\(\int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx\) [93]

Optimal result

Integrand size = 21, antiderivative size = 184 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {130 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^3}-\frac {268 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^2}+\frac {146 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))}-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4} \]

[Out]

130/693*sin(d*x+c)/a^6/d/(1+cos(d*x+c))^3-268/693*sin(d*x+c)/a^6/d/(1+cos(d*x+c))^2+146/693*sin(d*x+c)/a^6/d/(
1+cos(d*x+c))-1/11*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))^6-14/99*cos(d*x+c)^3*sin(d*x+c)/a/d/(a+a*cos(d*x
+c))^5-118/693*cos(d*x+c)^2*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^4

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2844, 3056, 3047, 3098, 2829, 2727} \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {146 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)}-\frac {268 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^2}+\frac {130 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^3}-\frac {118 \sin (c+d x) \cos ^2(c+d x)}{693 a^2 d (a \cos (c+d x)+a)^4}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac {14 \sin (c+d x) \cos ^3(c+d x)}{99 a d (a \cos (c+d x)+a)^5} \]

[In]

Int[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^6,x]

[Out]

(130*Sin[c + d*x])/(693*a^6*d*(1 + Cos[c + d*x])^3) - (268*Sin[c + d*x])/(693*a^6*d*(1 + Cos[c + d*x])^2) + (1
46*Sin[c + d*x])/(693*a^6*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^4*Sin[c + d*x])/(11*d*(a + a*Cos[c + d*x])^6)
- (14*Cos[c + d*x]^3*Sin[c + d*x])/(99*a*d*(a + a*Cos[c + d*x])^5) - (118*Cos[c + d*x]^2*Sin[c + d*x])/(693*a^
2*d*(a + a*Cos[c + d*x])^4)

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2829

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*
c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(a*f*(2*m + 1))), x] + Dist[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)
), Int[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 -
b^2, 0] && LtQ[m, -2^(-1)]

Rule 2844

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Dist[1/
(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n -
1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ
[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3056

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x]
)^n/(a*f*(2*m + 1))), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 3098

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_
.)*(x_)]^2), x_Symbol] :> Simp[(A*b - a*B + b*C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(a*f*(2*m + 1))), x] + D
ist[1/(a^2*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*A*(m + 1) + m*(b*B - a*C) + b*C*(2*m + 1)*Sin[e
+ f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {\int \frac {\cos ^3(c+d x) (4 a-10 a \cos (c+d x))}{(a+a \cos (c+d x))^5} \, dx}{11 a^2} \\ & = -\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {\int \frac {\cos ^2(c+d x) \left (42 a^2-76 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx}{99 a^4} \\ & = -\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac {\int \frac {\cos (c+d x) \left (236 a^3-414 a^3 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{693 a^6} \\ & = -\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac {\int \frac {236 a^3 \cos (c+d x)-414 a^3 \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{693 a^6} \\ & = \frac {130 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^3}-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}+\frac {\int \frac {-1950 a^4+2070 a^4 \cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{3465 a^8} \\ & = \frac {130 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^3}-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac {268 \sin (c+d x)}{693 d \left (a^3+a^3 \cos (c+d x)\right )^2}+\frac {146 \int \frac {1}{a+a \cos (c+d x)} \, dx}{693 a^5} \\ & = \frac {130 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^3}-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac {268 \sin (c+d x)}{693 d \left (a^3+a^3 \cos (c+d x)\right )^2}+\frac {146 \sin (c+d x)}{693 d \left (a^6+a^6 \cos (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.41 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\left (8+48 \cos (c+d x)+124 \cos ^2(c+d x)+184 \cos ^3(c+d x)+183 \cos ^4(c+d x)+146 \cos ^5(c+d x)\right ) \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^6} \]

[In]

Integrate[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^6,x]

[Out]

((8 + 48*Cos[c + d*x] + 124*Cos[c + d*x]^2 + 184*Cos[c + d*x]^3 + 183*Cos[c + d*x]^4 + 146*Cos[c + d*x]^5)*Sin
[c + d*x])/(693*a^6*d*(1 + Cos[c + d*x])^6)

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.45

method	result	size
parallelrisch	\(-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {55 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {110 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-22 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {55 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-11\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{352 a^{6} d}\)	\(83\)
derivativedivides	\(\frac {-\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}+\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d \,a^{6}}\)	\(84\)
default	\(\frac {-\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}+\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d \,a^{6}}\)	\(84\)
risch	\(\frac {2 i \left (693 \,{\mathrm e}^{10 i \left (d x +c \right )}+3465 \,{\mathrm e}^{9 i \left (d x +c \right )}+11550 \,{\mathrm e}^{8 i \left (d x +c \right )}+23100 \,{\mathrm e}^{7 i \left (d x +c \right )}+33726 \,{\mathrm e}^{6 i \left (d x +c \right )}+33726 \,{\mathrm e}^{5 i \left (d x +c \right )}+25080 \,{\mathrm e}^{4 i \left (d x +c \right )}+12540 \,{\mathrm e}^{3 i \left (d x +c \right )}+4565 \,{\mathrm e}^{2 i \left (d x +c \right )}+913 \,{\mathrm e}^{i \left (d x +c \right )}+146\right )}{693 d \,a^{6} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{11}}\)	\(135\)

[In]

int(cos(d*x+c)^5/(a+cos(d*x+c)*a)^6,x,method=_RETURNVERBOSE)

[Out]

-1/352*(tan(1/2*d*x+1/2*c)^10-55/9*tan(1/2*d*x+1/2*c)^8+110/7*tan(1/2*d*x+1/2*c)^6-22*tan(1/2*d*x+1/2*c)^4+55/
3*tan(1/2*d*x+1/2*c)^2-11)*tan(1/2*d*x+1/2*c)/a^6/d

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {{\left (146 \, \cos \left (d x + c\right )^{5} + 183 \, \cos \left (d x + c\right )^{4} + 184 \, \cos \left (d x + c\right )^{3} + 124 \, \cos \left (d x + c\right )^{2} + 48 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{693 \, {\left (a^{6} d \cos \left (d x + c\right )^{6} + 6 \, a^{6} d \cos \left (d x + c\right )^{5} + 15 \, a^{6} d \cos \left (d x + c\right )^{4} + 20 \, a^{6} d \cos \left (d x + c\right )^{3} + 15 \, a^{6} d \cos \left (d x + c\right )^{2} + 6 \, a^{6} d \cos \left (d x + c\right ) + a^{6} d\right )}} \]

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^6,x, algorithm="fricas")

[Out]

1/693*(146*cos(d*x + c)^5 + 183*cos(d*x + c)^4 + 184*cos(d*x + c)^3 + 124*cos(d*x + c)^2 + 48*cos(d*x + c) + 8
)*sin(d*x + c)/(a^6*d*cos(d*x + c)^6 + 6*a^6*d*cos(d*x + c)^5 + 15*a^6*d*cos(d*x + c)^4 + 20*a^6*d*cos(d*x + c
)^3 + 15*a^6*d*cos(d*x + c)^2 + 6*a^6*d*cos(d*x + c) + a^6*d)

Sympy [A] (verification not implemented)

Time = 11.97 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\begin {cases} - \frac {\tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{352 a^{6} d} + \frac {5 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{288 a^{6} d} - \frac {5 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{112 a^{6} d} + \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{6} d} - \frac {5 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{96 a^{6} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32 a^{6} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{6}} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**5/(a+a*cos(d*x+c))**6,x)

[Out]

Piecewise((-tan(c/2 + d*x/2)**11/(352*a**6*d) + 5*tan(c/2 + d*x/2)**9/(288*a**6*d) - 5*tan(c/2 + d*x/2)**7/(11
2*a**6*d) + tan(c/2 + d*x/2)**5/(16*a**6*d) - 5*tan(c/2 + d*x/2)**3/(96*a**6*d) + tan(c/2 + d*x/2)/(32*a**6*d)
, Ne(d, 0)), (x*cos(c)**5/(a*cos(c) + a)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\frac {693 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1155 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1386 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {990 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {385 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {63 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{22176 \, a^{6} d} \]

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^6,x, algorithm="maxima")

[Out]

1/22176*(693*sin(d*x + c)/(cos(d*x + c) + 1) - 1155*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1386*sin(d*x + c)^5/
(cos(d*x + c) + 1)^5 - 990*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 385*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 63*
sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/(a^6*d)

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.46 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=-\frac {63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 385 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 990 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1386 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 693 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{22176 \, a^{6} d} \]

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^6,x, algorithm="giac")

[Out]

-1/22176*(63*tan(1/2*d*x + 1/2*c)^11 - 385*tan(1/2*d*x + 1/2*c)^9 + 990*tan(1/2*d*x + 1/2*c)^7 - 1386*tan(1/2*
d*x + 1/2*c)^5 + 1155*tan(1/2*d*x + 1/2*c)^3 - 693*tan(1/2*d*x + 1/2*c))/(a^6*d)

Mupad [B] (verification not implemented)

Time = 15.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.41 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\frac {495\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}+\frac {495\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{16}+\frac {275\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}+\frac {55\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{8}+\frac {73\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{16}}{22176\,a^6\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}} \]

[In]

int(cos(c + d*x)^5/(a + a*cos(c + d*x))^6,x)

[Out]

((495*sin((3*c)/2 + (3*d*x)/2))/8 + (495*sin((5*c)/2 + (5*d*x)/2))/16 + (275*sin((7*c)/2 + (7*d*x)/2))/8 + (55
*sin((9*c)/2 + (9*d*x)/2))/8 + (73*sin((11*c)/2 + (11*d*x)/2))/16)/(22176*a^6*d*cos(c/2 + (d*x)/2)^11)