3.706 \(\int \frac{\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=183 \[ \frac{\cos ^{11}(c+d x)}{11 a d}-\frac{2 \cos ^9(c+d x)}{9 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}-\frac{3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{160 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{128 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{256 a d}+\frac{3 x}{256 a} \]

[Out]

(3*x)/(256*a) + Cos[c + d*x]^7/(7*a*d) - (2*Cos[c + d*x]^9)/(9*a*d) + Cos[c + d*x]^11/(11*a*d) + (3*Cos[c + d*
x]*Sin[c + d*x])/(256*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(128*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(160*a*d)
- (3*Cos[c + d*x]^7*Sin[c + d*x])/(80*a*d) - (Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*a*d)

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Rubi [A]  time = 0.240174, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2839, 2568, 2635, 8, 2565, 270} \[ \frac{\cos ^{11}(c+d x)}{11 a d}-\frac{2 \cos ^9(c+d x)}{9 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}-\frac{3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{160 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{128 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{256 a d}+\frac{3 x}{256 a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^8*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(3*x)/(256*a) + Cos[c + d*x]^7/(7*a*d) - (2*Cos[c + d*x]^9)/(9*a*d) + Cos[c + d*x]^11/(11*a*d) + (3*Cos[c + d*
x]*Sin[c + d*x])/(256*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(128*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(160*a*d)
- (3*Cos[c + d*x]^7*Sin[c + d*x])/(80*a*d) - (Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*a*d)

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 2568

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \cos ^6(c+d x) \sin ^4(c+d x) \, dx}{a}-\frac{\int \cos ^6(c+d x) \sin ^5(c+d x) \, dx}{a}\\ &=-\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}+\frac{3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{10 a}+\frac{\operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac{3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}-\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}+\frac{3 \int \cos ^6(c+d x) \, dx}{80 a}+\frac{\operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac{\cos ^7(c+d x)}{7 a d}-\frac{2 \cos ^9(c+d x)}{9 a d}+\frac{\cos ^{11}(c+d x)}{11 a d}+\frac{\cos ^5(c+d x) \sin (c+d x)}{160 a d}-\frac{3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}-\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}+\frac{\int \cos ^4(c+d x) \, dx}{32 a}\\ &=\frac{\cos ^7(c+d x)}{7 a d}-\frac{2 \cos ^9(c+d x)}{9 a d}+\frac{\cos ^{11}(c+d x)}{11 a d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{128 a d}+\frac{\cos ^5(c+d x) \sin (c+d x)}{160 a d}-\frac{3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}-\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}+\frac{3 \int \cos ^2(c+d x) \, dx}{128 a}\\ &=\frac{\cos ^7(c+d x)}{7 a d}-\frac{2 \cos ^9(c+d x)}{9 a d}+\frac{\cos ^{11}(c+d x)}{11 a d}+\frac{3 \cos (c+d x) \sin (c+d x)}{256 a d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{128 a d}+\frac{\cos ^5(c+d x) \sin (c+d x)}{160 a d}-\frac{3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}-\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}+\frac{3 \int 1 \, dx}{256 a}\\ &=\frac{3 x}{256 a}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{2 \cos ^9(c+d x)}{9 a d}+\frac{\cos ^{11}(c+d x)}{11 a d}+\frac{3 \cos (c+d x) \sin (c+d x)}{256 a d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{128 a d}+\frac{\cos ^5(c+d x) \sin (c+d x)}{160 a d}-\frac{3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}-\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}\\ \end{align*}

Mathematica [B]  time = 12.2312, size = 573, normalized size = 3.13 \[ \frac{\frac{97020 \sin ^2\left (\frac{1}{2} (c+d x)\right )}{d (a \sin (c+d x)+a)}+\frac{103950 \sin (c) \sin (d x)}{a d}-\frac{66990 \sin (3 c) \sin (3 d x)}{a d}+\frac{24948 \sin (5 c) \sin (5 d x)}{a d}-\frac{1980 \sin (7 c) \sin (7 d x)}{a d}-\frac{76230 \sin (2 (c+d x))}{a d}+\frac{27720 \sin (4 (c+d x))}{a d}-\frac{11550 \sin (6 (c+d x))}{a d}+\frac{3465 \sin (8 (c+d x))}{a d}+\frac{1386 \sin (10 (c+d x))}{a d}+\frac{48510 \sin (c+d x)}{a d (\sin (c+d x)+1)}-\frac{103950 \cos (c) \cos (d x)}{a d}+\frac{66990 \cos (3 c) \cos (3 d x)}{a d}-\frac{24948 \cos (5 c) \cos (5 d x)}{a d}+\frac{1980 \cos (7 c) \cos (7 d x)}{a d}+\frac{173250 \cos (c+d x)}{a d}-\frac{43890 \cos (3 (c+d x))}{a d}+\frac{18018 \cos (5 (c+d x))}{a d}-\frac{6930 \cos (7 (c+d x))}{a d}+\frac{770 \cos (9 (c+d x))}{a d}+\frac{630 \cos (11 (c+d x))}{a d}-\frac{20790 \sin \left (\frac{1}{2} (c+d x)\right )}{a d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{90090 \sin (2 c) \cos (2 d x)}{a d}-\frac{55440 \sin (4 c) \cos (4 d x)}{a d}+\frac{4620 \sin (6 c) \cos (6 d x)}{a d}+\frac{90090 \cos (2 c) \sin (2 d x)}{a d}-\frac{55440 \cos (4 c) \sin (4 d x)}{a d}+\frac{4620 \cos (6 c) \sin (6 d x)}{a d}-\frac{76230 \sin \left (\frac{d x}{2}\right )}{a d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{97020 c}{a d}+\frac{83160 x}{a}}{7096320} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

((97020*c)/(a*d) + (83160*x)/a - (103950*Cos[c]*Cos[d*x])/(a*d) + (66990*Cos[3*c]*Cos[3*d*x])/(a*d) - (24948*C
os[5*c]*Cos[5*d*x])/(a*d) + (1980*Cos[7*c]*Cos[7*d*x])/(a*d) + (173250*Cos[c + d*x])/(a*d) - (43890*Cos[3*(c +
d*x)])/(a*d) + (18018*Cos[5*(c + d*x)])/(a*d) - (6930*Cos[7*(c + d*x)])/(a*d) + (770*Cos[9*(c + d*x)])/(a*d)
+ (630*Cos[11*(c + d*x)])/(a*d) + (90090*Cos[2*d*x]*Sin[2*c])/(a*d) - (55440*Cos[4*d*x]*Sin[4*c])/(a*d) + (462
0*Cos[6*d*x]*Sin[6*c])/(a*d) + (103950*Sin[c]*Sin[d*x])/(a*d) + (90090*Cos[2*c]*Sin[2*d*x])/(a*d) - (66990*Sin
[3*c]*Sin[3*d*x])/(a*d) - (55440*Cos[4*c]*Sin[4*d*x])/(a*d) + (24948*Sin[5*c]*Sin[5*d*x])/(a*d) + (4620*Cos[6*
c]*Sin[6*d*x])/(a*d) - (1980*Sin[7*c]*Sin[7*d*x])/(a*d) - (76230*Sin[(d*x)/2])/(a*d*(Cos[c/2] + Sin[c/2])*(Cos
[(c + d*x)/2] + Sin[(c + d*x)/2])) - (20790*Sin[(c + d*x)/2])/(a*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (4
8510*Sin[c + d*x])/(a*d*(1 + Sin[c + d*x])) + (97020*Sin[(c + d*x)/2]^2)/(d*(a + a*Sin[c + d*x])) - (76230*Sin
[2*(c + d*x)])/(a*d) + (27720*Sin[4*(c + d*x)])/(a*d) - (11550*Sin[6*(c + d*x)])/(a*d) + (3465*Sin[8*(c + d*x)
])/(a*d) + (1386*Sin[10*(c + d*x)])/(a*d))/7096320

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Maple [B]  time = 0.119, size = 653, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c)),x)

[Out]

16/693/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11-3/128/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)+16/63/d/a/(1+t
an(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^2-1/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^3+80/63/d/
a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^4+3323/640/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)
^5-48/7/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^6-54/5/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+
1/2*c)^7+240/7/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^8+841/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan
(1/2*d*x+1/2*c)^9-48/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^10+176/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^
11*tan(1/2*d*x+1/2*c)^12-841/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^13-80/3/d/a/(1+tan(1/2*d*x+
1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^14+54/5/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^15+32/3/d/a/(1+tan(
1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^16-3323/640/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^17+1/4/
d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^19+3/128/d/a/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)
^21+3/128/a/d*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.72599, size = 842, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/443520*((10395*sin(d*x + c)/(cos(d*x + c) + 1) - 112640*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 110880*sin(d*
x + c)^3/(cos(d*x + c) + 1)^3 - 563200*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 2302839*sin(d*x + c)^5/(cos(d*x +
c) + 1)^5 + 3041280*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 4790016*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 15206
400*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 5828130*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 21288960*sin(d*x + c)^
10/(cos(d*x + c) + 1)^10 - 26019840*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 5828130*sin(d*x + c)^13/(cos(d*x +
c) + 1)^13 + 11827200*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 4790016*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 -
4730880*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 + 2302839*sin(d*x + c)^17/(cos(d*x + c) + 1)^17 - 110880*sin(d*
x + c)^19/(cos(d*x + c) + 1)^19 - 10395*sin(d*x + c)^21/(cos(d*x + c) + 1)^21 - 10240)/(a + 11*a*sin(d*x + c)^
2/(cos(d*x + c) + 1)^2 + 55*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 165*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6
+ 330*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 462*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 462*a*sin(d*x + c)
^12/(cos(d*x + c) + 1)^12 + 330*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 165*a*sin(d*x + c)^16/(cos(d*x + c)
+ 1)^16 + 55*a*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 + 11*a*sin(d*x + c)^20/(cos(d*x + c) + 1)^20 + a*sin(d*x
+ c)^22/(cos(d*x + c) + 1)^22) - 10395*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

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Fricas [A]  time = 1.22871, size = 294, normalized size = 1.61 \begin{align*} \frac{80640 \, \cos \left (d x + c\right )^{11} - 197120 \, \cos \left (d x + c\right )^{9} + 126720 \, \cos \left (d x + c\right )^{7} + 10395 \, d x + 693 \,{\left (128 \, \cos \left (d x + c\right )^{9} - 176 \, \cos \left (d x + c\right )^{7} + 8 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/887040*(80640*cos(d*x + c)^11 - 197120*cos(d*x + c)^9 + 126720*cos(d*x + c)^7 + 10395*d*x + 693*(128*cos(d*x
+ c)^9 - 176*cos(d*x + c)^7 + 8*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c))/(a*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**8*sin(d*x+c)**4/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.32072, size = 365, normalized size = 1.99 \begin{align*} \frac{\frac{10395 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (10395 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{21} + 110880 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{19} - 2302839 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{17} + 4730880 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{16} + 4790016 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} - 11827200 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} - 5828130 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 26019840 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} - 21288960 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 5828130 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 15206400 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 4790016 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 3041280 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 2302839 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 563200 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 110880 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 112640 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 10395 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 10240\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{11} a}}{887040 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/887040*(10395*(d*x + c)/a + 2*(10395*tan(1/2*d*x + 1/2*c)^21 + 110880*tan(1/2*d*x + 1/2*c)^19 - 2302839*tan(
1/2*d*x + 1/2*c)^17 + 4730880*tan(1/2*d*x + 1/2*c)^16 + 4790016*tan(1/2*d*x + 1/2*c)^15 - 11827200*tan(1/2*d*x
+ 1/2*c)^14 - 5828130*tan(1/2*d*x + 1/2*c)^13 + 26019840*tan(1/2*d*x + 1/2*c)^12 - 21288960*tan(1/2*d*x + 1/2
*c)^10 + 5828130*tan(1/2*d*x + 1/2*c)^9 + 15206400*tan(1/2*d*x + 1/2*c)^8 - 4790016*tan(1/2*d*x + 1/2*c)^7 - 3
041280*tan(1/2*d*x + 1/2*c)^6 + 2302839*tan(1/2*d*x + 1/2*c)^5 + 563200*tan(1/2*d*x + 1/2*c)^4 - 110880*tan(1/
2*d*x + 1/2*c)^3 + 112640*tan(1/2*d*x + 1/2*c)^2 - 10395*tan(1/2*d*x + 1/2*c) + 10240)/((tan(1/2*d*x + 1/2*c)^
2 + 1)^11*a))/d