(a^3*b - a*b^3)*cos(3*d*x + 3*c) + 3*(a^2*b^2 - b^4)*sin(4*d*x
+ 4*c) - 3*(a^2*b^2 - b^4)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 6*(8*(a^3*b - a*b^3)*cos(3*d*x + 3*c) - 3*(a^
2*b^2 - b^4)*sin(2*d*x + 2*c))*sin(4*d*x + 4*c) - 16*(a^3*b - a*b^3 - 3*(a^3*b - a*b^3)*cos(2*d*x + 2*c))*sin(
3*d*x + 3*c)), x) + 2*(b*sin(d*x + c) - a)*sin(2*d*x + 2*c))/((a^2 - b^2)*d*cos(2*d*x + 2*c)^2 + (a^2 - b^2)*d
*sin(2*d*x + 2*c)^2 + 2*(a^2 - b^2)*d*cos(2*d*x + 2*c) + (a^2 - b^2)*d)

Giac [F]

\[ \int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{b \sin \left (d x + c\right )^{3} + a} \,d x } \]

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 17.20 (sec) , antiderivative size = 19737, normalized size of antiderivative = 66.01 \[ \int \frac {\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^2*(a + b*sin(c + d*x)^3)),x)

[Out]

(a^2*symsum(log((8192*(a^2*b^20*cos(c/2 + (d*x)/2) - 12*a*b^21*sin(c/2 + (d*x)/2) - 7*a^4*b^18*cos(c/2 + (d*x)
/2) + 21*a^6*b^16*cos(c/2 + (d*x)/2) - 35*a^8*b^14*cos(c/2 + (d*x)/2) + 35*a^10*b^12*cos(c/2 + (d*x)/2) - 21*a
^12*b^10*cos(c/2 + (d*x)/2) + 7*a^14*b^8*cos(c/2 + (d*x)/2) - a^16*b^6*cos(c/2 + (d*x)/2) + 84*a^3*b^19*sin(c/
2 + (d*x)/2) - 252*a^5*b^17*sin(c/2 + (d*x)/2) + 420*a^7*b^15*sin(c/2 + (d*x)/2) - 420*a^9*b^13*sin(c/2 + (d*x
)/2) + 252*a^11*b^11*sin(c/2 + (d*x)/2) - 84*a^13*b^9*sin(c/2 + (d*x)/2) + 12*a^15*b^7*sin(c/2 + (d*x)/2) - 19
8*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d
^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^3*b^20*sin(c/2 + (d*x)/2) + 714*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 +
729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^5*b^18*sin
(c/2 + (d*x)/2) - 1470*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^
4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^7*b^16*sin(c/2 + (d*x)/2) + 1890*root(2187*a^8*b^2*d^6
- 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b
^4, d, k)*a^9*b^14*sin(c/2 + (d*x)/2) - 1554*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*
a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^11*b^12*sin(c/2 + (d*x)/2) + 798
*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^
4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^13*b^10*sin(c/2 + (d*x)/2) - 234*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 +
729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^15*b^8*sin
(c/2 + (d*x)/2) + 30*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*
d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^17*b^6*sin(c/2 + (d*x)/2) + 36*root(2187*a^8*b^2*d^6 - 2
187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4,
d, k)^2*a^2*b^22*cos(c/2 + (d*x)/2) - 369*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^1
0*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^4*b^20*cos(c/2 + (d*x)/2) + 1575*
root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4
- 81*a^2*b^4*d^2 - b^4, d, k)^2*a^6*b^18*cos(c/2 + (d*x)/2) - 3717*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 +
729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^8*b^16*
cos(c/2 + (d*x)/2) + 5355*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4
*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^10*b^14*cos(c/2 + (d*x)/2) - 4851*root(2187*a^8*b
^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d
^2 - b^4, d, k)^2*a^12*b^12*cos(c/2 + (d*x)/2) + 2709*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d
^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^14*b^10*cos(c/2 + (d*
x)/2) - 855*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729
*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^16*b^8*cos(c/2 + (d*x)/2) + 117*root(2187*a^8*b^2*d^6 - 2187*a^
6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^
2*a^18*b^6*cos(c/2 + (d*x)/2) - 1188*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6
- 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^4*b^21*cos(c/2 + (d*x)/2) + 7803*root(
2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81
*a^2*b^4*d^2 - b^4, d, k)^3*a^6*b^19*cos(c/2 + (d*x)/2) - 21357*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729
*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^8*b^17*cos(
c/2 + (d*x)/2) + 30807*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^
4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^10*b^15*cos(c/2 + (d*x)/2) - 23625*root(2187*a^8*b^2
*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2
- b^4, d, k)^3*a^12*b^13*cos(c/2 + (d*x)/2) + 6993*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6
- 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^14*b^11*cos(c/2 + (d*x)
/2) + 2457*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*
a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^16*b^9*cos(c/2 + (d*x)/2) - 2403*root(2187*a^8*b^2*d^6 - 2187*a^
6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^
3*a^18*b^7*cos(c/2 + (d*x)/2) + 513*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6
- 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^20*b^5*cos(c/2 + (d*x)/2) + 567*root(21
87*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a
^2*b^4*d^2 - b^4, d, k)^4*a^4*b^22*cos(c/2 + (d*x)/2) - 4374*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^
4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^6*b^20*cos(c/2
+ (d*x)/2) + 14580*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d
^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^8*b^18*cos(c/2 + (d*x)/2) - 27216*root(2187*a^8*b^2*d^6
- 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b
^4, d, k)^4*a^10*b^16*cos(c/2 + (d*x)/2) + 30618*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 -
729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^12*b^14*cos(c/2 + (d*x)/2)
- 20412*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^
6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^14*b^12*cos(c/2 + (d*x)/2) + 6804*root(2187*a^8*b^2*d^6 - 2187*a^6
*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4
*a^16*b^10*cos(c/2 + (d*x)/2) - 729*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6
- 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^20*b^6*cos(c/2 + (d*x)/2) + 162*root(21
87*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a
^2*b^4*d^2 - b^4, d, k)^4*a^22*b^4*cos(c/2 + (d*x)/2) + 972*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4
*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^4*b^23*cos(c/2
+ (d*x)/2) - 9477*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4
- 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^6*b^21*cos(c/2 + (d*x)/2) + 41553*root(2187*a^8*b^2*d^6 -
2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4
, d, k)^5*a^8*b^19*cos(c/2 + (d*x)/2) - 107892*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 72
9*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^10*b^17*cos(c/2 + (d*x)/2) +
183708*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6
*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^12*b^15*cos(c/2 + (d*x)/2) - 214326*root(2187*a^8*b^2*d^6 - 2187*a^
6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^
5*a^14*b^13*cos(c/2 + (d*x)/2) + 173502*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*
d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^16*b^11*cos(c/2 + (d*x)/2) - 96228*
root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4
- 81*a^2*b^4*d^2 - b^4, d, k)^5*a^18*b^9*cos(c/2 + (d*x)/2) + 34992*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6
+ 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^20*b^7
*cos(c/2 + (d*x)/2) - 7533*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^
4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^22*b^5*cos(c/2 + (d*x)/2) + 729*root(2187*a^8*b^
2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^
2 - b^4, d, k)^5*a^24*b^3*cos(c/2 + (d*x)/2) - 396*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6
- 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^3*b^21*sin(c/2 + (d*x)/2
) + 2736*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^
6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^5*b^19*sin(c/2 + (d*x)/2) - 8064*root(2187*a^8*b^2*d^6 - 2187*a^6*
b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*
a^7*b^17*sin(c/2 + (d*x)/2) + 13104*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6
- 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^9*b^15*sin(c/2 + (d*x)/2) - 12600*root(
2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81
*a^2*b^4*d^2 - b^4, d, k)^2*a^11*b^13*sin(c/2 + (d*x)/2) + 7056*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729
*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^13*b^11*sin
(c/2 + (d*x)/2) - 2016*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^
4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^15*b^9*sin(c/2 + (d*x)/2) + 144*root(2187*a^8*b^2*d^
6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 -
b^4, d, k)^2*a^17*b^7*sin(c/2 + (d*x)/2) + 36*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729
*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^19*b^5*sin(c/2 + (d*x)/2) + 6
48*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*
d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^3*b^22*sin(c/2 + (d*x)/2) - 3456*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^
6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^5*b^
20*sin(c/2 + (d*x)/2) + 6021*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*
a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^7*b^18*sin(c/2 + (d*x)/2) + 189*root(2187*a^8*
b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*
d^2 - b^4, d, k)^3*a^9*b^16*sin(c/2 + (d*x)/2) - 15687*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*
d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^11*b^14*sin(c/2 + (d
*x)/2) + 25137*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 -
729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^13*b^12*sin(c/2 + (d*x)/2) - 19089*root(2187*a^8*b^2*d^6 - 2
187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4,
d, k)^3*a^15*b^10*sin(c/2 + (d*x)/2) + 7479*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a
^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^17*b^8*sin(c/2 + (d*x)/2) - 126
9*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d
^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^19*b^6*sin(c/2 + (d*x)/2) + 27*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 +
729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^21*b^4*
sin(c/2 + (d*x)/2) + 648*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*
b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^3*b^23*sin(c/2 + (d*x)/2) - 4860*root(2187*a^8*b^2
*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2
- b^4, d, k)^4*a^5*b^21*sin(c/2 + (d*x)/2) + 15552*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6
- 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^7*b^19*sin(c/2 + (d*x)/
2) - 27216*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*
a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^9*b^17*sin(c/2 + (d*x)/2) + 27216*root(2187*a^8*b^2*d^6 - 2187*a
^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)
^4*a^11*b^15*sin(c/2 + (d*x)/2) - 13608*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*
d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^13*b^13*sin(c/2 + (d*x)/2) + 3888*r
oot(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4
- 81*a^2*b^4*d^2 - b^4, d, k)^4*a^17*b^9*sin(c/2 + (d*x)/2) - 1944*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 +
729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^19*b^7*s
in(c/2 + (d*x)/2) + 324*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b
^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^21*b^5*sin(c/2 + (d*x)/2) + 243*root(2187*a^8*b^2*d
^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 -
b^4, d, k)^5*a^5*b^22*sin(c/2 + (d*x)/2) - 2187*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 -
729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^7*b^20*sin(c/2 + (d*x)/2)
+ 8748*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*
b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^9*b^18*sin(c/2 + (d*x)/2) - 20412*root(2187*a^8*b^2*d^6 - 2187*a^6*b
^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a
^11*b^16*sin(c/2 + (d*x)/2) + 30618*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6
- 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^13*b^14*sin(c/2 + (d*x)/2) - 30618*root
(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 8
1*a^2*b^4*d^2 - b^4, d, k)^5*a^15*b^12*sin(c/2 + (d*x)/2) + 20412*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 7
29*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^17*b^10*s
in(c/2 + (d*x)/2) - 8748*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*
b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^19*b^8*sin(c/2 + (d*x)/2) + 2187*root(2187*a^8*b^2
*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2
- b^4, d, k)^5*a^21*b^6*sin(c/2 + (d*x)/2) - 243*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 -
729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^23*b^4*sin(c/2 + (d*x)/2)
+ 24*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b
^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a*b^22*sin(c/2 + (d*x)/2) - 33*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 +
729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^2*b^21*co
s(c/2 + (d*x)/2) + 231*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^
4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^4*b^19*cos(c/2 + (d*x)/2) - 693*root(2187*a^8*b^2*d^6
- 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^
4, d, k)*a^6*b^17*cos(c/2 + (d*x)/2) + 1155*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a
^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^8*b^15*cos(c/2 + (d*x)/2) - 1155*
root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4
- 81*a^2*b^4*d^2 - b^4, d, k)*a^10*b^13*cos(c/2 + (d*x)/2) + 693*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 7
29*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^12*b^11*cos
(c/2 + (d*x)/2) - 231*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4
*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^14*b^9*cos(c/2 + (d*x)/2) + 33*root(2187*a^8*b^2*d^6 -
2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4,
d, k)*a^16*b^7*cos(c/2 + (d*x)/2)))/cos(c/2 + (d*x)/2))*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^
6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k), k, 1, 6))/(d*(a^2 - b
^2)) - (b^2*symsum(log((8192*(a^2*b^20*cos(c/2 + (d*x)/2) - 12*a*b^21*sin(c/2 + (d*x)/2) - 7*a^4*b^18*cos(c/2
+ (d*x)/2) + 21*a^6*b^16*cos(c/2 + (d*x)/2) - 35*a^8*b^14*cos(c/2 + (d*x)/2) + 35*a^10*b^12*cos(c/2 + (d*x)/2)
- 21*a^12*b^10*cos(c/2 + (d*x)/2) + 7*a^14*b^8*cos(c/2 + (d*x)/2) - a^16*b^6*cos(c/2 + (d*x)/2) + 84*a^3*b^19
*sin(c/2 + (d*x)/2) - 252*a^5*b^17*sin(c/2 + (d*x)/2) + 420*a^7*b^15*sin(c/2 + (d*x)/2) - 420*a^9*b^13*sin(c/2
+ (d*x)/2) + 252*a^11*b^11*sin(c/2 + (d*x)/2) - 84*a^13*b^9*sin(c/2 + (d*x)/2) + 12*a^15*b^7*sin(c/2 + (d*x)/
2) - 198*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^
6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^3*b^20*sin(c/2 + (d*x)/2) + 714*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4
*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^5*b
^18*sin(c/2 + (d*x)/2) - 1470*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458
*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^7*b^16*sin(c/2 + (d*x)/2) + 1890*root(2187*a^8*
b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*
d^2 - b^4, d, k)*a^9*b^14*sin(c/2 + (d*x)/2) - 1554*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6
- 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^11*b^12*sin(c/2 + (d*x)/2
) + 798*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6
*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^13*b^10*sin(c/2 + (d*x)/2) - 234*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4
*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^15*
b^8*sin(c/2 + (d*x)/2) + 30*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a
^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^17*b^6*sin(c/2 + (d*x)/2) + 36*root(2187*a^8*b^2*
d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2
- b^4, d, k)^2*a^2*b^22*cos(c/2 + (d*x)/2) - 369*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 -
729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^4*b^20*cos(c/2 + (d*x)/2)
+ 1575*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*
b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^6*b^18*cos(c/2 + (d*x)/2) - 3717*root(2187*a^8*b^2*d^6 - 2187*a^6*b^
4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^
8*b^16*cos(c/2 + (d*x)/2) + 5355*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1
458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^10*b^14*cos(c/2 + (d*x)/2) - 4851*root(218
7*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^
2*b^4*d^2 - b^4, d, k)^2*a^12*b^12*cos(c/2 + (d*x)/2) + 2709*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^
4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^14*b^10*cos(c/
2 + (d*x)/2) - 855*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^
4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^16*b^8*cos(c/2 + (d*x)/2) + 117*root(2187*a^8*b^2*d^6 -
2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4,
d, k)^2*a^18*b^6*cos(c/2 + (d*x)/2) - 1188*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a
^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^4*b^21*cos(c/2 + (d*x)/2) + 780
3*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d
^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^6*b^19*cos(c/2 + (d*x)/2) - 21357*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^
6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^8*b^
17*cos(c/2 + (d*x)/2) + 30807*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458
*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^10*b^15*cos(c/2 + (d*x)/2) - 23625*root(2187*
a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*
b^4*d^2 - b^4, d, k)^3*a^12*b^13*cos(c/2 + (d*x)/2) + 6993*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*
b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^14*b^11*cos(c/2
+ (d*x)/2) + 2457*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4
- 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^16*b^9*cos(c/2 + (d*x)/2) - 2403*root(2187*a^8*b^2*d^6 -
2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4,
d, k)^3*a^18*b^7*cos(c/2 + (d*x)/2) + 513*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^
10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^20*b^5*cos(c/2 + (d*x)/2) + 567*
root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4
- 81*a^2*b^4*d^2 - b^4, d, k)^4*a^4*b^22*cos(c/2 + (d*x)/2) - 4374*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 +
729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^6*b^20*
cos(c/2 + (d*x)/2) + 14580*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^
4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^8*b^18*cos(c/2 + (d*x)/2) - 27216*root(2187*a^8*
b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*
d^2 - b^4, d, k)^4*a^10*b^16*cos(c/2 + (d*x)/2) + 30618*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6
*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^12*b^14*cos(c/2 + (
d*x)/2) - 20412*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 -
729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^14*b^12*cos(c/2 + (d*x)/2) + 6804*root(2187*a^8*b^2*d^6 - 2
187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4,
d, k)^4*a^16*b^10*cos(c/2 + (d*x)/2) - 729*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^
10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^20*b^6*cos(c/2 + (d*x)/2) + 162*
root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4
- 81*a^2*b^4*d^2 - b^4, d, k)^4*a^22*b^4*cos(c/2 + (d*x)/2) + 972*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 +
729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^4*b^23*c
os(c/2 + (d*x)/2) - 9477*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*
b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^6*b^21*cos(c/2 + (d*x)/2) + 41553*root(2187*a^8*b^
2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^
2 - b^4, d, k)^5*a^8*b^19*cos(c/2 + (d*x)/2) - 107892*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d
^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^10*b^17*cos(c/2 + (d*
x)/2) + 183708*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 -
729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^12*b^15*cos(c/2 + (d*x)/2) - 214326*root(2187*a^8*b^2*d^6 -
2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4,
d, k)^5*a^14*b^13*cos(c/2 + (d*x)/2) + 173502*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 72
9*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^16*b^11*cos(c/2 + (d*x)/2) -
96228*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*
b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^18*b^9*cos(c/2 + (d*x)/2) + 34992*root(2187*a^8*b^2*d^6 - 2187*a^6*b
^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a
^20*b^7*cos(c/2 + (d*x)/2) - 7533*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 -
1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^22*b^5*cos(c/2 + (d*x)/2) + 729*root(2187
*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2
*b^4*d^2 - b^4, d, k)^5*a^24*b^3*cos(c/2 + (d*x)/2) - 396*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b
^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^3*b^21*sin(c/2 +
(d*x)/2) + 2736*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 -
729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^5*b^19*sin(c/2 + (d*x)/2) - 8064*root(2187*a^8*b^2*d^6 - 21
87*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d
, k)^2*a^7*b^17*sin(c/2 + (d*x)/2) + 13104*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^
10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^9*b^15*sin(c/2 + (d*x)/2) - 1260
0*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d
^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^11*b^13*sin(c/2 + (d*x)/2) + 7056*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^
6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^13*b
^11*sin(c/2 + (d*x)/2) - 2016*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458
*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^15*b^9*sin(c/2 + (d*x)/2) + 144*root(2187*a^8
*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4
*d^2 - b^4, d, k)^2*a^17*b^7*sin(c/2 + (d*x)/2) + 36*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^
6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^2*a^19*b^5*sin(c/2 + (d*x)
/2) + 648*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a
^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^3*b^22*sin(c/2 + (d*x)/2) - 3456*root(2187*a^8*b^2*d^6 - 2187*a^6
*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3
*a^5*b^20*sin(c/2 + (d*x)/2) + 6021*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6
- 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^7*b^18*sin(c/2 + (d*x)/2) + 189*root(21
87*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a
^2*b^4*d^2 - b^4, d, k)^3*a^9*b^16*sin(c/2 + (d*x)/2) - 15687*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a
^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^11*b^14*sin(c
/2 + (d*x)/2) + 25137*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4
*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^13*b^12*sin(c/2 + (d*x)/2) - 19089*root(2187*a^8*b^2*
d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2
- b^4, d, k)^3*a^15*b^10*sin(c/2 + (d*x)/2) + 7479*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6
- 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^17*b^8*sin(c/2 + (d*x)/2
) - 1269*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^
6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^19*b^6*sin(c/2 + (d*x)/2) + 27*root(2187*a^8*b^2*d^6 - 2187*a^6*b^
4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^3*a^
21*b^4*sin(c/2 + (d*x)/2) + 648*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 14
58*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^3*b^23*sin(c/2 + (d*x)/2) - 4860*root(2187*
a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*
b^4*d^2 - b^4, d, k)^4*a^5*b^21*sin(c/2 + (d*x)/2) + 15552*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*
b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^7*b^19*sin(c/2 +
(d*x)/2) - 27216*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4
- 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^9*b^17*sin(c/2 + (d*x)/2) + 27216*root(2187*a^8*b^2*d^6 -
2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4
, d, k)^4*a^11*b^15*sin(c/2 + (d*x)/2) - 13608*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 72
9*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^13*b^13*sin(c/2 + (d*x)/2) +
3888*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b
^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^17*b^9*sin(c/2 + (d*x)/2) - 1944*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4
*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^1
9*b^7*sin(c/2 + (d*x)/2) + 324*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 145
8*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^4*a^21*b^5*sin(c/2 + (d*x)/2) + 243*root(2187*a^
8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^
4*d^2 - b^4, d, k)^5*a^5*b^22*sin(c/2 + (d*x)/2) - 2187*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6
*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^7*b^20*sin(c/2 + (d
*x)/2) + 8748*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 7
29*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^9*b^18*sin(c/2 + (d*x)/2) - 20412*root(2187*a^8*b^2*d^6 - 218
7*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d,
k)^5*a^11*b^16*sin(c/2 + (d*x)/2) + 30618*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^
10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^13*b^14*sin(c/2 + (d*x)/2) - 306
18*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*
d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^15*b^12*sin(c/2 + (d*x)/2) + 20412*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*
d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^17
*b^10*sin(c/2 + (d*x)/2) - 8748*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 14
58*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^19*b^8*sin(c/2 + (d*x)/2) + 2187*root(2187*
a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*
b^4*d^2 - b^4, d, k)^5*a^21*b^6*sin(c/2 + (d*x)/2) - 243*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^
6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)^5*a^23*b^4*sin(c/2 + (
d*x)/2) + 24*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 72
9*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a*b^22*sin(c/2 + (d*x)/2) - 33*root(2187*a^8*b^2*d^6 - 2187*a^6*b^
4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^2*
b^21*cos(c/2 + (d*x)/2) + 231*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458
*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^4*b^19*cos(c/2 + (d*x)/2) - 693*root(2187*a^8*b
^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d
^2 - b^4, d, k)*a^6*b^17*cos(c/2 + (d*x)/2) + 1155*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6
- 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^8*b^15*cos(c/2 + (d*x)/2)
- 1155*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*
b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^10*b^13*cos(c/2 + (d*x)/2) + 693*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*
d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^12*b
^11*cos(c/2 + (d*x)/2) - 231*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*
a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k)*a^14*b^9*cos(c/2 + (d*x)/2) + 33*root(2187*a^8*b^2
*d^6 - 2187*a^6*b^4*d^6 + 729*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2
- b^4, d, k)*a^16*b^7*cos(c/2 + (d*x)/2)))/cos(c/2 + (d*x)/2))*root(2187*a^8*b^2*d^6 - 2187*a^6*b^4*d^6 + 729
*a^4*b^6*d^6 - 729*a^10*d^6 - 1458*a^4*b^4*d^4 - 729*a^6*b^2*d^4 - 81*a^2*b^4*d^2 - b^4, d, k), k, 1, 6))/(d*(
a^2 - b^2)) - b/(d*cos(c + d*x)*(a^2 - b^2)) + (a*sin(c + d*x))/(d*cos(c + d*x)*(a^2 - b^2))