\(\int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [78]

Optimal result

Integrand size = 24, antiderivative size = 82 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {4 i (a+i a \tan (c+d x))^{11}}{11 a^3 d}+\frac {i (a+i a \tan (c+d x))^{12}}{3 a^4 d}-\frac {i (a+i a \tan (c+d x))^{13}}{13 a^5 d} \] Output:

-4/11*I*(a+I*a*tan(d*x+c))^11/a^3/d+1/3*I*(a+I*a*tan(d*x+c))^12/a^4/d-1/13 
*I*(a+I*a*tan(d*x+c))^13/a^5/d
 

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.54 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 (-i+\tan (c+d x))^{11} \left (-46+77 i \tan (c+d x)+33 \tan ^2(c+d x)\right )}{429 d} \] Input:

Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^8,x]
 

Output:

(a^8*(-I + Tan[c + d*x])^11*(-46 + (77*I)*Tan[c + d*x] + 33*Tan[c + d*x]^2 
))/(429*d)
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3968, 49, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sec (c+d x)^6 (a+i a \tan (c+d x))^8dx\)

\(\Big \downarrow \) 3968

\(\displaystyle -\frac {i \int (a-i a \tan (c+d x))^2 (i \tan (c+d x) a+a)^{10}d(i a \tan (c+d x))}{a^5 d}\)

\(\Big \downarrow \) 49

\(\displaystyle -\frac {i \int \left ((i \tan (c+d x) a+a)^{12}-4 a (i \tan (c+d x) a+a)^{11}+4 a^2 (i \tan (c+d x) a+a)^{10}\right )d(i a \tan (c+d x))}{a^5 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {i \left (\frac {4}{11} a^2 (a+i a \tan (c+d x))^{11}+\frac {1}{13} (a+i a \tan (c+d x))^{13}-\frac {1}{3} a (a+i a \tan (c+d x))^{12}\right )}{a^5 d}\)

Input:

Int[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^8,x]
 

Output:

((-I)*((4*a^2*(a + I*a*Tan[c + d*x])^11)/11 - (a*(a + I*a*Tan[c + d*x])^12 
)/3 + (a + I*a*Tan[c + d*x])^13/13))/(a^5*d)
 

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3968

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (70 ) = 140\).

Time = 1.23 (sec) , antiderivative size = 475, normalized size of antiderivative = 5.79

\[\frac {a^{8} \left (\frac {\sin \left (d x +c \right )^{9}}{13 \cos \left (d x +c \right )^{13}}+\frac {4 \sin \left (d x +c \right )^{9}}{143 \cos \left (d x +c \right )^{11}}+\frac {8 \sin \left (d x +c \right )^{9}}{1287 \cos \left (d x +c \right )^{9}}\right )+56 i a^{8} \left (\frac {\sin \left (d x +c \right )^{6}}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{6}}{20 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{6}}{60 \cos \left (d x +c \right )^{6}}\right )-28 a^{8} \left (\frac {\sin \left (d x +c \right )^{7}}{11 \cos \left (d x +c \right )^{11}}+\frac {4 \sin \left (d x +c \right )^{7}}{99 \cos \left (d x +c \right )^{9}}+\frac {8 \sin \left (d x +c \right )^{7}}{693 \cos \left (d x +c \right )^{7}}\right )-8 i a^{8} \left (\frac {\sin \left (d x +c \right )^{8}}{12 \cos \left (d x +c \right )^{12}}+\frac {\sin \left (d x +c \right )^{8}}{30 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{8}}{120 \cos \left (d x +c \right )^{8}}\right )+70 a^{8} \left (\frac {\sin \left (d x +c \right )^{5}}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \sin \left (d x +c \right )^{5}}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{5}}{315 \cos \left (d x +c \right )^{5}}\right )-56 i a^{8} \left (\frac {\sin \left (d x +c \right )^{4}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{24 \cos \left (d x +c \right )^{4}}\right )-28 a^{8} \left (\frac {\sin \left (d x +c \right )^{3}}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \sin \left (d x +c \right )^{3}}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{3}}\right )+\frac {4 i a^{8}}{3 \cos \left (d x +c \right )^{6}}-a^{8} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\]

Input:

int(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^8,x)
 

Output:

1/d*(a^8*(1/13*sin(d*x+c)^9/cos(d*x+c)^13+4/143*sin(d*x+c)^9/cos(d*x+c)^11 
+8/1287*sin(d*x+c)^9/cos(d*x+c)^9)+56*I*a^8*(1/10*sin(d*x+c)^6/cos(d*x+c)^ 
10+1/20*sin(d*x+c)^6/cos(d*x+c)^8+1/60*sin(d*x+c)^6/cos(d*x+c)^6)-28*a^8*( 
1/11*sin(d*x+c)^7/cos(d*x+c)^11+4/99*sin(d*x+c)^7/cos(d*x+c)^9+8/693*sin(d 
*x+c)^7/cos(d*x+c)^7)-8*I*a^8*(1/12*sin(d*x+c)^8/cos(d*x+c)^12+1/30*sin(d* 
x+c)^8/cos(d*x+c)^10+1/120*sin(d*x+c)^8/cos(d*x+c)^8)+70*a^8*(1/9*sin(d*x+ 
c)^5/cos(d*x+c)^9+4/63*sin(d*x+c)^5/cos(d*x+c)^7+8/315*sin(d*x+c)^5/cos(d* 
x+c)^5)-56*I*a^8*(1/8*sin(d*x+c)^4/cos(d*x+c)^8+1/12*sin(d*x+c)^4/cos(d*x+ 
c)^6+1/24*sin(d*x+c)^4/cos(d*x+c)^4)-28*a^8*(1/7*sin(d*x+c)^3/cos(d*x+c)^7 
+4/35*sin(d*x+c)^3/cos(d*x+c)^5+8/105*sin(d*x+c)^3/cos(d*x+c)^3)+4/3*I*a^8 
/cos(d*x+c)^6-a^8*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c))
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (64) = 128\).

Time = 0.09 (sec) , antiderivative size = 307, normalized size of antiderivative = 3.74 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {4096 \, {\left (-286 i \, a^{8} e^{\left (20 i \, d x + 20 i \, c\right )} - 715 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} - 1287 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 1716 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 1716 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 1287 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 715 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 286 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 78 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 13 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{8}\right )}}{429 \, {\left (d e^{\left (26 i \, d x + 26 i \, c\right )} + 13 \, d e^{\left (24 i \, d x + 24 i \, c\right )} + 78 \, d e^{\left (22 i \, d x + 22 i \, c\right )} + 286 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 715 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 1287 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 1716 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 1716 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 1287 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 715 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 286 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 78 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 13 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:

integrate(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
 

Output:

-4096/429*(-286*I*a^8*e^(20*I*d*x + 20*I*c) - 715*I*a^8*e^(18*I*d*x + 18*I 
*c) - 1287*I*a^8*e^(16*I*d*x + 16*I*c) - 1716*I*a^8*e^(14*I*d*x + 14*I*c) 
- 1716*I*a^8*e^(12*I*d*x + 12*I*c) - 1287*I*a^8*e^(10*I*d*x + 10*I*c) - 71 
5*I*a^8*e^(8*I*d*x + 8*I*c) - 286*I*a^8*e^(6*I*d*x + 6*I*c) - 78*I*a^8*e^( 
4*I*d*x + 4*I*c) - 13*I*a^8*e^(2*I*d*x + 2*I*c) - I*a^8)/(d*e^(26*I*d*x + 
26*I*c) + 13*d*e^(24*I*d*x + 24*I*c) + 78*d*e^(22*I*d*x + 22*I*c) + 286*d* 
e^(20*I*d*x + 20*I*c) + 715*d*e^(18*I*d*x + 18*I*c) + 1287*d*e^(16*I*d*x + 
 16*I*c) + 1716*d*e^(14*I*d*x + 14*I*c) + 1716*d*e^(12*I*d*x + 12*I*c) + 1 
287*d*e^(10*I*d*x + 10*I*c) + 715*d*e^(8*I*d*x + 8*I*c) + 286*d*e^(6*I*d*x 
 + 6*I*c) + 78*d*e^(4*I*d*x + 4*I*c) + 13*d*e^(2*I*d*x + 2*I*c) + d)
 

Sympy [F]

\[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=a^{8} \left (\int \left (- 28 \tan ^{2}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int 70 \tan ^{4}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 28 \tan ^{6}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int \tan ^{8}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int 8 i \tan {\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 56 i \tan ^{3}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int 56 i \tan ^{5}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 8 i \tan ^{7}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int \sec ^{6}{\left (c + d x \right )}\, dx\right ) \] Input:

integrate(sec(d*x+c)**6*(a+I*a*tan(d*x+c))**8,x)
 

Output:

a**8*(Integral(-28*tan(c + d*x)**2*sec(c + d*x)**6, x) + Integral(70*tan(c 
 + d*x)**4*sec(c + d*x)**6, x) + Integral(-28*tan(c + d*x)**6*sec(c + d*x) 
**6, x) + Integral(tan(c + d*x)**8*sec(c + d*x)**6, x) + Integral(8*I*tan( 
c + d*x)*sec(c + d*x)**6, x) + Integral(-56*I*tan(c + d*x)**3*sec(c + d*x) 
**6, x) + Integral(56*I*tan(c + d*x)**5*sec(c + d*x)**6, x) + Integral(-8* 
I*tan(c + d*x)**7*sec(c + d*x)**6, x) + Integral(sec(c + d*x)**6, x))
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (64) = 128\).

Time = 0.03 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.11 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {33 \, a^{8} \tan \left (d x + c\right )^{13} - 286 i \, a^{8} \tan \left (d x + c\right )^{12} - 1014 \, a^{8} \tan \left (d x + c\right )^{11} + 1716 i \, a^{8} \tan \left (d x + c\right )^{10} + 715 \, a^{8} \tan \left (d x + c\right )^{9} + 2574 i \, a^{8} \tan \left (d x + c\right )^{8} + 5148 \, a^{8} \tan \left (d x + c\right )^{7} - 3432 i \, a^{8} \tan \left (d x + c\right )^{6} + 1287 \, a^{8} \tan \left (d x + c\right )^{5} - 4290 i \, a^{8} \tan \left (d x + c\right )^{4} - 3718 \, a^{8} \tan \left (d x + c\right )^{3} + 1716 i \, a^{8} \tan \left (d x + c\right )^{2} + 429 \, a^{8} \tan \left (d x + c\right )}{429 \, d} \] Input:

integrate(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
 

Output:

1/429*(33*a^8*tan(d*x + c)^13 - 286*I*a^8*tan(d*x + c)^12 - 1014*a^8*tan(d 
*x + c)^11 + 1716*I*a^8*tan(d*x + c)^10 + 715*a^8*tan(d*x + c)^9 + 2574*I* 
a^8*tan(d*x + c)^8 + 5148*a^8*tan(d*x + c)^7 - 3432*I*a^8*tan(d*x + c)^6 + 
 1287*a^8*tan(d*x + c)^5 - 4290*I*a^8*tan(d*x + c)^4 - 3718*a^8*tan(d*x + 
c)^3 + 1716*I*a^8*tan(d*x + c)^2 + 429*a^8*tan(d*x + c))/d
 

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (64) = 128\).

Time = 0.32 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.11 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {33 \, a^{8} \tan \left (d x + c\right )^{13} - 286 i \, a^{8} \tan \left (d x + c\right )^{12} - 1014 \, a^{8} \tan \left (d x + c\right )^{11} + 1716 i \, a^{8} \tan \left (d x + c\right )^{10} + 715 \, a^{8} \tan \left (d x + c\right )^{9} + 2574 i \, a^{8} \tan \left (d x + c\right )^{8} + 5148 \, a^{8} \tan \left (d x + c\right )^{7} - 3432 i \, a^{8} \tan \left (d x + c\right )^{6} + 1287 \, a^{8} \tan \left (d x + c\right )^{5} - 4290 i \, a^{8} \tan \left (d x + c\right )^{4} - 3718 \, a^{8} \tan \left (d x + c\right )^{3} + 1716 i \, a^{8} \tan \left (d x + c\right )^{2} + 429 \, a^{8} \tan \left (d x + c\right )}{429 \, d} \] Input:

integrate(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
 

Output:

1/429*(33*a^8*tan(d*x + c)^13 - 286*I*a^8*tan(d*x + c)^12 - 1014*a^8*tan(d 
*x + c)^11 + 1716*I*a^8*tan(d*x + c)^10 + 715*a^8*tan(d*x + c)^9 + 2574*I* 
a^8*tan(d*x + c)^8 + 5148*a^8*tan(d*x + c)^7 - 3432*I*a^8*tan(d*x + c)^6 + 
 1287*a^8*tan(d*x + c)^5 - 4290*I*a^8*tan(d*x + c)^4 - 3718*a^8*tan(d*x + 
c)^3 + 1716*I*a^8*tan(d*x + c)^2 + 429*a^8*tan(d*x + c))/d
 

Mupad [B] (verification not implemented)

Time = 2.47 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.32 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8\,\sin \left (c+d\,x\right )\,\left (2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\,\left (-184\,{\sin \left (c+d\,x\right )}^2-184\,{\sin \left (2\,c+2\,d\,x\right )}^2+\frac {\sin \left (2\,c+2\,d\,x\right )\,9867{}\mathrm {i}}{256}-184\,{\sin \left (3\,c+3\,d\,x\right )}^2-184\,{\sin \left (4\,c+4\,d\,x\right )}^2+\frac {\sin \left (4\,c+4\,d\,x\right )\,69069{}\mathrm {i}}{1024}-28\,{\sin \left (5\,c+5\,d\,x\right )}^2-2\,{\sin \left (6\,c+6\,d\,x\right )}^2+\frac {\sin \left (6\,c+6\,d\,x\right )\,42757{}\mathrm {i}}{512}+\frac {\sin \left (8\,c+8\,d\,x\right )\,23023{}\mathrm {i}}{256}+\frac {\sin \left (10\,c+10\,d\,x\right )\,7007{}\mathrm {i}}{512}+\frac {\sin \left (12\,c+12\,d\,x\right )\,1001{}\mathrm {i}}{1024}+429\right )}{429\,d\,{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^7} \] Input:

int((a + a*tan(c + d*x)*1i)^8/cos(c + d*x)^6,x)
 

Output:

(a^8*sin(c + d*x)*(2*sin(c/2 + (d*x)/2)^2 - 1)*((sin(2*c + 2*d*x)*9867i)/2 
56 + (sin(4*c + 4*d*x)*69069i)/1024 + (sin(6*c + 6*d*x)*42757i)/512 + (sin 
(8*c + 8*d*x)*23023i)/256 + (sin(10*c + 10*d*x)*7007i)/512 + (sin(12*c + 1 
2*d*x)*1001i)/1024 - 184*sin(2*c + 2*d*x)^2 - 184*sin(3*c + 3*d*x)^2 - 184 
*sin(4*c + 4*d*x)^2 - 28*sin(5*c + 5*d*x)^2 - 2*sin(6*c + 6*d*x)^2 - 184*s 
in(c + d*x)^2 + 429))/(429*d*(sin(c + d*x)^2 - 1)^7)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 246, normalized size of antiderivative = 3.00 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {\sin \left (d x +c \right ) a^{8} \left (-2002 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{11} i +12012 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9} i -30030 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} i +30888 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} i -12870 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} i +1716 \cos \left (d x +c \right ) \sin \left (d x +c \right ) i +2048 \sin \left (d x +c \right )^{12}-13312 \sin \left (d x +c \right )^{10}+36608 \sin \left (d x +c \right )^{8}-45760 \sin \left (d x +c \right )^{6}+26312 \sin \left (d x +c \right )^{4}-6292 \sin \left (d x +c \right )^{2}+429\right )}{429 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{12}-6 \sin \left (d x +c \right )^{10}+15 \sin \left (d x +c \right )^{8}-20 \sin \left (d x +c \right )^{6}+15 \sin \left (d x +c \right )^{4}-6 \sin \left (d x +c \right )^{2}+1\right )} \] Input:

int(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^8,x)
 

Output:

(sin(c + d*x)*a**8*( - 2002*cos(c + d*x)*sin(c + d*x)**11*i + 12012*cos(c 
+ d*x)*sin(c + d*x)**9*i - 30030*cos(c + d*x)*sin(c + d*x)**7*i + 30888*co 
s(c + d*x)*sin(c + d*x)**5*i - 12870*cos(c + d*x)*sin(c + d*x)**3*i + 1716 
*cos(c + d*x)*sin(c + d*x)*i + 2048*sin(c + d*x)**12 - 13312*sin(c + d*x)* 
*10 + 36608*sin(c + d*x)**8 - 45760*sin(c + d*x)**6 + 26312*sin(c + d*x)** 
4 - 6292*sin(c + d*x)**2 + 429))/(429*cos(c + d*x)*d*(sin(c + d*x)**12 - 6 
*sin(c + d*x)**10 + 15*sin(c + d*x)**8 - 20*sin(c + d*x)**6 + 15*sin(c + d 
*x)**4 - 6*sin(c + d*x)**2 + 1))