3.55 \(\int \cot ^{12}(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=213 \[ -\frac{4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^{11}(c+d x)}{11 d}+\frac{19 a^3 \csc ^9(c+d x)}{9 d}-\frac{36 a^3 \csc ^7(c+d x)}{7 d}+\frac{34 a^3 \csc ^5(c+d x)}{5 d}-\frac{16 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x \]

[Out]

a^3*x + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/
(7*d) + (a^3*Cot[c + d*x]^9)/(9*d) - (4*a^3*Cot[c + d*x]^11)/(11*d) + (3*a^3*Csc[c + d*x])/d - (16*a^3*Csc[c +
d*x]^3)/(3*d) + (34*a^3*Csc[c + d*x]^5)/(5*d) - (36*a^3*Csc[c + d*x]^7)/(7*d) + (19*a^3*Csc[c + d*x]^9)/(9*d)
- (4*a^3*Csc[c + d*x]^11)/(11*d)

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Rubi [A]  time = 0.221298, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3886, 3473, 8, 2606, 194, 2607, 30, 270} \[ -\frac{4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^{11}(c+d x)}{11 d}+\frac{19 a^3 \csc ^9(c+d x)}{9 d}-\frac{36 a^3 \csc ^7(c+d x)}{7 d}+\frac{34 a^3 \csc ^5(c+d x)}{5 d}-\frac{16 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^12*(a + a*Sec[c + d*x])^3,x]

[Out]

a^3*x + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/
(7*d) + (a^3*Cot[c + d*x]^9)/(9*d) - (4*a^3*Cot[c + d*x]^11)/(11*d) + (3*a^3*Csc[c + d*x])/d - (16*a^3*Csc[c +
d*x]^3)/(3*d) + (34*a^3*Csc[c + d*x]^5)/(5*d) - (36*a^3*Csc[c + d*x]^7)/(7*d) + (19*a^3*Csc[c + d*x]^9)/(9*d)
- (4*a^3*Csc[c + d*x]^11)/(11*d)

Rule 3886

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

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

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 194

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

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 \cot ^{12}(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cot ^{12}(c+d x)+3 a^3 \cot ^{11}(c+d x) \csc (c+d x)+3 a^3 \cot ^{10}(c+d x) \csc ^2(c+d x)+a^3 \cot ^9(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^{12}(c+d x) \, dx+a^3 \int \cot ^9(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^{11}(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^{10}(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^3 \cot ^{11}(c+d x)}{11 d}-a^3 \int \cot ^{10}(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^4 \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^{10} \, dx,x,-\cot (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right )^5 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^{11}(c+d x)}{11 d}+a^3 \int \cot ^8(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (x^2-4 x^4+6 x^6-4 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (-1+5 x^2-10 x^4+10 x^6-5 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{3 a^3 \csc (c+d x)}{d}-\frac{16 a^3 \csc ^3(c+d x)}{3 d}+\frac{34 a^3 \csc ^5(c+d x)}{5 d}-\frac{36 a^3 \csc ^7(c+d x)}{7 d}+\frac{19 a^3 \csc ^9(c+d x)}{9 d}-\frac{4 a^3 \csc ^{11}(c+d x)}{11 d}-a^3 \int \cot ^6(c+d x) \, dx\\ &=\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{3 a^3 \csc (c+d x)}{d}-\frac{16 a^3 \csc ^3(c+d x)}{3 d}+\frac{34 a^3 \csc ^5(c+d x)}{5 d}-\frac{36 a^3 \csc ^7(c+d x)}{7 d}+\frac{19 a^3 \csc ^9(c+d x)}{9 d}-\frac{4 a^3 \csc ^{11}(c+d x)}{11 d}+a^3 \int \cot ^4(c+d x) \, dx\\ &=-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{3 a^3 \csc (c+d x)}{d}-\frac{16 a^3 \csc ^3(c+d x)}{3 d}+\frac{34 a^3 \csc ^5(c+d x)}{5 d}-\frac{36 a^3 \csc ^7(c+d x)}{7 d}+\frac{19 a^3 \csc ^9(c+d x)}{9 d}-\frac{4 a^3 \csc ^{11}(c+d x)}{11 d}-a^3 \int \cot ^2(c+d x) \, dx\\ &=\frac{a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{3 a^3 \csc (c+d x)}{d}-\frac{16 a^3 \csc ^3(c+d x)}{3 d}+\frac{34 a^3 \csc ^5(c+d x)}{5 d}-\frac{36 a^3 \csc ^7(c+d x)}{7 d}+\frac{19 a^3 \csc ^9(c+d x)}{9 d}-\frac{4 a^3 \csc ^{11}(c+d x)}{11 d}+a^3 \int 1 \, dx\\ &=a^3 x+\frac{a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{3 a^3 \csc (c+d x)}{d}-\frac{16 a^3 \csc ^3(c+d x)}{3 d}+\frac{34 a^3 \csc ^5(c+d x)}{5 d}-\frac{36 a^3 \csc ^7(c+d x)}{7 d}+\frac{19 a^3 \csc ^9(c+d x)}{9 d}-\frac{4 a^3 \csc ^{11}(c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 6.01919, size = 268, normalized size = 1.26 \[ -\frac{a^3 \tan \left (\frac{c}{2}\right ) (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (20 \cot ^2\left (\frac{c}{2}\right ) (-4528480 \cos (c+d x)+2388316 \cos (2 (c+d x))-750112 \cos (3 (c+d x))+112229 \cos (4 (c+d x))+2786111) \csc ^{10}\left (\frac{1}{2} (c+d x)\right )+7392 \csc \left (\frac{c}{2}\right ) \left (-3060 \sin \left (c+\frac{d x}{2}\right )+2860 \sin \left (c+\frac{3 d x}{2}\right )-855 \sin \left (2 c+\frac{3 d x}{2}\right )+743 \sin \left (2 c+\frac{5 d x}{2}\right )+4370 \sin \left (\frac{d x}{2}\right )\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right )-5 \cot \left (\frac{c}{2}\right ) \left (\csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) (54812150 \cos (c+d x)-32118776 \cos (2 (c+d x))[phone] \cos (3 (c+d x))-3023754 \cos (4 (c+d x))+347267 \cos (5 (c+d x))-32611198) \csc ^{11}\left (\frac{1}{2} (c+d x)\right )[phone] d x\right )\right )}{3633315840 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^12*(a + a*Sec[c + d*x])^3,x]

[Out]

-(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(20*(2786111 - 4528480*Cos[c + d*x] + 2388316*Cos[2*(c + d*x)] -
750112*Cos[3*(c + d*x)] + 112229*Cos[4*(c + d*x)])*Cot[c/2]^2*Csc[(c + d*x)/2]^10 - 5*Cot[c/2]*(90832896*d*x
+ (-32611198 + 54812150*Cos[c + d*x] - 32118776*Cos[2*(c + d*x)] + 12626567*Cos[3*(c + d*x)] - 3023754*Cos[4*(
c + d*x)] + 347267*Cos[5*(c + d*x)])*Csc[c/2]*Csc[(c + d*x)/2]^11*Sin[(d*x)/2]) + 7392*Csc[c/2]*Sec[(c + d*x)/
2]^5*(4370*Sin[(d*x)/2] - 3060*Sin[c + (d*x)/2] + 2860*Sin[c + (3*d*x)/2] - 855*Sin[2*c + (3*d*x)/2] + 743*Sin
[2*c + (5*d*x)/2]))*Tan[c/2])/(3633315840*d)

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Maple [B]  time = 0.145, size = 425, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^12*(a+a*sec(d*x+c))^3,x)

[Out]

1/d*(a^3*(-1/11*cot(d*x+c)^11+1/9*cot(d*x+c)^9-1/7*cot(d*x+c)^7+1/5*cot(d*x+c)^5-1/3*cot(d*x+c)^3+cot(d*x+c)+d
*x+c)+3*a^3*(-1/11/sin(d*x+c)^11*cos(d*x+c)^12+1/99/sin(d*x+c)^9*cos(d*x+c)^12-1/231/sin(d*x+c)^7*cos(d*x+c)^1
2+1/231/sin(d*x+c)^5*cos(d*x+c)^12-1/99/sin(d*x+c)^3*cos(d*x+c)^12+1/11/sin(d*x+c)*cos(d*x+c)^12+1/11*(256/63+
cos(d*x+c)^10+10/9*cos(d*x+c)^8+80/63*cos(d*x+c)^6+32/21*cos(d*x+c)^4+128/63*cos(d*x+c)^2)*sin(d*x+c))-3/11*a^
3/sin(d*x+c)^11*cos(d*x+c)^11+a^3*(-1/11/sin(d*x+c)^11*cos(d*x+c)^10-1/99/sin(d*x+c)^9*cos(d*x+c)^10+1/693/sin
(d*x+c)^7*cos(d*x+c)^10-1/1155/sin(d*x+c)^5*cos(d*x+c)^10+1/693/sin(d*x+c)^3*cos(d*x+c)^10-1/99/sin(d*x+c)*cos
(d*x+c)^10-1/99*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35*cos(d*x+c)^4+64/35*cos(d*x+c)^2)*sin(d*x+c)))

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Maxima [A]  time = 1.65026, size = 286, normalized size = 1.34 \begin{align*} \frac{{\left (3465 \, d x + 3465 \, c + \frac{3465 \, \tan \left (d x + c\right )^{10} - 1155 \, \tan \left (d x + c\right )^{8} + 693 \, \tan \left (d x + c\right )^{6} - 495 \, \tan \left (d x + c\right )^{4} + 385 \, \tan \left (d x + c\right )^{2} - 315}{\tan \left (d x + c\right )^{11}}\right )} a^{3} + \frac{15 \,{\left (693 \, \sin \left (d x + c\right )^{10} - 1155 \, \sin \left (d x + c\right )^{8} + 1386 \, \sin \left (d x + c\right )^{6} - 990 \, \sin \left (d x + c\right )^{4} + 385 \, \sin \left (d x + c\right )^{2} - 63\right )} a^{3}}{\sin \left (d x + c\right )^{11}} - \frac{{\left (1155 \, \sin \left (d x + c\right )^{8} - 2772 \, \sin \left (d x + c\right )^{6} + 2970 \, \sin \left (d x + c\right )^{4} - 1540 \, \sin \left (d x + c\right )^{2} + 315\right )} a^{3}}{\sin \left (d x + c\right )^{11}} - \frac{945 \, a^{3}}{\tan \left (d x + c\right )^{11}}}{3465 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^12*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/3465*((3465*d*x + 3465*c + (3465*tan(d*x + c)^10 - 1155*tan(d*x + c)^8 + 693*tan(d*x + c)^6 - 495*tan(d*x +
c)^4 + 385*tan(d*x + c)^2 - 315)/tan(d*x + c)^11)*a^3 + 15*(693*sin(d*x + c)^10 - 1155*sin(d*x + c)^8 + 1386*s
in(d*x + c)^6 - 990*sin(d*x + c)^4 + 385*sin(d*x + c)^2 - 63)*a^3/sin(d*x + c)^11 - (1155*sin(d*x + c)^8 - 277
2*sin(d*x + c)^6 + 2970*sin(d*x + c)^4 - 1540*sin(d*x + c)^2 + 315)*a^3/sin(d*x + c)^11 - 945*a^3/tan(d*x + c)
^11)/d

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Fricas [A]  time = 1.22568, size = 805, normalized size = 3.78 \begin{align*} \frac{7453 \, a^{3} \cos \left (d x + c\right )^{8} - 11964 \, a^{3} \cos \left (d x + c\right )^{7} - 11866 \, a^{3} \cos \left (d x + c\right )^{6} + 30542 \, a^{3} \cos \left (d x + c\right )^{5} + 90 \, a^{3} \cos \left (d x + c\right )^{4} - 26438 \, a^{3} \cos \left (d x + c\right )^{3} + 8539 \, a^{3} \cos \left (d x + c\right )^{2} + 7671 \, a^{3} \cos \left (d x + c\right ) - 3712 \, a^{3} + 3465 \,{\left (a^{3} d x \cos \left (d x + c\right )^{7} - 3 \, a^{3} d x \cos \left (d x + c\right )^{6} + a^{3} d x \cos \left (d x + c\right )^{5} + 5 \, a^{3} d x \cos \left (d x + c\right )^{4} - 5 \, a^{3} d x \cos \left (d x + c\right )^{3} - a^{3} d x \cos \left (d x + c\right )^{2} + 3 \, a^{3} d x \cos \left (d x + c\right ) - a^{3} d x\right )} \sin \left (d x + c\right )}{3465 \,{\left (d \cos \left (d x + c\right )^{7} - 3 \, d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5} + 5 \, d \cos \left (d x + c\right )^{4} - 5 \, d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} + 3 \, d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^12*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/3465*(7453*a^3*cos(d*x + c)^8 - 11964*a^3*cos(d*x + c)^7 - 11866*a^3*cos(d*x + c)^6 + 30542*a^3*cos(d*x + c)
^5 + 90*a^3*cos(d*x + c)^4 - 26438*a^3*cos(d*x + c)^3 + 8539*a^3*cos(d*x + c)^2 + 7671*a^3*cos(d*x + c) - 3712
*a^3 + 3465*(a^3*d*x*cos(d*x + c)^7 - 3*a^3*d*x*cos(d*x + c)^6 + a^3*d*x*cos(d*x + c)^5 + 5*a^3*d*x*cos(d*x +
c)^4 - 5*a^3*d*x*cos(d*x + c)^3 - a^3*d*x*cos(d*x + c)^2 + 3*a^3*d*x*cos(d*x + c) - a^3*d*x)*sin(d*x + c))/((d
*cos(d*x + c)^7 - 3*d*cos(d*x + c)^6 + d*cos(d*x + c)^5 + 5*d*cos(d*x + c)^4 - 5*d*cos(d*x + c)^3 - d*cos(d*x
+ c)^2 + 3*d*cos(d*x + c) - d)*sin(d*x + c))

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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(cot(d*x+c)**12*(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.58861, size = 217, normalized size = 1.02 \begin{align*} -\frac{693 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 11550 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 887040 \,{\left (d x + c\right )} a^{3} + 159390 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{5 \,{\left (264726 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 59136 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 18018 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 4554 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 770 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 63 \, a^{3}\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11}}}{887040 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^12*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/887040*(693*a^3*tan(1/2*d*x + 1/2*c)^5 - 11550*a^3*tan(1/2*d*x + 1/2*c)^3 - 887040*(d*x + c)*a^3 + 159390*a
^3*tan(1/2*d*x + 1/2*c) - 5*(264726*a^3*tan(1/2*d*x + 1/2*c)^10 - 59136*a^3*tan(1/2*d*x + 1/2*c)^8 + 18018*a^3
*tan(1/2*d*x + 1/2*c)^6 - 4554*a^3*tan(1/2*d*x + 1/2*c)^4 + 770*a^3*tan(1/2*d*x + 1/2*c)^2 - 63*a^3)/tan(1/2*d
*x + 1/2*c)^11)/d