\(\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx\) [698]
Optimal result
Integrand size = 29, antiderivative size = 184 \[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {2 a^2 \sin ^{2+n}(c+d x)}{d (2+n)}-\frac {2 a^2 \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {6 a^2 \sin ^{4+n}(c+d x)}{d (4+n)}+\frac {6 a^2 \sin ^{6+n}(c+d x)}{d (6+n)}+\frac {2 a^2 \sin ^{7+n}(c+d x)}{d (7+n)}-\frac {2 a^2 \sin ^{8+n}(c+d x)}{d (8+n)}-\frac {a^2 \sin ^{9+n}(c+d x)}{d (9+n)}
\]
[Out]
a^2*sin(d*x+c)^(1+n)/d/(1+n)+2*a^2*sin(d*x+c)^(2+n)/d/(2+n)-2*a^2*sin(d*x+c)^(3+n)/d/(3+n)-6*a^2*sin(d*x+c)^(4
+n)/d/(4+n)+6*a^2*sin(d*x+c)^(6+n)/d/(6+n)+2*a^2*sin(d*x+c)^(7+n)/d/(7+n)-2*a^2*sin(d*x+c)^(8+n)/d/(8+n)-a^2*s
in(d*x+c)^(9+n)/d/(9+n)
Rubi [A] (verified)
Time = 0.14 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2915, 90}
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac {2 a^2 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {6 a^2 \sin ^{n+4}(c+d x)}{d (n+4)}+\frac {6 a^2 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac {2 a^2 \sin ^{n+7}(c+d x)}{d (n+7)}-\frac {2 a^2 \sin ^{n+8}(c+d x)}{d (n+8)}-\frac {a^2 \sin ^{n+9}(c+d x)}{d (n+9)}
\]
[In]
Int[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]
[Out]
(a^2*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (2*a^2*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (2*a^2*Sin[c + d*x]^(3 + n
))/(d*(3 + n)) - (6*a^2*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (6*a^2*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (2*a^2*
Sin[c + d*x]^(7 + n))/(d*(7 + n)) - (2*a^2*Sin[c + d*x]^(8 + n))/(d*(8 + n)) - (a^2*Sin[c + d*x]^(9 + n))/(d*(
9 + n))
Rule 90
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))
Rule 2915
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int (a-x)^3 \left (\frac {x}{a}\right )^n (a+x)^5 \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \left (a^8 \left (\frac {x}{a}\right )^n+2 a^8 \left (\frac {x}{a}\right )^{1+n}-2 a^8 \left (\frac {x}{a}\right )^{2+n}-6 a^8 \left (\frac {x}{a}\right )^{3+n}+6 a^8 \left (\frac {x}{a}\right )^{5+n}+2 a^8 \left (\frac {x}{a}\right )^{6+n}-2 a^8 \left (\frac {x}{a}\right )^{7+n}-a^8 \left (\frac {x}{a}\right )^{8+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {a^2 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {2 a^2 \sin ^{2+n}(c+d x)}{d (2+n)}-\frac {2 a^2 \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {6 a^2 \sin ^{4+n}(c+d x)}{d (4+n)}+\frac {6 a^2 \sin ^{6+n}(c+d x)}{d (6+n)}+\frac {2 a^2 \sin ^{7+n}(c+d x)}{d (7+n)}-\frac {2 a^2 \sin ^{8+n}(c+d x)}{d (8+n)}-\frac {a^2 \sin ^{9+n}(c+d x)}{d (9+n)} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.53 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.68
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \sin ^{1+n}(c+d x) \left (\frac {1}{1+n}+\frac {2 \sin (c+d x)}{2+n}-\frac {2 \sin ^2(c+d x)}{3+n}-\frac {6 \sin ^3(c+d x)}{4+n}+\frac {6 \sin ^5(c+d x)}{6+n}+\frac {2 \sin ^6(c+d x)}{7+n}-\frac {2 \sin ^7(c+d x)}{8+n}-\frac {\sin ^8(c+d x)}{9+n}\right )}{d}
\]
[In]
Integrate[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]
[Out]
(a^2*Sin[c + d*x]^(1 + n)*((1 + n)^(-1) + (2*Sin[c + d*x])/(2 + n) - (2*Sin[c + d*x]^2)/(3 + n) - (6*Sin[c + d
*x]^3)/(4 + n) + (6*Sin[c + d*x]^5)/(6 + n) + (2*Sin[c + d*x]^6)/(7 + n) - (2*Sin[c + d*x]^7)/(8 + n) - Sin[c
+ d*x]^8/(9 + n)))/d
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(378\) vs. \(2(184)=368\).
Time = 16.28 (sec) , antiderivative size = 379, normalized size of antiderivative =
2.06
| | |
| method | result | size |
| | |
| parallelrisch |
\(\frac {13 \left (\sin ^{n}\left (d x +c \right )\right ) \left (\frac {8 \left (n -4\right ) \left (9+n \right ) \left (7+n \right ) \left (3+n \right ) \left (1+n \right ) \left (n^{2}+22 n +168\right ) \cos \left (2 d x +2 c \right )}{13}-\frac {8 \left (9+n \right ) \left (7+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \left (n^{2}+22 n +168\right ) \cos \left (4 d x +4 c \right )}{13}-\frac {8 \left (n +12\right ) \left (9+n \right ) \left (7+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (6 d x +6 c \right )}{13}-\frac {2 \left (9+n \right ) \left (7+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (8 d x +8 c \right )}{13}+\frac {22 \left (4+n \right ) \left (8+n \right ) \left (6+n \right ) \left (2+n \right ) \left (1+n \right ) \left (n^{2}+\frac {218}{11} n +\frac {1323}{11}\right ) \sin \left (3 d x +3 c \right )}{13}+\frac {10 \left (4+n \right ) \left (3+n \right ) \left (8+n \right ) \left (6+n \right ) \left (2+n \right ) \left (n +\frac {63}{5}\right ) \left (1+n \right ) \sin \left (5 d x +5 c \right )}{13}+\frac {\left (n -9\right ) \left (8+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (7 d x +7 c \right )}{26}-\frac {\left (8+n \right ) \left (7+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (9 d x +9 c \right )}{26}+\left (4+n \right ) \left (8+n \right ) \left (n^{3}+\frac {295}{13} n^{2}+\frac {2547}{13} n +\frac {14553}{13}\right ) \left (6+n \right ) \left (2+n \right ) \sin \left (d x +c \right )+\frac {10 \left (3+n \right ) \left (7+n \right ) \left (9+n \right ) \left (1+n \right ) \left (n^{3}+\frac {108}{5} n^{2}+\frac {892}{5} n +\frac {4464}{5}\right )}{13}\right ) a^{2}}{128 \left (2+n \right ) \left (1+n \right ) \left (9+n \right ) \left (8+n \right ) \left (7+n \right ) \left (3+n \right ) \left (6+n \right ) \left (4+n \right ) d}\) |
\(379\) |
| | |
|
|
|
|
[In]
int(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
[Out]
13/128*sin(d*x+c)^n*(8/13*(n-4)*(9+n)*(7+n)*(3+n)*(1+n)*(n^2+22*n+168)*cos(2*d*x+2*c)-8/13*(9+n)*(7+n)*(3+n)*(
2+n)*(1+n)*(n^2+22*n+168)*cos(4*d*x+4*c)-8/13*(n+12)*(9+n)*(7+n)*(4+n)*(3+n)*(2+n)*(1+n)*cos(6*d*x+6*c)-2/13*(
9+n)*(7+n)*(6+n)*(4+n)*(3+n)*(2+n)*(1+n)*cos(8*d*x+8*c)+22/13*(4+n)*(8+n)*(6+n)*(2+n)*(1+n)*(n^2+218/11*n+1323
/11)*sin(3*d*x+3*c)+10/13*(4+n)*(3+n)*(8+n)*(6+n)*(2+n)*(n+63/5)*(1+n)*sin(5*d*x+5*c)+1/26*(n-9)*(8+n)*(6+n)*(
4+n)*(3+n)*(2+n)*(1+n)*sin(7*d*x+7*c)-1/26*(8+n)*(7+n)*(6+n)*(4+n)*(3+n)*(2+n)*(1+n)*sin(9*d*x+9*c)+(4+n)*(8+n
)*(n^3+295/13*n^2+2547/13*n+14553/13)*(6+n)*(2+n)*sin(d*x+c)+10/13*(3+n)*(7+n)*(9+n)*(1+n)*(n^3+108/5*n^2+892/
5*n+4464/5))*a^2/(2+n)/(1+n)/(9+n)/(8+n)/(7+n)/(3+n)/(6+n)/(4+n)/d
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 628 vs. \(2 (184) = 368\).
Time = 0.32 (sec) , antiderivative size = 628, normalized size of antiderivative = 3.41
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {{\left (2 \, {\left (a^{2} n^{7} + 32 \, a^{2} n^{6} + 414 \, a^{2} n^{5} + 2788 \, a^{2} n^{4} + 10469 \, a^{2} n^{3} + 21708 \, a^{2} n^{2} + 22716 \, a^{2} n + 9072 \, a^{2}\right )} \cos \left (d x + c\right )^{8} - 2 \, {\left (a^{2} n^{7} + 26 \, a^{2} n^{6} + 258 \, a^{2} n^{5} + 1240 \, a^{2} n^{4} + 3029 \, a^{2} n^{3} + 3534 \, a^{2} n^{2} + 1512 \, a^{2} n\right )} \cos \left (d x + c\right )^{6} - 96 \, a^{2} n^{4} - 1920 \, a^{2} n^{3} - 12 \, {\left (a^{2} n^{6} + 22 \, a^{2} n^{5} + 170 \, a^{2} n^{4} + 560 \, a^{2} n^{3} + 789 \, a^{2} n^{2} + 378 \, a^{2} n\right )} \cos \left (d x + c\right )^{4} - 12480 \, a^{2} n^{2} - 28800 \, a^{2} n - 48 \, {\left (a^{2} n^{5} + 20 \, a^{2} n^{4} + 130 \, a^{2} n^{3} + 300 \, a^{2} n^{2} + 189 \, a^{2} n\right )} \cos \left (d x + c\right )^{2} - 18144 \, a^{2} + {\left ({\left (a^{2} n^{7} + 31 \, a^{2} n^{6} + 391 \, a^{2} n^{5} + 2581 \, a^{2} n^{4} + 9544 \, a^{2} n^{3} + 19564 \, a^{2} n^{2} + 20304 \, a^{2} n + 8064 \, a^{2}\right )} \cos \left (d x + c\right )^{8} - 2 \, {\left (a^{2} n^{7} + 29 \, a^{2} n^{6} + 343 \, a^{2} n^{5} + 2135 \, a^{2} n^{4} + 7504 \, a^{2} n^{3} + 14756 \, a^{2} n^{2} + 14832 \, a^{2} n + 5760 \, a^{2}\right )} \cos \left (d x + c\right )^{6} - 96 \, a^{2} n^{4} - 1920 \, a^{2} n^{3} - 12 \, {\left (a^{2} n^{6} + 24 \, a^{2} n^{5} + 223 \, a^{2} n^{4} + 1020 \, a^{2} n^{3} + 2404 \, a^{2} n^{2} + 2736 \, a^{2} n + 1152 \, a^{2}\right )} \cos \left (d x + c\right )^{4} - 13440 \, a^{2} n^{2} - 38400 \, a^{2} n - 48 \, {\left (a^{2} n^{5} + 21 \, a^{2} n^{4} + 160 \, a^{2} n^{3} + 540 \, a^{2} n^{2} + 784 \, a^{2} n + 384 \, a^{2}\right )} \cos \left (d x + c\right )^{2} - 36864 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 40 \, d n^{7} + 670 \, d n^{6} + 6100 \, d n^{5} + 32773 \, d n^{4} + 105460 \, d n^{3} + 196380 \, d n^{2} + 190800 \, d n + 72576 \, d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x, algorithm="fricas")
[Out]
-(2*(a^2*n^7 + 32*a^2*n^6 + 414*a^2*n^5 + 2788*a^2*n^4 + 10469*a^2*n^3 + 21708*a^2*n^2 + 22716*a^2*n + 9072*a^
2)*cos(d*x + c)^8 - 2*(a^2*n^7 + 26*a^2*n^6 + 258*a^2*n^5 + 1240*a^2*n^4 + 3029*a^2*n^3 + 3534*a^2*n^2 + 1512*
a^2*n)*cos(d*x + c)^6 - 96*a^2*n^4 - 1920*a^2*n^3 - 12*(a^2*n^6 + 22*a^2*n^5 + 170*a^2*n^4 + 560*a^2*n^3 + 789
*a^2*n^2 + 378*a^2*n)*cos(d*x + c)^4 - 12480*a^2*n^2 - 28800*a^2*n - 48*(a^2*n^5 + 20*a^2*n^4 + 130*a^2*n^3 +
300*a^2*n^2 + 189*a^2*n)*cos(d*x + c)^2 - 18144*a^2 + ((a^2*n^7 + 31*a^2*n^6 + 391*a^2*n^5 + 2581*a^2*n^4 + 95
44*a^2*n^3 + 19564*a^2*n^2 + 20304*a^2*n + 8064*a^2)*cos(d*x + c)^8 - 2*(a^2*n^7 + 29*a^2*n^6 + 343*a^2*n^5 +
2135*a^2*n^4 + 7504*a^2*n^3 + 14756*a^2*n^2 + 14832*a^2*n + 5760*a^2)*cos(d*x + c)^6 - 96*a^2*n^4 - 1920*a^2*n
^3 - 12*(a^2*n^6 + 24*a^2*n^5 + 223*a^2*n^4 + 1020*a^2*n^3 + 2404*a^2*n^2 + 2736*a^2*n + 1152*a^2)*cos(d*x + c
)^4 - 13440*a^2*n^2 - 38400*a^2*n - 48*(a^2*n^5 + 21*a^2*n^4 + 160*a^2*n^3 + 540*a^2*n^2 + 784*a^2*n + 384*a^2
)*cos(d*x + c)^2 - 36864*a^2)*sin(d*x + c))*sin(d*x + c)^n/(d*n^8 + 40*d*n^7 + 670*d*n^6 + 6100*d*n^5 + 32773*
d*n^4 + 105460*d*n^3 + 196380*d*n^2 + 190800*d*n + 72576*d)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 29818 vs. \(2 (158) = 316\).
Time = 48.25 (sec) , antiderivative size = 29818, normalized size of antiderivative = 162.05
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)**7*sin(d*x+c)**n*(a+a*sin(d*x+c))**2,x)
[Out]
Piecewise((x*(a*sin(c) + a)**2*sin(c)**n*cos(c)**7, Eq(d, 0)), (-a**2*log(sin(c + d*x))/d + 32*a**2/(35*d*sin(
c + d*x)) - a**2*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) + a**2/(8*d*sin(c + d*x)**2) - 16*a**2*cos(c + d*x)**2/
(35*d*sin(c + d*x)**3) + a**2*cos(c + d*x)**4/(4*d*sin(c + d*x)**4) - a**2*cos(c + d*x)**2/(8*d*sin(c + d*x)**
4) + 12*a**2*cos(c + d*x)**4/(35*d*sin(c + d*x)**5) - a**2*cos(c + d*x)**6/(6*d*sin(c + d*x)**6) + a**2*cos(c
+ d*x)**4/(8*d*sin(c + d*x)**6) - 2*a**2*cos(c + d*x)**6/(7*d*sin(c + d*x)**7) - a**2*cos(c + d*x)**6/(8*d*sin
(c + d*x)**8), Eq(n, -9)), (-2*a**2*log(sin(c + d*x))/d - 16*a**2*sin(c + d*x)/(5*d) - 8*a**2*cos(c + d*x)**2/
(5*d*sin(c + d*x)) + 16*a**2/(35*d*sin(c + d*x)) - a**2*cos(c + d*x)**2/(d*sin(c + d*x)**2) + 2*a**2*cos(c + d
*x)**4/(5*d*sin(c + d*x)**3) - 8*a**2*cos(c + d*x)**2/(35*d*sin(c + d*x)**3) + a**2*cos(c + d*x)**4/(2*d*sin(c
+ d*x)**4) - a**2*cos(c + d*x)**6/(5*d*sin(c + d*x)**5) + 6*a**2*cos(c + d*x)**4/(35*d*sin(c + d*x)**5) - a**
2*cos(c + d*x)**6/(3*d*sin(c + d*x)**6) - a**2*cos(c + d*x)**6/(7*d*sin(c + d*x)**7), Eq(n, -8)), (-3840*a**2*
log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**10/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 +
1920*d*tan(c/2 + d*x/2)**6) - 7680*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**8/(1920*d*tan(c/2 + d*x
/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) - 3840*a**2*log(tan(c/2 + d*x/2)**2 + 1)*ta
n(c/2 + d*x/2)**6/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) + 38
40*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**10/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 +
1920*d*tan(c/2 + d*x/2)**6) + 7680*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**8/(1920*d*tan(c/2 + d*x/2)**1
0 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) + 3840*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2
)**6/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) - 5*a**2*tan(c/2
+ d*x/2)**16/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) - 24*a**2
*tan(c/2 + d*x/2)**15/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6)
+ 20*a**2*tan(c/2 + d*x/2)**14/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*
x/2)**6) + 312*a**2*tan(c/2 + d*x/2)**13/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*ta
n(c/2 + d*x/2)**6) + 220*a**2*tan(c/2 + d*x/2)**12/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 +
1920*d*tan(c/2 + d*x/2)**6) - 3864*a**2*tan(c/2 + d*x/2)**11/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 +
d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) - 21000*a**2*tan(c/2 + d*x/2)**9/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d
*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) - 4230*a**2*tan(c/2 + d*x/2)**8/(1920*d*tan(c/2 + d*x/2)**1
0 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) - 21000*a**2*tan(c/2 + d*x/2)**7/(1920*d*tan(c/2
+ d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) - 3864*a**2*tan(c/2 + d*x/2)**5/(1920*
d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) + 220*a**2*tan(c/2 + d*x/2)*
*4/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) + 312*a**2*tan(c/2
+ d*x/2)**3/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) + 20*a**2*
tan(c/2 + d*x/2)**2/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6) -
24*a**2*tan(c/2 + d*x/2)/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**
6) - 5*a**2/(1920*d*tan(c/2 + d*x/2)**10 + 3840*d*tan(c/2 + d*x/2)**8 + 1920*d*tan(c/2 + d*x/2)**6), Eq(n, -7)
), (-2880*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**11/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2
+ d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) - 8640*a**2*log(tan(c/2 + d*x/2)**2 + 1)
*tan(c/2 + d*x/2)**9/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 4
80*d*tan(c/2 + d*x/2)**5) - 8640*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**7/(480*d*tan(c/2 + d*x/2)
**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) - 2880*a**2*log(ta
n(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**5/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*t
an(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) + 2880*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**11/(480*d*
tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) +
8640*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**9/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 +
1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) + 8640*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**7
/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2
)**5) + 2880*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**5/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x
/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) - 3*a**2*tan(c/2 + d*x/2)**16/(480*d*tan(c/2
+ d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) - 15*a**2*
tan(c/2 + d*x/2)**15/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 4
80*d*tan(c/2 + d*x/2)**5) + 16*a**2*tan(c/2 + d*x/2)**14/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)
**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) + 255*a**2*tan(c/2 + d*x/2)**13/(480*d*tan(c/2 +
d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) + 156*a**2*
tan(c/2 + d*x/2)**12/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 4
80*d*tan(c/2 + d*x/2)**5) + 2352*a**2*tan(c/2 + d*x/2)**10/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/
2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) - 3300*a**2*tan(c/2 + d*x/2)**9/(480*d*tan(c/2
+ d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) + 3150*a*
*2*tan(c/2 + d*x/2)**8/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 +
480*d*tan(c/2 + d*x/2)**5) - 3300*a**2*tan(c/2 + d*x/2)**7/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x
/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) + 2352*a**2*tan(c/2 + d*x/2)**6/(480*d*tan(c/
2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) + 156*a*
*2*tan(c/2 + d*x/2)**4/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 +
480*d*tan(c/2 + d*x/2)**5) + 255*a**2*tan(c/2 + d*x/2)**3/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/
2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) + 16*a**2*tan(c/2 + d*x/2)**2/(480*d*tan(c/2 +
d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*tan(c/2 + d*x/2)**5) - 15*a**2*t
an(c/2 + d*x/2)/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/2)**7 + 480*d*
tan(c/2 + d*x/2)**5) - 3*a**2/(480*d*tan(c/2 + d*x/2)**11 + 1440*d*tan(c/2 + d*x/2)**9 + 1440*d*tan(c/2 + d*x/
2)**7 + 480*d*tan(c/2 + d*x/2)**5), Eq(n, -6)), (-3*a**2*tan(c/2 + d*x/2)**16/(192*d*tan(c/2 + d*x/2)**12 + 76
8*d*tan(c/2 + d*x/2)**10 + 1152*d*tan(c/2 + d*x/2)**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4)
- 16*a**2*tan(c/2 + d*x/2)**15/(192*d*tan(c/2 + d*x/2)**12 + 768*d*tan(c/2 + d*x/2)**10 + 1152*d*tan(c/2 + d*
x/2)**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 24*a**2*tan(c/2 + d*x/2)**14/(192*d*tan(c/2
+ d*x/2)**12 + 768*d*tan(c/2 + d*x/2)**10 + 1152*d*tan(c/2 + d*x/2)**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*ta
n(c/2 + d*x/2)**4) + 464*a**2*tan(c/2 + d*x/2)**13/(192*d*tan(c/2 + d*x/2)**12 + 768*d*tan(c/2 + d*x/2)**10 +
1152*d*tan(c/2 + d*x/2)**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 4848*a**2*tan(c/2 + d*x/
2)**11/(192*d*tan(c/2 + d*x/2)**12 + 768*d*tan(c/2 + d*x/2)**10 + 1152*d*tan(c/2 + d*x/2)**8 + 768*d*tan(c/2 +
d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 504*a**2*tan(c/2 + d*x/2)**10/(192*d*tan(c/2 + d*x/2)**12 + 768*d*ta
n(c/2 + d*x/2)**10 + 1152*d*tan(c/2 + d*x/2)**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 110
88*a**2*tan(c/2 + d*x/2)**9/(192*d*tan(c/2 + d*x/2)**12 + 768*d*tan(c/2 + d*x/2)**10 + 1152*d*tan(c/2 + d*x/2)
**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 294*a**2*tan(c/2 + d*x/2)**8/(192*d*tan(c/2 + d
*x/2)**12 + 768*d*tan(c/2 + d*x/2)**10 + 1152*d*tan(c/2 + d*x/2)**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/
2 + d*x/2)**4) + 11088*a**2*tan(c/2 + d*x/2)**7/(192*d*tan(c/2 + d*x/2)**12 + 768*d*tan(c/2 + d*x/2)**10 + 115
2*d*tan(c/2 + d*x/2)**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 504*a**2*tan(c/2 + d*x/2)**
6/(192*d*tan(c/2 + d*x/2)**12 + 768*d*tan(c/2 + d*x/2)**10 + 1152*d*tan(c/2 + d*x/2)**8 + 768*d*tan(c/2 + d*x/
2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 4848*a**2*tan(c/2 + d*x/2)**5/(192*d*tan(c/2 + d*x/2)**12 + 768*d*tan(c/2
+ d*x/2)**10 + 1152*d*tan(c/2 + d*x/2)**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 464*a**2
*tan(c/2 + d*x/2)**3/(192*d*tan(c/2 + d*x/2)**12 + 768*d*tan(c/2 + d*x/2)**10 + 1152*d*tan(c/2 + d*x/2)**8 + 7
68*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 24*a**2*tan(c/2 + d*x/2)**2/(192*d*tan(c/2 + d*x/2)**1
2 + 768*d*tan(c/2 + d*x/2)**10 + 1152*d*tan(c/2 + d*x/2)**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/
2)**4) - 16*a**2*tan(c/2 + d*x/2)/(192*d*tan(c/2 + d*x/2)**12 + 768*d*tan(c/2 + d*x/2)**10 + 1152*d*tan(c/2 +
d*x/2)**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) - 3*a**2/(192*d*tan(c/2 + d*x/2)**12 + 768*
d*tan(c/2 + d*x/2)**10 + 1152*d*tan(c/2 + d*x/2)**8 + 768*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4),
Eq(n, -5)), (720*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**13/(120*d*tan(c/2 + d*x/2)**13 + 600*d*ta
n(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120
*d*tan(c/2 + d*x/2)**3) + 3600*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**11/(120*d*tan(c/2 + d*x/2)*
*13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d
*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 7200*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**9/(120*d*tan(
c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*
d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 7200*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**
7/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x
/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 3600*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c
/2 + d*x/2)**5/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*
tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 720*a**2*log(tan(c/2 + d*x/2)**
2 + 1)*tan(c/2 + d*x/2)**3/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)*
*9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) - 720*a**2*log(tan(c/
2 + d*x/2))*tan(c/2 + d*x/2)**13/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d
*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) - 3600*a**2*log
(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**11/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(
c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) - 7200*a
**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**9/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*
d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) -
7200*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**7/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 +
1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)*
*3) - 3600*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**5/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)
**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d
*x/2)**3) - 720*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**3/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d
*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/
2 + d*x/2)**3) - 5*a**2*tan(c/2 + d*x/2)**16/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d
*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) - 3
0*a**2*tan(c/2 + d*x/2)**15/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)
**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 80*a**2*tan(c/2 +
d*x/2)**14/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(
c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 580*a**2*tan(c/2 + d*x/2)**12/(120*
d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7
+ 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 1860*a**2*tan(c/2 + d*x/2)**11/(120*d*tan(c/2 + d*x
/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2
+ d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 2160*a**2*tan(c/2 + d*x/2)**10/(120*d*tan(c/2 + d*x/2)**13 + 600*d
*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 +
120*d*tan(c/2 + d*x/2)**3) + 4410*a**2*tan(c/2 + d*x/2)**9/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2
)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 +
d*x/2)**3) + 2562*a**2*tan(c/2 + d*x/2)**8/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*t
an(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 441
0*a**2*tan(c/2 + d*x/2)**7/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)*
*9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 2160*a**2*tan(c/2 +
d*x/2)**6/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(
c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 1860*a**2*tan(c/2 + d*x/2)**5/(120*
d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7
+ 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 580*a**2*tan(c/2 + d*x/2)**4/(120*d*tan(c/2 + d*x/2
)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 +
d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3) + 80*a**2*tan(c/2 + d*x/2)**2/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(
c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d
*tan(c/2 + d*x/2)**3) - 30*a**2*tan(c/2 + d*x/2)/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 12
00*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3)
- 5*a**2/(120*d*tan(c/2 + d*x/2)**13 + 600*d*tan(c/2 + d*x/2)**11 + 1200*d*tan(c/2 + d*x/2)**9 + 1200*d*tan(c
/2 + d*x/2)**7 + 600*d*tan(c/2 + d*x/2)**5 + 120*d*tan(c/2 + d*x/2)**3), Eq(n, -4)), (240*a**2*log(tan(c/2 + d
*x/2)**2 + 1)*tan(c/2 + d*x/2)**14/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 +
d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(
c/2 + d*x/2)**2) + 1440*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**12/(120*d*tan(c/2 + d*x/2)**14 + 7
20*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)
**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 3600*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2
+ d*x/2)**10/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*t
an(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 480
0*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**8/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**
12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d
*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 3600*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**6/(120*d*tan(
c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 180
0*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 1440*a**2*log(tan(c/2 + d*x
/2)**2 + 1)*tan(c/2 + d*x/2)**4/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*
x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2
+ d*x/2)**2) + 240*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**2/(120*d*tan(c/2 + d*x/2)**14 + 720*d*
tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 +
720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 240*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**14/
(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/
2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 1440*a**2*log(ta
n(c/2 + d*x/2))*tan(c/2 + d*x/2)**12/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2
+ d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*ta
n(c/2 + d*x/2)**2) - 3600*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**10/(120*d*tan(c/2 + d*x/2)**14 + 720*d*
tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 +
720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 4800*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**8/
(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/
2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 3600*a**2*log(ta
n(c/2 + d*x/2))*tan(c/2 + d*x/2)**6/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2
+ d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan
(c/2 + d*x/2)**2) - 1440*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**4/(120*d*tan(c/2 + d*x/2)**14 + 720*d*ta
n(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 7
20*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 240*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(12
0*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)*
*8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 15*a**2*tan(c/2 + d
*x/2)**16/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(
c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 120*a*
*2*tan(c/2 + d*x/2)**15/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10
+ 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2
)**2) - 2280*a**2*tan(c/2 + d*x/2)**13/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c
/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*
tan(c/2 + d*x/2)**2) + 300*a**2*tan(c/2 + d*x/2)**12/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12
+ 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/
2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 7800*a**2*tan(c/2 + d*x/2)**11/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/
2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d
*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 1920*a**2*tan(c/2 + d*x/2)**10/(120*d*tan(c/2 + d*x/2)**14
+ 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*
x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 14376*a**2*tan(c/2 + d*x/2)**9/(120*d*tan(c
/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800
*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 1990*a**2*tan(c/2 + d*x/2)**
8/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*
x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 14376*a**2*tan
(c/2 + d*x/2)**7/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400
*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) +
1920*a**2*tan(c/2 + d*x/2)**6/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x
/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2
+ d*x/2)**2) - 7800*a**2*tan(c/2 + d*x/2)**5/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)**12 + 1800*d
*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 + d*x/2)**4 +
120*d*tan(c/2 + d*x/2)**2) + 300*a**2*tan(c/2 + d*x/2)**4/(120*d*tan(c/2 + d*x/2)**14 + 720*d*tan(c/2 + d*x/2)
**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 720*d*tan(c/2 +
d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 2280*a**2*tan(c/2 + d*x/2)**3/(120*d*tan(c/2 + d*x/2)**14 + 720*d*ta
n(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 + 7
20*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 120*a**2*tan(c/2 + d*x/2)/(120*d*tan(c/2 + d*x/2)**14
+ 720*d*tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x
/2)**6 + 720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 15*a**2/(120*d*tan(c/2 + d*x/2)**14 + 720*d*
tan(c/2 + d*x/2)**12 + 1800*d*tan(c/2 + d*x/2)**10 + 2400*d*tan(c/2 + d*x/2)**8 + 1800*d*tan(c/2 + d*x/2)**6 +
720*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2), Eq(n, -3)), (-420*a**2*log(tan(c/2 + d*x/2)**2 + 1)*t
an(c/2 + d*x/2)**15/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 +
7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)
**3 + 210*d*tan(c/2 + d*x/2)) - 2940*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**13/(210*d*tan(c/2 + d
*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*ta
n(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 8820*a
**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**11/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**1
3 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d
*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 14700*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(
c/2 + d*x/2)**9/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350
*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3
+ 210*d*tan(c/2 + d*x/2)) - 14700*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**7/(210*d*tan(c/2 + d*x/2
)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/
2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 8820*a**2*
log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**5/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4
410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)
**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 2940*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 +
d*x/2)**3/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan
(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*
d*tan(c/2 + d*x/2)) - 420*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)/(210*d*tan(c/2 + d*x/2)**15 + 147
0*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)*
*7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) + 420*a**2*log(tan(c/2
+ d*x/2))*tan(c/2 + d*x/2)**15/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*
x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c
/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) + 2940*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**13/(210*d*tan(c/2
+ d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*
d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) + 88
20*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**11/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 +
4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/
2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) + 14700*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/
2)**9/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2
+ d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*ta
n(c/2 + d*x/2)) + 14700*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**7/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*ta
n(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4
410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) + 8820*a**2*log(tan(c/2 + d*x
/2))*tan(c/2 + d*x/2)**5/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**
11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d
*x/2)**3 + 210*d*tan(c/2 + d*x/2)) + 2940*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**3/(210*d*tan(c/2 + d*x/
2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c
/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) + 420*a**2*
log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(
c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470
*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 105*a**2*tan(c/2 + d*x/2)**16/(210*d*tan(c/2 + d*x/2)**15 +
1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x
/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 1680*a**2*tan(c/2
+ d*x/2)**14/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d
*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 +
210*d*tan(c/2 + d*x/2)) - 2520*a**2*tan(c/2 + d*x/2)**13/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)
**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2
+ d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 7980*a**2*tan(c/2 + d*x/2)**12/(210*d*tan
(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7
350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2))
- 7560*a**2*tan(c/2 + d*x/2)**11/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 +
d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan
(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 15792*a**2*tan(c/2 + d*x/2)**10/(210*d*tan(c/2 + d*x/2)**15 + 147
0*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)*
*7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 14560*a**2*tan(c/2 +
d*x/2)**9/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan
(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*
d*tan(c/2 + d*x/2)) - 22614*a**2*tan(c/2 + d*x/2)**8/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13
+ 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*
x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 14560*a**2*tan(c/2 + d*x/2)**7/(210*d*tan(c/2
+ d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*
d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 15
792*a**2*tan(c/2 + d*x/2)**6/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/
2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2
+ d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 7560*a**2*tan(c/2 + d*x/2)**5/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*ta
n(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4
410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 7980*a**2*tan(c/2 + d*x/2)*
*4/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 +
d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c
/2 + d*x/2)) - 2520*a**2*tan(c/2 + d*x/2)**3/(210*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*
d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5
+ 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 1680*a**2*tan(c/2 + d*x/2)**2/(210*d*tan(c/2 + d*x/2)
**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2)**9 + 7350*d*tan(c/2
+ d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 + d*x/2)) - 105*a**2/(2
10*d*tan(c/2 + d*x/2)**15 + 1470*d*tan(c/2 + d*x/2)**13 + 4410*d*tan(c/2 + d*x/2)**11 + 7350*d*tan(c/2 + d*x/2
)**9 + 7350*d*tan(c/2 + d*x/2)**7 + 4410*d*tan(c/2 + d*x/2)**5 + 1470*d*tan(c/2 + d*x/2)**3 + 210*d*tan(c/2 +
d*x/2)), Eq(n, -2)), (-105*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**16/(105*d*tan(c/2 + d*x/2)**16
+ 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*
x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) - 840*a
**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**14/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14
+ 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d
*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) - 2940*a**2*log(tan(c/2 + d*x/2)**2
+ 1)*tan(c/2 + d*x/2)**12/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)*
*12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 +
d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) - 5880*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**10/
(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/
2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2
+ d*x/2)**2 + 105*d) - 7350*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**8/(105*d*tan(c/2 + d*x/2)**16
+ 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*
x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) - 5880*
a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**6/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14
+ 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d
*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) - 2940*a**2*log(tan(c/2 + d*x/2)**2
+ 1)*tan(c/2 + d*x/2)**4/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**
12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 +
d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) - 840*a**2*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**2/(10
5*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)*
*10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d
*x/2)**2 + 105*d) - 105*a**2*log(tan(c/2 + d*x/2)**2 + 1)/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)
**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2
+ d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 105*a**2*log(tan(c/2 + d*x/2)
)*tan(c/2 + d*x/2)**16/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12
+ 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x
/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 840*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**14/(105*d*tan(c
/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 735
0*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2
+ 105*d) + 2940*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**12/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 +
d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*t
an(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 5880*a**2*log(tan(c/2 +
d*x/2))*tan(c/2 + d*x/2)**10/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/
2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/
2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 7350*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**8/(105*
d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**1
0 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x
/2)**2 + 105*d) + 5880*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**6/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(
c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 58
80*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 2940*a**2*log(tan
(c/2 + d*x/2))*tan(c/2 + d*x/2)**4/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 +
d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*t
an(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 840*a**2*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(
105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2
)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 +
d*x/2)**2 + 105*d) + 105*a**2*log(tan(c/2 + d*x/2))/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14
+ 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*
x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 420*a**2*tan(c/2 + d*x/2)**15/(105
*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**
10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*
x/2)**2 + 105*d) - 420*a**2*tan(c/2 + d*x/2)**14/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 29
40*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)
**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 1260*a**2*tan(c/2 + d*x/2)**13/(105*d*
tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10
+ 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2
)**2 + 105*d) - 2520*a**2*tan(c/2 + d*x/2)**12/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940
*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**
6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 4452*a**2*tan(c/2 + d*x/2)**11/(105*d*ta
n(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 +
7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)*
*2 + 105*d) - 4060*a**2*tan(c/2 + d*x/2)**10/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d
*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6
+ 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 6156*a**2*tan(c/2 + d*x/2)**9/(105*d*tan(c
/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 735
0*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2
+ 105*d) - 7280*a**2*tan(c/2 + d*x/2)**8/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan
(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 29
40*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 6156*a**2*tan(c/2 + d*x/2)**7/(105*d*tan(c/2 +
d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*
tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 10
5*d) - 4060*a**2*tan(c/2 + d*x/2)**6/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2
+ d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d
*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 4452*a**2*tan(c/2 + d*x/2)**5/(105*d*tan(c/2 + d*x
/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(
c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d)
- 2520*a**2*tan(c/2 + d*x/2)**4/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d
*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan
(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 1260*a**2*tan(c/2 + d*x/2)**3/(105*d*tan(c/2 + d*x/2)*
*16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2
+ d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) - 4
20*a**2*tan(c/2 + d*x/2)**2/(105*d*tan(c/2 + d*x/2)**16 + 840*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)
**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)**8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2
+ d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d) + 420*a**2*tan(c/2 + d*x/2)/(105*d*tan(c/2 + d*x/2)**16 + 840
*d*tan(c/2 + d*x/2)**14 + 2940*d*tan(c/2 + d*x/2)**12 + 5880*d*tan(c/2 + d*x/2)**10 + 7350*d*tan(c/2 + d*x/2)*
*8 + 5880*d*tan(c/2 + d*x/2)**6 + 2940*d*tan(c/2 + d*x/2)**4 + 840*d*tan(c/2 + d*x/2)**2 + 105*d), Eq(n, -1)),
(a**2*n**8*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 6
3273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 2*a**2*n**8*sin(c + d
*x)**2*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*
d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + a**2*n**8*sin(c + d*x)*sin(c + d*x)**n*cos
(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1
172700*d*n**2 + 1026576*d*n + 362880*d) + 6*a**2*n**7*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9
+ 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 10265
76*d*n + 362880*d) + 12*a**2*n**7*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*
n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) +
42*a**2*n**7*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 +
63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 6*a**2*n**7*sin(c +
d*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 26932
5*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 86*a**2*n**7*sin(c + d*x)**2*sin(c + d*x
)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*
n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 44*a**2*n**7*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(d
*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 +
1026576*d*n + 362880*d) + 24*a**2*n**6*sin(c + d*x)**7*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 +
870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 36288
0*d) + 48*a**2*n**6*sin(c + d*x)**6*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*
n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 222*a**2*n**6
*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5
+ 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 24*a**2*n**6*sin(c + d*x)**5*sin
(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 7
23680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 468*a**2*n**6*sin(c + d*x)**4*sin(c + d*x)**n*cos(c
+ d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172
700*d*n**2 + 1026576*d*n + 362880*d) + 744*a**2*n**6*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 +
45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 102657
6*d*n + 362880*d) + 246*a**2*n**6*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*
n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) +
1568*a**2*n**6*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6
+ 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 826*a**2*n**6*sin
(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 2693
25*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 48*a**2*n**5*sin(c + d*x)**9*sin(c + d*
x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*
d*n**2 + 1026576*d*n + 362880*d) + 96*a**2*n**5*sin(c + d*x)**8*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n*
*7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 7
20*a**2*n**5*sin(c + d*x)**7*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 +
63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 48*a**2*n**5*sin(c +
d*x)**7*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 72368
0*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 1584*a**2*n**5*sin(c + d*x)**6*sin(c + d*x)**n*cos(c + d
*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700
*d*n**2 + 1026576*d*n + 362880*d) + 3354*a**2*n**5*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 4
5*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*
d*n + 362880*d) + 864*a**2*n**5*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n*
*7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 7
536*a**2*n**5*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 +
63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 7218*a**2*n**5*sin(
c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 26
9325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 4218*a**2*n**5*sin(c + d*x)**3*sin(c
+ d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 7236
80*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 15764*a**2*n**5*sin(c + d*x)**2*sin(c + d*x)**n*cos(c +
d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 11727
00*d*n**2 + 1026576*d*n + 362880*d) + 8624*a**2*n**5*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45
*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d
*n + 362880*d) + 1008*a**2*n**4*sin(c + d*x)**9*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6
+ 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 2400*a**2*n**4*si
n(c + d*x)**8*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 +
723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 8376*a**2*n**4*sin(c + d*x)**7*sin(c + d*x)**n*cos(
c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 11
72700*d*n**2 + 1026576*d*n + 362880*d) + 1392*a**2*n**4*sin(c + d*x)**7*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 +
870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 36288
0*d) + 20640*a**2*n**4*sin(c + d*x)**6*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450
*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 26538*a**2
*n**4*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d
*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 12552*a**2*n**4*sin(c + d*x
)**5*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*
n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 64440*a**2*n**4*sin(c + d*x)**4*sin(c + d*x)
**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n
**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 41619*a**2*n**4*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)*
*6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n
**2 + 1026576*d*n + 362880*d) + 39090*a**2*n**4*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d
*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n
+ 362880*d) + 95018*a**2*n**4*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**
7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 54
649*a**2*n**4*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63
273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 7680*a**2*n**3*sin(c +
d*x)**9*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 72368
0*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 22080*a**2*n**3*sin(c + d*x)**8*sin(c + d*x)**n/(d*n**9
+ 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 10265
76*d*n + 362880*d) + 47520*a**2*n**3*sin(c + d*x)**7*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870
*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d
) + 15360*a**2*n**3*sin(c + d*x)**7*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n
**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 133920*a**2*n**3*sin(c + d*x)
**6*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n
**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 117024*a**2*n**3*sin(c + d*x)**5*sin(c + d*x)
**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n
**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 93600*a**2*n**3*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)*
*2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n
**2 + 1026576*d*n + 362880*d) + 312348*a**2*n**3*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*
d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*
n + 362880*d) + 144468*a**2*n**3*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n
**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) +
210624*a**2*n**3*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**
6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 348614*a**2*n**3
*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5
+ 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 214676*a**2*n**3*sin(c + d*x)*si
n(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 +
723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 25920*a**2*n**2*sin(c + d*x)**9*sin(c + d*x)**n/(d*
n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 +
1026576*d*n + 362880*d) + 91200*a**2*n**2*sin(c + d*x)**8*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9
450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 135456*
a**2*n**2*sin(c + d*x)**7*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 632
73*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 79680*a**2*n**2*sin(c +
d*x)**7*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 72368
0*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 445872*a**2*n**2*sin(c + d*x)**6*sin(c + d*x)**n*cos(c +
d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 11727
00*d*n**2 + 1026576*d*n + 362880*d) + 281688*a**2*n**2*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9
+ 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026
576*d*n + 362880*d) + 374496*a**2*n**2*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 8
70*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880
*d) + 842292*a**2*n**2*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450
*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 290276*a**
2*n**2*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*
d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 656184*a**2*n**2*sin(c + d
*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*
d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 750132*a**2*n**2*sin(c + d*x)**2*sin(c + d
*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*
d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 509004*a**2*n**2*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)
**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*
n**2 + 1026576*d*n + 362880*d) + 37632*a**2*n*sin(c + d*x)**9*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7
+ 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 162
144*a**2*n*sin(c + d*x)**8*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269
325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 178560*a**2*n*sin(c + d*x)**7*sin(c +
d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680
*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 191232*a**2*n*sin(c + d*x)**7*sin(c + d*x)**n/(d*n**9 + 4
5*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*
d*n + 362880*d) + 693936*a**2*n*sin(c + d*x)**6*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n*
*7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 3
33216*a**2*n*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 +
63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 752256*a**2*n*sin(c
+ d*x)**5*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 2693
25*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 1131624*a**2*n*sin(c + d*x)**4*sin(c +
d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680
*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 301872*a**2*n*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x
)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d
*n**2 + 1026576*d*n + 362880*d) + 1085472*a**2*n*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*
d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*
n + 362880*d) + 845136*a**2*n*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7
+ 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 663
696*a**2*n*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273
*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 18432*a**2*sin(c + d*x)**
9*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**
3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 90720*a**2*sin(c + d*x)**8*sin(c + d*x)**n/(d*n**9 + 45*d*n**8
+ 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362
880*d) + 82944*a**2*sin(c + d*x)**7*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*
n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 165888*a**2*s
in(c + d*x)**7*sin(c + d*x)**n/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 +
723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 362880*a**2*sin(c + d*x)**6*sin(c + d*x)**n*cos(c
+ d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172
700*d*n**2 + 1026576*d*n + 362880*d) + 145152*a**2*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 4
5*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*
d*n + 362880*d) + 580608*a**2*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**9 + 45*d*n**8 + 870*d*n**7
+ 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 544
320*a**2*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 6327
3*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 120960*a**2*sin(c + d*x)
**3*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n
**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d) + 725760*a**2*sin(c + d*x)**3*sin(c + d*x)**n*c
os(c + d*x)**4/(d*n**9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 +
1172700*d*n**2 + 1026576*d*n + 362880*d) + 362880*a**2*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**
9 + 45*d*n**8 + 870*d*n**7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 102
6576*d*n + 362880*d) + 362880*a**2*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(d*n**9 + 45*d*n**8 + 870*d*n*
*7 + 9450*d*n**6 + 63273*d*n**5 + 269325*d*n**4 + 723680*d*n**3 + 1172700*d*n**2 + 1026576*d*n + 362880*d), Tr
ue))
Maxima [A] (verification not implemented)
none
Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.90
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {\frac {a^{2} \sin \left (d x + c\right )^{n + 9}}{n + 9} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 8}}{n + 8} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 7}}{n + 7} - \frac {6 \, a^{2} \sin \left (d x + c\right )^{n + 6}}{n + 6} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 3}}{n + 3} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a^{2} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x, algorithm="maxima")
[Out]
-(a^2*sin(d*x + c)^(n + 9)/(n + 9) + 2*a^2*sin(d*x + c)^(n + 8)/(n + 8) - 2*a^2*sin(d*x + c)^(n + 7)/(n + 7) -
6*a^2*sin(d*x + c)^(n + 6)/(n + 6) + 6*a^2*sin(d*x + c)^(n + 4)/(n + 4) + 2*a^2*sin(d*x + c)^(n + 3)/(n + 3)
- 2*a^2*sin(d*x + c)^(n + 2)/(n + 2) - a^2*sin(d*x + c)^(n + 1)/(n + 1))/d
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (184) = 368\).
Time = 0.42 (sec) , antiderivative size = 1019, normalized size of antiderivative = 5.54
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x, algorithm="giac")
[Out]
-((n^3*sin(d*x + c)^n*sin(d*x + c)^9 + 15*n^2*sin(d*x + c)^n*sin(d*x + c)^9 - 3*n^3*sin(d*x + c)^n*sin(d*x + c
)^7 + 71*n*sin(d*x + c)^n*sin(d*x + c)^9 - 51*n^2*sin(d*x + c)^n*sin(d*x + c)^7 + 105*sin(d*x + c)^n*sin(d*x +
c)^9 + 3*n^3*sin(d*x + c)^n*sin(d*x + c)^5 - 261*n*sin(d*x + c)^n*sin(d*x + c)^7 + 57*n^2*sin(d*x + c)^n*sin(
d*x + c)^5 - 405*sin(d*x + c)^n*sin(d*x + c)^7 - n^3*sin(d*x + c)^n*sin(d*x + c)^3 + 333*n*sin(d*x + c)^n*sin(
d*x + c)^5 - 21*n^2*sin(d*x + c)^n*sin(d*x + c)^3 + 567*sin(d*x + c)^n*sin(d*x + c)^5 - 143*n*sin(d*x + c)^n*s
in(d*x + c)^3 - 315*sin(d*x + c)^n*sin(d*x + c)^3)*a^2/(n^4 + 24*n^3 + 206*n^2 + 744*n + 945) + 2*(n^3*sin(d*x
+ c)^n*sin(d*x + c)^8 + 12*n^2*sin(d*x + c)^n*sin(d*x + c)^8 - 3*n^3*sin(d*x + c)^n*sin(d*x + c)^6 + 44*n*sin
(d*x + c)^n*sin(d*x + c)^8 - 42*n^2*sin(d*x + c)^n*sin(d*x + c)^6 + 48*sin(d*x + c)^n*sin(d*x + c)^8 + 3*n^3*s
in(d*x + c)^n*sin(d*x + c)^4 - 168*n*sin(d*x + c)^n*sin(d*x + c)^6 + 48*n^2*sin(d*x + c)^n*sin(d*x + c)^4 - 19
2*sin(d*x + c)^n*sin(d*x + c)^6 - n^3*sin(d*x + c)^n*sin(d*x + c)^2 + 228*n*sin(d*x + c)^n*sin(d*x + c)^4 - 18
*n^2*sin(d*x + c)^n*sin(d*x + c)^2 + 288*sin(d*x + c)^n*sin(d*x + c)^4 - 104*n*sin(d*x + c)^n*sin(d*x + c)^2 -
192*sin(d*x + c)^n*sin(d*x + c)^2)*a^2/(n^4 + 20*n^3 + 140*n^2 + 400*n + 384) + (n^3*sin(d*x + c)^n*sin(d*x +
c)^7 + 9*n^2*sin(d*x + c)^n*sin(d*x + c)^7 - 3*n^3*sin(d*x + c)^n*sin(d*x + c)^5 + 23*n*sin(d*x + c)^n*sin(d*
x + c)^7 - 33*n^2*sin(d*x + c)^n*sin(d*x + c)^5 + 15*sin(d*x + c)^n*sin(d*x + c)^7 + 3*n^3*sin(d*x + c)^n*sin(
d*x + c)^3 - 93*n*sin(d*x + c)^n*sin(d*x + c)^5 + 39*n^2*sin(d*x + c)^n*sin(d*x + c)^3 - 63*sin(d*x + c)^n*sin
(d*x + c)^5 - n^3*sin(d*x + c)^n*sin(d*x + c) + 141*n*sin(d*x + c)^n*sin(d*x + c)^3 - 15*n^2*sin(d*x + c)^n*si
n(d*x + c) + 105*sin(d*x + c)^n*sin(d*x + c)^3 - 71*n*sin(d*x + c)^n*sin(d*x + c) - 105*sin(d*x + c)^n*sin(d*x
+ c))*a^2/(n^4 + 16*n^3 + 86*n^2 + 176*n + 105))/d
Mupad [B] (verification not implemented)
Time = 19.26 (sec) , antiderivative size = 1142, normalized size of antiderivative = 6.21
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Too large to display}
\]
[In]
int(cos(c + d*x)^7*sin(c + d*x)^n*(a + a*sin(c + d*x))^2,x)
[Out]
(a^2*sin(c + d*x)^n*(n*1507788i + n^2*868332i + n^3*238585i + n^4*37844i + n^5*3702i + n^6*208i + n^7*5i + 843
696i))/(64*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 7
2576i)) - (a^2*sin(c + d*x)^n*sin(9*c + 9*d*x)*(20304*n + 19564*n^2 + 9544*n^3 + 2581*n^4 + 391*n^5 + 31*n^6 +
n^7 + 8064)*1i)/(256*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i +
n^8*1i + 72576i)) + (a^2*sin(c + d*x)*sin(c + d*x)^n*(6799248*n + 3169500*n^2 + 770632*n^3 + 111993*n^4 + 102
67*n^5 + 555*n^6 + 13*n^7 + 5588352)*1i)/(128*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100
i + n^6*670i + n^7*40i + n^8*1i + 72576i)) - (a^2*sin(c + d*x)^n*cos(8*c + 8*d*x)*(n*22716i + n^2*21708i + n^3
*10469i + n^4*2788i + n^5*414i + n^6*32i + n^7*1i + 9072i))/(64*d*(n*190800i + n^2*196380i + n^3*105460i + n^4
*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 72576i)) - (a^2*sin(c + d*x)^n*cos(6*c + 6*d*x)*(n*43920i
+ n^2*39882i + n^3*17909i + n^4*4336i + n^5*570i + n^6*38i + n^7*1i + 18144i))/(16*d*(n*190800i + n^2*196380i
+ n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 72576i)) - (a^2*sin(c + d*x)^n*cos(4*c
+ 4*d*x)*(n*140868i + n^2*111816i + n^3*41669i + n^4*7996i + n^5*822i + n^6*44i + n^7*1i + 63504i))/(16*d*(n*1
90800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 72576i)) + (a^2*si
n(c + d*x)^n*cos(2*c + 2*d*x)*(n^3*2549i - n^2*59958i - n*186480i + n^4*3568i + n^5*570i + n^6*38i + n^7*1i -
127008i))/(16*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i
+ 72576i)) - (a^2*sin(c + d*x)^n*sin(7*c + 7*d*x)*(23472*n + 18900*n^2 + 6776*n^3 + 987*n^4 - 7*n^5 - 15*n^6 -
n^7 + 10368)*1i)/(256*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i
+ n^8*1i + 72576i)) + (a^2*sin(c + d*x)^n*sin(5*c + 5*d*x)*(178128*n + 165132*n^2 + 76280*n^3 + 19149*n^4 + 26
27*n^5 + 183*n^6 + 5*n^7 + 72576)*1i)/(64*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i +
n^6*670i + n^7*40i + n^8*1i + 72576i)) + (a^2*sin(c + d*x)^n*sin(3*c + 3*d*x)*(1120944*n + 889556*n^2 + 338024
*n^3 + 68603*n^4 + 7661*n^5 + 449*n^6 + 11*n^7 + 508032)*1i)/(64*d*(n*190800i + n^2*196380i + n^3*105460i + n^
4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 72576i))