\(\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx\) [565]
Optimal result
Integrand size = 29, antiderivative size = 181 \[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {3 a^3 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac {a^3 \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {5 a^3 \sin ^{4+n}(c+d x)}{d (4+n)}-\frac {5 a^3 \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {a^3 \sin ^{6+n}(c+d x)}{d (6+n)}+\frac {3 a^3 \sin ^{7+n}(c+d x)}{d (7+n)}+\frac {a^3 \sin ^{8+n}(c+d x)}{d (8+n)}
\]
[Out]
a^3*sin(d*x+c)^(1+n)/d/(1+n)+3*a^3*sin(d*x+c)^(2+n)/d/(2+n)+a^3*sin(d*x+c)^(3+n)/d/(3+n)-5*a^3*sin(d*x+c)^(4+n
)/d/(4+n)-5*a^3*sin(d*x+c)^(5+n)/d/(5+n)+a^3*sin(d*x+c)^(6+n)/d/(6+n)+3*a^3*sin(d*x+c)^(7+n)/d/(7+n)+a^3*sin(d
*x+c)^(8+n)/d/(8+n)
Rubi [A] (verified)
Time = 0.13 (sec) , antiderivative size = 181, 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}
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac {a^3 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {5 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac {5 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac {3 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}+\frac {a^3 \sin ^{n+8}(c+d x)}{d (n+8)}
\]
[In]
Int[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]
[Out]
(a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (a^3*Sin[c + d*x]^(3 + n))
/(d*(3 + n)) - (5*a^3*Sin[c + d*x]^(4 + n))/(d*(4 + n)) - (5*a^3*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (a^3*Sin[
c + d*x]^(6 + n))/(d*(6 + n)) + (3*a^3*Sin[c + d*x]^(7 + n))/(d*(7 + n)) + (a^3*Sin[c + d*x]^(8 + n))/(d*(8 +
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)^2 \left (\frac {x}{a}\right )^n (a+x)^5 \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \left (a^7 \left (\frac {x}{a}\right )^n+3 a^7 \left (\frac {x}{a}\right )^{1+n}+a^7 \left (\frac {x}{a}\right )^{2+n}-5 a^7 \left (\frac {x}{a}\right )^{3+n}-5 a^7 \left (\frac {x}{a}\right )^{4+n}+a^7 \left (\frac {x}{a}\right )^{5+n}+3 a^7 \left (\frac {x}{a}\right )^{6+n}+a^7 \left (\frac {x}{a}\right )^{7+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {a^3 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {3 a^3 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac {a^3 \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {5 a^3 \sin ^{4+n}(c+d x)}{d (4+n)}-\frac {5 a^3 \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {a^3 \sin ^{6+n}(c+d x)}{d (6+n)}+\frac {3 a^3 \sin ^{7+n}(c+d x)}{d (7+n)}+\frac {a^3 \sin ^{8+n}(c+d x)}{d (8+n)} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.43 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.68
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^{1+n}(c+d x) \left (\frac {1}{1+n}+\frac {3 \sin (c+d x)}{2+n}+\frac {\sin ^2(c+d x)}{3+n}-\frac {5 \sin ^3(c+d x)}{4+n}-\frac {5 \sin ^4(c+d x)}{5+n}+\frac {\sin ^5(c+d x)}{6+n}+\frac {3 \sin ^6(c+d x)}{7+n}+\frac {\sin ^7(c+d x)}{8+n}\right )}{d}
\]
[In]
Integrate[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]
[Out]
(a^3*Sin[c + d*x]^(1 + n)*((1 + n)^(-1) + (3*Sin[c + d*x])/(2 + n) + Sin[c + d*x]^2/(3 + n) - (5*Sin[c + d*x]^
3)/(4 + n) - (5*Sin[c + d*x]^4)/(5 + n) + Sin[c + d*x]^5/(6 + n) + (3*Sin[c + d*x]^6)/(7 + n) + Sin[c + d*x]^7
/(8 + n)))/d
Maple [A] (verified)
Time = 12.31 (sec) , antiderivative size = 244, normalized size of antiderivative =
1.35
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {a^{3} \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}+\frac {a^{3} \left (\sin ^{6}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (6+n \right )}+\frac {a^{3} \left (\sin ^{8}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (8+n \right )}+\frac {3 a^{3} \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}-\frac {5 a^{3} \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}-\frac {5 a^{3} \left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {3 a^{3} \left (\sin ^{7}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (7+n \right )}\) |
\(244\) |
| default |
\(\frac {a^{3} \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}+\frac {a^{3} \left (\sin ^{6}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (6+n \right )}+\frac {a^{3} \left (\sin ^{8}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (8+n \right )}+\frac {3 a^{3} \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}-\frac {5 a^{3} \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}-\frac {5 a^{3} \left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {3 a^{3} \left (\sin ^{7}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (7+n \right )}\) |
\(244\) |
| parallelrisch |
\(\frac {17 \left (\sin ^{n}\left (d x +c \right )\right ) \left (\frac {6 \left (1+n \right ) \left (5+n \right ) \left (7+n \right ) \left (n^{3}+\frac {38}{3} n^{2}-\frac {368}{3} n -1056\right ) \left (3+n \right ) \cos \left (2 d x +2 c \right )}{17}-\frac {14 \left (1+n \right ) \left (2+n \right ) \left (n^{2}+\frac {138}{7} n +\frac {600}{7}\right ) \left (5+n \right ) \left (7+n \right ) \left (3+n \right ) \cos \left (4 d x +4 c \right )}{17}-\frac {6 \left (n +\frac {20}{3}\right ) \left (1+n \right ) \left (2+n \right ) \left (5+n \right ) \left (7+n \right ) \left (4+n \right ) \left (3+n \right ) \cos \left (6 d x +6 c \right )}{17}+\frac {\left (7+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (8 d x +8 c \right )}{34}+\frac {21 \left (n +\frac {85}{7}\right ) \left (8+n \right ) \left (1+n \right ) \left (6+n \right ) \left (2+n \right ) \left (n +\frac {7}{3}\right ) \left (4+n \right ) \sin \left (3 d x +3 c \right )}{17}+\frac {\left (n -35\right ) \left (8+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (5 d x +5 c \right )}{17}-\frac {3 \left (8+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (7 d x +7 c \right )}{17}+\left (n^{3}+\frac {329}{17} n^{2}+\frac {3015}{17} n +\frac {5775}{17}\right ) \left (8+n \right ) \left (6+n \right ) \left (2+n \right ) \left (4+n \right ) \sin \left (d x +c \right )+\frac {27 \left (n^{3}+\frac {596}{27} n^{2}+204 n +\frac {18064}{27}\right ) \left (1+n \right ) \left (5+n \right ) \left (7+n \right ) \left (3+n \right )}{34}\right ) a^{3}}{64 \left (3+n \right ) \left (1+n \right ) \left (4+n \right ) \left (2+n \right ) \left (7+n \right ) \left (5+n \right ) \left (8+n \right ) d \left (6+n \right )}\) |
\(347\) |
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[In]
int(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
[Out]
a^3/d/(1+n)*sin(d*x+c)*exp(n*ln(sin(d*x+c)))+a^3/d/(3+n)*sin(d*x+c)^3*exp(n*ln(sin(d*x+c)))+a^3/d/(6+n)*sin(d*
x+c)^6*exp(n*ln(sin(d*x+c)))+a^3/d/(8+n)*sin(d*x+c)^8*exp(n*ln(sin(d*x+c)))+3*a^3/d/(2+n)*sin(d*x+c)^2*exp(n*l
n(sin(d*x+c)))-5*a^3/d/(4+n)*sin(d*x+c)^4*exp(n*ln(sin(d*x+c)))-5*a^3/d/(5+n)*sin(d*x+c)^5*exp(n*ln(sin(d*x+c)
))+3*a^3/d/(7+n)*sin(d*x+c)^7*exp(n*ln(sin(d*x+c)))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (181) = 362\).
Time = 0.31 (sec) , antiderivative size = 616, normalized size of antiderivative = 3.40
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {{\left ({\left (a^{3} n^{7} + 28 \, a^{3} n^{6} + 322 \, a^{3} n^{5} + 1960 \, a^{3} n^{4} + 6769 \, a^{3} n^{3} + 13132 \, a^{3} n^{2} + 13068 \, a^{3} n + 5040 \, a^{3}\right )} \cos \left (d x + c\right )^{8} + 32 \, a^{3} n^{5} + 720 \, a^{3} n^{4} - {\left (5 \, a^{3} n^{7} + 142 \, a^{3} n^{6} + 1654 \, a^{3} n^{5} + 10180 \, a^{3} n^{4} + 35485 \, a^{3} n^{3} + 69358 \, a^{3} n^{2} + 69416 \, a^{3} n + 26880 \, a^{3}\right )} \cos \left (d x + c\right )^{6} + 6080 \, a^{3} n^{3} + 23520 \, a^{3} n^{2} + 2 \, {\left (2 \, a^{3} n^{7} + 49 \, a^{3} n^{6} + 470 \, a^{3} n^{5} + 2230 \, a^{3} n^{4} + 5438 \, a^{3} n^{3} + 6361 \, a^{3} n^{2} + 2730 \, a^{3} n\right )} \cos \left (d x + c\right )^{4} + 39968 \, a^{3} n + 21840 \, a^{3} + 8 \, {\left (2 \, a^{3} n^{6} + 45 \, a^{3} n^{5} + 380 \, a^{3} n^{4} + 1470 \, a^{3} n^{3} + 2498 \, a^{3} n^{2} + 1365 \, a^{3} n\right )} \cos \left (d x + c\right )^{2} + {\left (32 \, a^{3} n^{5} + 720 \, a^{3} n^{4} - 3 \, {\left (a^{3} n^{7} + 29 \, a^{3} n^{6} + 343 \, a^{3} n^{5} + 2135 \, a^{3} n^{4} + 7504 \, a^{3} n^{3} + 14756 \, a^{3} n^{2} + 14832 \, a^{3} n + 5760 \, a^{3}\right )} \cos \left (d x + c\right )^{6} + 6080 \, a^{3} n^{3} + 24000 \, a^{3} n^{2} + 2 \, {\left (2 \, a^{3} n^{7} + 53 \, a^{3} n^{6} + 566 \, a^{3} n^{5} + 3155 \, a^{3} n^{4} + 9908 \, a^{3} n^{3} + 17492 \, a^{3} n^{2} + 15984 \, a^{3} n + 5760 \, a^{3}\right )} \cos \left (d x + c\right )^{4} + 44288 \, a^{3} n + 30720 \, a^{3} + 8 \, {\left (2 \, a^{3} n^{6} + 47 \, a^{3} n^{5} + 425 \, a^{3} n^{4} + 1880 \, a^{3} n^{3} + 4268 \, a^{3} n^{2} + 4688 \, a^{3} n + 1920 \, a^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 36 \, d n^{7} + 546 \, d n^{6} + 4536 \, d n^{5} + 22449 \, d n^{4} + 67284 \, d n^{3} + 118124 \, d n^{2} + 109584 \, d n + 40320 \, d}
\]
[In]
integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="fricas")
[Out]
((a^3*n^7 + 28*a^3*n^6 + 322*a^3*n^5 + 1960*a^3*n^4 + 6769*a^3*n^3 + 13132*a^3*n^2 + 13068*a^3*n + 5040*a^3)*c
os(d*x + c)^8 + 32*a^3*n^5 + 720*a^3*n^4 - (5*a^3*n^7 + 142*a^3*n^6 + 1654*a^3*n^5 + 10180*a^3*n^4 + 35485*a^3
*n^3 + 69358*a^3*n^2 + 69416*a^3*n + 26880*a^3)*cos(d*x + c)^6 + 6080*a^3*n^3 + 23520*a^3*n^2 + 2*(2*a^3*n^7 +
49*a^3*n^6 + 470*a^3*n^5 + 2230*a^3*n^4 + 5438*a^3*n^3 + 6361*a^3*n^2 + 2730*a^3*n)*cos(d*x + c)^4 + 39968*a^
3*n + 21840*a^3 + 8*(2*a^3*n^6 + 45*a^3*n^5 + 380*a^3*n^4 + 1470*a^3*n^3 + 2498*a^3*n^2 + 1365*a^3*n)*cos(d*x
+ c)^2 + (32*a^3*n^5 + 720*a^3*n^4 - 3*(a^3*n^7 + 29*a^3*n^6 + 343*a^3*n^5 + 2135*a^3*n^4 + 7504*a^3*n^3 + 147
56*a^3*n^2 + 14832*a^3*n + 5760*a^3)*cos(d*x + c)^6 + 6080*a^3*n^3 + 24000*a^3*n^2 + 2*(2*a^3*n^7 + 53*a^3*n^6
+ 566*a^3*n^5 + 3155*a^3*n^4 + 9908*a^3*n^3 + 17492*a^3*n^2 + 15984*a^3*n + 5760*a^3)*cos(d*x + c)^4 + 44288*
a^3*n + 30720*a^3 + 8*(2*a^3*n^6 + 47*a^3*n^5 + 425*a^3*n^4 + 1880*a^3*n^3 + 4268*a^3*n^2 + 4688*a^3*n + 1920*
a^3)*cos(d*x + c)^2)*sin(d*x + c))*sin(d*x + c)^n/(d*n^8 + 36*d*n^7 + 546*d*n^6 + 4536*d*n^5 + 22449*d*n^4 + 6
7284*d*n^3 + 118124*d*n^2 + 109584*d*n + 40320*d)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 23312 vs. \(2 (155) = 310\).
Time = 29.06 (sec) , antiderivative size = 23312, normalized size of antiderivative = 128.80
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)**5*sin(d*x+c)**n*(a+a*sin(d*x+c))**3,x)
[Out]
Piecewise((x*(a*sin(c) + a)**3*sin(c)**n*cos(c)**5, Eq(d, 0)), (a**3*log(sin(c + d*x))/d - 8*a**3/(5*d*sin(c +
d*x)) + a**3*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) - a**3/(2*d*sin(c + d*x)**2) + 4*a**3*cos(c + d*x)**2/(5*d
*sin(c + d*x)**3) - 8*a**3/(105*d*sin(c + d*x)**3) - a**3*cos(c + d*x)**4/(4*d*sin(c + d*x)**4) + a**3*cos(c +
d*x)**2/(2*d*sin(c + d*x)**4) - 3*a**3*cos(c + d*x)**4/(5*d*sin(c + d*x)**5) + 4*a**3*cos(c + d*x)**2/(35*d*s
in(c + d*x)**5) - a**3*cos(c + d*x)**4/(2*d*sin(c + d*x)**6) - a**3*cos(c + d*x)**4/(7*d*sin(c + d*x)**7), Eq(
n, -8)), (3*a**3*log(sin(c + d*x))/d + 8*a**3*sin(c + d*x)/(3*d) + 4*a**3*cos(c + d*x)**2/(3*d*sin(c + d*x)) -
8*a**3/(5*d*sin(c + d*x)) + 3*a**3*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) - a**3/(6*d*sin(c + d*x)**2) - a**3*
cos(c + d*x)**4/(3*d*sin(c + d*x)**3) + 4*a**3*cos(c + d*x)**2/(5*d*sin(c + d*x)**3) - 3*a**3*cos(c + d*x)**4/
(4*d*sin(c + d*x)**4) + a**3*cos(c + d*x)**2/(6*d*sin(c + d*x)**4) - 3*a**3*cos(c + d*x)**4/(5*d*sin(c + d*x)*
*5) - a**3*cos(c + d*x)**4/(6*d*sin(c + d*x)**6), Eq(n, -7)), (-960*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2
+ d*x/2)**9/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) - 1920*a**3*l
og(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**7/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*
d*tan(c/2 + d*x/2)**5) - 960*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**5/(960*d*tan(c/2 + d*x/2)**9
+ 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) + 960*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**9
/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) + 1920*a**3*log(tan(c/2
+ d*x/2))*tan(c/2 + d*x/2)**7/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)
**5) + 960*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**5/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)
**7 + 960*d*tan(c/2 + d*x/2)**5) - 6*a**3*tan(c/2 + d*x/2)**14/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d
*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) - 45*a**3*tan(c/2 + d*x/2)**13/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c
/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) - 82*a**3*tan(c/2 + d*x/2)**12/(960*d*tan(c/2 + d*x/2)**9 + 1920*d
*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) + 330*a**3*tan(c/2 + d*x/2)**11/(960*d*tan(c/2 + d*x/2)**9 +
1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) + 2074*a**3*tan(c/2 + d*x/2)**10/(960*d*tan(c/2 + d*x
/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) - 330*a**3*tan(c/2 + d*x/2)**9/(960*d*tan(c/2
+ d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) + 12350*a**3*tan(c/2 + d*x/2)**8/(960*d
*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) + 510*a**3*tan(c/2 + d*x/2)**7/
(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) + 12350*a**3*tan(c/2 + d*
x/2)**6/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) - 330*a**3*tan(c/
2 + d*x/2)**5/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) + 2074*a**3
*tan(c/2 + d*x/2)**4/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5) + 33
0*a**3*tan(c/2 + d*x/2)**3/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5
) - 82*a**3*tan(c/2 + d*x/2)**2/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/
2)**5) - 45*a**3*tan(c/2 + d*x/2)/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*
x/2)**5) - 6*a**3/(960*d*tan(c/2 + d*x/2)**9 + 1920*d*tan(c/2 + d*x/2)**7 + 960*d*tan(c/2 + d*x/2)**5), Eq(n,
-6)), (960*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**10/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2
+ d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 2880*a**3*log(tan(c/2 + d*x/2)**2 + 1)*
tan(c/2 + d*x/2)**8/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*
d*tan(c/2 + d*x/2)**4) + 2880*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**6/(192*d*tan(c/2 + d*x/2)**1
0 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 960*a**3*log(tan(c/2
+ d*x/2)**2 + 1)*tan(c/2 + d*x/2)**4/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 +
d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) - 960*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**10/(192*d*tan(c/2 +
d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) - 2880*a**3*l
og(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**8/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c
/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) - 2880*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**6/(192*d*tan(c
/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) - 960*a**
3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**4/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*ta
n(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) - 3*a**3*tan(c/2 + d*x/2)**14/(192*d*tan(c/2 + d*x/2)**10 + 576
*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) - 24*a**3*tan(c/2 + d*x/2)**13
/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)*
*4) - 45*a**3*tan(c/2 + d*x/2)**12/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d
*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 336*a**3*tan(c/2 + d*x/2)**11/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(
c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 1944*a**3*tan(c/2 + d*x/2)**9/(192*
d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) +
1356*a**3*tan(c/2 + d*x/2)**8/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)
**6 + 192*d*tan(c/2 + d*x/2)**4) + 3680*a**3*tan(c/2 + d*x/2)**7/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 +
d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 1356*a**3*tan(c/2 + d*x/2)**6/(192*d*tan
(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 1944*
a**3*tan(c/2 + d*x/2)**5/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 +
192*d*tan(c/2 + d*x/2)**4) + 336*a**3*tan(c/2 + d*x/2)**3/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2
)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) - 45*a**3*tan(c/2 + d*x/2)**2/(192*d*tan(c/2 + d
*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) - 24*a**3*tan(c
/2 + d*x/2)/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/
2 + d*x/2)**4) - 3*a**3/(192*d*tan(c/2 + d*x/2)**10 + 576*d*tan(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 +
192*d*tan(c/2 + d*x/2)**4), Eq(n, -5)), (120*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**11/(24*d*tan(
c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(
c/2 + d*x/2)**3) + 480*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**9/(24*d*tan(c/2 + d*x/2)**11 + 96*d
*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 720*
a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**7/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 +
144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 480*a**3*log(tan(c/2 + d*x/
2)**2 + 1)*tan(c/2 + d*x/2)**5/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)*
*7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 120*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*
x/2)**3/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x
/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 120*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**11/(24*d*tan(c/2 + d*x/2
)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2
)**3) - 480*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**9/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)*
*9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 720*a**3*log(tan(c/2 +
d*x/2))*tan(c/2 + d*x/2)**7/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7
+ 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 480*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**5/(
24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 +
24*d*tan(c/2 + d*x/2)**3) - 120*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**3/(24*d*tan(c/2 + d*x/2)**11 + 96
*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - a*
*3*tan(c/2 + d*x/2)**14/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96
*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 9*a**3*tan(c/2 + d*x/2)**13/(24*d*tan(c/2 + d*x/2)**11 +
96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) -
19*a**3*tan(c/2 + d*x/2)**12/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7
+ 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 321*a**3*tan(c/2 + d*x/2)**10/(24*d*tan(c/2 + d*x/2)
**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)
**3) + 129*a**3*tan(c/2 + d*x/2)**9/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*
x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 683*a**3*tan(c/2 + d*x/2)**8/(24*d*tan(c/2 +
d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 +
d*x/2)**3) + 336*a**3*tan(c/2 + d*x/2)**7/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/
2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 683*a**3*tan(c/2 + d*x/2)**6/(24*d*tan(
c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(
c/2 + d*x/2)**3) + 129*a**3*tan(c/2 + d*x/2)**5/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*
tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 321*a**3*tan(c/2 + d*x/2)**4/(24*
d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*
d*tan(c/2 + d*x/2)**3) - 19*a**3*tan(c/2 + d*x/2)**2/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 1
44*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 9*a**3*tan(c/2 + d*x/2)/(24*
d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*tan(c/2 + d*x/2)**5 + 24*
d*tan(c/2 + d*x/2)**3) - a**3/(24*d*tan(c/2 + d*x/2)**11 + 96*d*tan(c/2 + d*x/2)**9 + 144*d*tan(c/2 + d*x/2)**
7 + 96*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3), Eq(n, -4)), (-120*a**3*log(tan(c/2 + d*x/2)**2 + 1)*
tan(c/2 + d*x/2)**12/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1
200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 600*a**3*log(tan(c/2 + d*
x/2)**2 + 1)*tan(c/2 + d*x/2)**10/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 +
d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 1200*a**3*lo
g(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**8/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200
*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) -
1200*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**6/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/
2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 +
d*x/2)**2) - 600*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**4/(120*d*tan(c/2 + d*x/2)**12 + 600*d*ta
n(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120
*d*tan(c/2 + d*x/2)**2) - 120*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**2/(120*d*tan(c/2 + d*x/2)**1
2 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x
/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 120*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**12/(120*d*tan(c/2 + d*x
/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2
+ d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 600*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**10/(120*d*tan(c/2
+ d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*t
an(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 1200*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**8/(120*d*t
an(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 6
00*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 1200*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**6/(1
20*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)*
*6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 600*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)*
*4/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*
x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) + 120*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*
x/2)**2/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2
+ d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 15*a**3*tan(c/2 + d*x/2)**14/(120*d*ta
n(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 60
0*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 180*a**3*tan(c/2 + d*x/2)**13/(120*d*tan(c/2 + d*x/2)**
12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*
x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 2280*a**3*tan(c/2 + d*x/2)**11/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(
c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d
*tan(c/2 + d*x/2)**2) - 990*a**3*tan(c/2 + d*x/2)**10/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10
+ 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2
)**2) - 7180*a**3*tan(c/2 + d*x/2)**9/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/
2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 1635*a**
3*tan(c/2 + d*x/2)**8/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 +
1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 9392*a**3*tan(c/2 + d*x/
2)**7/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 +
d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 1635*a**3*tan(c/2 + d*x/2)**6/(120*d*tan
(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600
*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 7180*a**3*tan(c/2 + d*x/2)**5/(120*d*tan(c/2 + d*x/2)**1
2 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x
/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 990*a**3*tan(c/2 + d*x/2)**4/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2
+ d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*ta
n(c/2 + d*x/2)**2) - 2280*a**3*tan(c/2 + d*x/2)**3/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 +
1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**
2) - 180*a**3*tan(c/2 + d*x/2)/(120*d*tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x
/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 + 600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2) - 15*a**3/(120*d*
tan(c/2 + d*x/2)**12 + 600*d*tan(c/2 + d*x/2)**10 + 1200*d*tan(c/2 + d*x/2)**8 + 1200*d*tan(c/2 + d*x/2)**6 +
600*d*tan(c/2 + d*x/2)**4 + 120*d*tan(c/2 + d*x/2)**2), Eq(n, -3)), (-90*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan
(c/2 + d*x/2)**13/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*
tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 540*a**
3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**11/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 +
450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3
+ 30*d*tan(c/2 + d*x/2)) - 1350*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**9/(30*d*tan(c/2 + d*x/2)**
13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/
2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 1800*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 +
d*x/2)**7/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2
+ d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 1350*a**3*log(t
an(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**5/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*ta
n(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*t
an(c/2 + d*x/2)) - 540*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**3/(30*d*tan(c/2 + d*x/2)**13 + 180*
d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 1
80*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 90*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)/(30*
d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 +
450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) + 90*a**3*log(tan(c/2 + d*x/2))
*tan(c/2 + d*x/2)**13/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 60
0*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) + 540
*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**11/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450
*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 3
0*d*tan(c/2 + d*x/2)) + 1350*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**9/(30*d*tan(c/2 + d*x/2)**13 + 180*d
*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 18
0*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) + 1800*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**7/(30*d*t
an(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450
*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) + 1350*a**3*log(tan(c/2 + d*x/2))*
tan(c/2 + d*x/2)**5/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*
d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) + 540*a
**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**3/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*
tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d
*tan(c/2 + d*x/2)) + 90*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2
+ d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(
c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 15*a**3*tan(c/2 + d*x/2)**14/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan
(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*
tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 45*a**3*tan(c/2 + d*x/2)**12/(30*d*tan(c/2 + d*x/2)**13 + 180*d
*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 18
0*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 300*a**3*tan(c/2 + d*x/2)**11/(30*d*tan(c/2 + d*x/2)**13 +
180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5
+ 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 415*a**3*tan(c/2 + d*x/2)**10/(30*d*tan(c/2 + d*x/2)**
13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 + d*x/
2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 1080*a**3*tan(c/2 + d*x/2)**9/(30*d*tan(c/2 + d*x
/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c/2 +
d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 549*a**3*tan(c/2 + d*x/2)**8/(30*d*tan(c/2 +
d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*tan(c
/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 1240*a**3*tan(c/2 + d*x/2)**7/(30*d*tan(
c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450*d*
tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 549*a**3*tan(c/2 + d*x/2)**6/(30*d*
tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 45
0*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 1080*a**3*tan(c/2 + d*x/2)**5/(
30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7
+ 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 415*a**3*tan(c/2 + d*x/2)*
*4/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2
)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 300*a**3*tan(c/2 + d*x
/2)**3/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d
*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 45*a**3*tan(c/2 +
d*x/2)**2/(30*d*tan(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2
+ d*x/2)**7 + 450*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 15*a**3/(30*d*t
an(c/2 + d*x/2)**13 + 180*d*tan(c/2 + d*x/2)**11 + 450*d*tan(c/2 + d*x/2)**9 + 600*d*tan(c/2 + d*x/2)**7 + 450
*d*tan(c/2 + d*x/2)**5 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)), Eq(n, -2)), (-105*a**3*log(tan(c/
2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**14/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(
c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*
d*tan(c/2 + d*x/2)**2 + 105*d) - 735*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**12/(105*d*tan(c/2 + d
*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan
(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) - 2205*a**3*log(tan(c/2 + d
*x/2)**2 + 1)*tan(c/2 + d*x/2)**10/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 +
d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan
(c/2 + d*x/2)**2 + 105*d) - 3675*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**8/(105*d*tan(c/2 + d*x/2)
**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2
+ d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) - 3675*a**3*log(tan(c/2 + d*x/2)
**2 + 1)*tan(c/2 + d*x/2)**6/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2
)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 +
d*x/2)**2 + 105*d) - 2205*a**3*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**4/(105*d*tan(c/2 + d*x/2)**14 +
735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/
2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) - 735*a**3*log(tan(c/2 + d*x/2)**2 + 1
)*tan(c/2 + d*x/2)**2/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 +
3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)
**2 + 105*d) - 105*a**3*log(tan(c/2 + d*x/2)**2 + 1)/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12
+ 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x
/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 105*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**14/(105*d*tan(c
/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675
*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 735*a**3*log(tan(c/
2 + d*x/2))*tan(c/2 + d*x/2)**12/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d
*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c
/2 + d*x/2)**2 + 105*d) + 2205*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**10/(105*d*tan(c/2 + d*x/2)**14 + 7
35*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)
**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 3675*a**3*log(tan(c/2 + d*x/2))*tan(c/
2 + d*x/2)**8/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*
tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 10
5*d) + 3675*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**6/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2
)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2
+ d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 2205*a**3*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**4/(105*d
*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8
+ 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 735*a**3*log(
tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/
2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*
tan(c/2 + d*x/2)**2 + 105*d) + 105*a**3*log(tan(c/2 + d*x/2))/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*
x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(
c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 630*a**3*tan(c/2 + d*x/2)**13/(105*d*tan(c/2 + d*x/2)**
14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 +
d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 210*a**3*tan(c/2 + d*x/2)**12/(1
05*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)
**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 2380*a**3
*tan(c/2 + d*x/2)**11/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 +
3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)
**2 + 105*d) - 1050*a**3*tan(c/2 + d*x/2)**10/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*
d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4
+ 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 4522*a**3*tan(c/2 + d*x/2)**9/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c
/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205
*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) - 840*a**3*tan(c/2 + d*x/2)**8/(105*d*tan(c/2 + d*
x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(
c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 7464*a**3*tan(c/2 + d*x/2)
**7/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 +
d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) - 840
*a**3*tan(c/2 + d*x/2)**6/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**
10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*
x/2)**2 + 105*d) + 4522*a**3*tan(c/2 + d*x/2)**5/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 22
05*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)*
*4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) - 1050*a**3*tan(c/2 + d*x/2)**4/(105*d*tan(c/2 + d*x/2)**14 + 735*d*ta
n(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2
205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 2380*a**3*tan(c/2 + d*x/2)**3/(105*d*tan(c/2
+ d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*
tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 210*a**3*tan(c/2 + d*x
/2)**2/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**10 + 3675*d*tan(c/2
+ d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) +
630*a**3*tan(c/2 + d*x/2)/(105*d*tan(c/2 + d*x/2)**14 + 735*d*tan(c/2 + d*x/2)**12 + 2205*d*tan(c/2 + d*x/2)**
10 + 3675*d*tan(c/2 + d*x/2)**8 + 3675*d*tan(c/2 + d*x/2)**6 + 2205*d*tan(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*
x/2)**2 + 105*d), Eq(n, -1)), (a**3*n**7*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 +
546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 3*a**3*n**7*
sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4
+ 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 3*a**3*n**7*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d
*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d
*n + 40320*d) + a**3*n**7*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536
*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 4*a**3*n**6*sin(c + d*x)**6*si
n(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 1
18124*d*n**2 + 109584*d*n + 40320*d) + 12*a**3*n**6*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 +
36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 3
2*a**3*n**6*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 2
2449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 12*a**3*n**6*sin(c + d*x)**4*sin(c + d*x)
**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n*
*2 + 109584*d*n + 40320*d) + 99*a**3*n**6*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7
+ 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 4*a**3*n**6
*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4
+ 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 102*a**3*n**6*sin(c + d*x)**2*sin(c + d*x)**n*cos(c
+ d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 10958
4*d*n + 40320*d) + 35*a**3*n**6*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6
+ 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 8*a**3*n**5*sin(c + d*x)
**8*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n*
*2 + 109584*d*n + 40320*d) + 24*a**3*n**5*sin(c + d*x)**7*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4
536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 104*a**3*n**5*sin(c + d*x)*
*6*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**
3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 24*a**3*n**5*sin(c + d*x)**6*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 +
546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 336*a**3*n**
5*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**
4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 8*a**3*n**5*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**8
+ 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d)
+ 418*a**3*n**5*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5
+ 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 360*a**3*n**5*sin(c + d*x)**4*sin(c +
d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124
*d*n**2 + 109584*d*n + 40320*d) + 1341*a**3*n**5*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*
d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 128*
a**3*n**5*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 224
49*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 1434*a**3*n**5*sin(c + d*x)**2*sin(c + d*x)
**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n*
*2 + 109584*d*n + 40320*d) + 511*a**3*n**5*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 +
546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 144*a**3*n**4
*sin(c + d*x)**8*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3
+ 118124*d*n**2 + 109584*d*n + 40320*d) + 504*a**3*n**4*sin(c + d*x)**7*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 +
546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 1048*a**3*n**
4*sin(c + d*x)**6*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**
4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 576*a**3*n**4*sin(c + d*x)**6*sin(c + d*x)**n/(d*n*
*8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d
) + 3684*a**3*n**4*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n
**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 216*a**3*n**4*sin(c + d*x)**5*sin(
c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109
584*d*n + 40320*d) + 2864*a**3*n**4*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*
d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 4296*a**3*n**4*si
n(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 +
67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 9585*a**3*n**4*sin(c + d*x)**3*sin(c + d*x)**n*cos(c +
d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*
d*n + 40320*d) + 1660*a**3*n**4*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n*
*6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 10740*a**3*n**4*sin(c
+ d*x)**2*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 672
84*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 4025*a**3*n**4*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**
4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n +
40320*d) + 944*a**3*n**3*sin(c + d*x)**8*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 2244
9*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 3840*a**3*n**3*sin(c + d*x)**7*sin(c + d*x)*
*n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n +
40320*d) + 5168*a**3*n**3*sin(c + d*x)**6*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 +
4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 5136*a**3*n**3*sin(c + d*x
)**6*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n
**2 + 109584*d*n + 40320*d) + 19920*a**3*n**3*sin(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n
**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 2240*a*
*3*n**3*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*
d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 10993*a**3*n**3*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)*
*4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n +
40320*d) + 25776*a**3*n**3*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 +
4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 38592*a**3*n**3*sin(c + d
*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d
*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 11120*a**3*n**3*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**
2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n +
40320*d) + 45867*a**3*n**3*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 +
4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 18424*a**3*n**3*sin(c + d*
x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**
3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 2784*a**3*n**2*sin(c + d*x)**8*sin(c + d*x)**n/(d*n**8 + 36*d*n**7
+ 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 12960*a**3
*n**2*sin(c + d*x)**7*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*
n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 12964*a**3*n**2*sin(c + d*x)**6*sin(c + d*x)**n*cos(c + d*x)**2
/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 4
0320*d) + 20736*a**3*n**2*sin(c + d*x)**6*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 224
49*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 54768*a**3*n**2*sin(c + d*x)**5*sin(c + d*x
)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n
**2 + 109584*d*n + 40320*d) + 11040*a**3*n**2*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6
+ 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 23312*a**3*n**2*sin(c +
d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284
*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 80364*a**3*n**2*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)
**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n
+ 40320*d) + 86076*a**3*n**2*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6
+ 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 40336*a**3*n**2*sin(c +
d*x)**3*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*
d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 110118*a**3*n**2*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)
**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n
+ 40320*d) + 48860*a**3*n**2*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4
536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 3656*a**3*n*sin(c + d*x)**8
*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2
+ 109584*d*n + 40320*d) + 18816*a**3*n*sin(c + d*x)**7*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536
*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 15464*a**3*n*sin(c + d*x)**6*s
in(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 +
118124*d*n**2 + 109584*d*n + 40320*d) + 36312*a**3*n*sin(c + d*x)**6*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546
*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 70464*a**3*n*sin
(c + d*x)**5*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 6
7284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 25472*a**3*n*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**8 + 3
6*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 24
876*a**3*n*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22
449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 119016*a**3*n*sin(c + d*x)**4*sin(c + d*x)
**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n*
*2 + 109584*d*n + 40320*d) + 96144*a**3*n*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7
+ 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 74432*a**3*
n*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**
4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 134136*a**3*n*sin(c + d*x)**2*sin(c + d*x)**n*cos(c
+ d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 1095
84*d*n + 40320*d) + 69264*a**3*n*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6
+ 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 1680*a**3*sin(c + d*x)*
*8*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**
2 + 109584*d*n + 40320*d) + 9216*a**3*sin(c + d*x)**7*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*
d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 6720*a**3*sin(c + d*x)**6*sin(c
+ d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 1181
24*d*n**2 + 109584*d*n + 40320*d) + 20160*a**3*sin(c + d*x)**6*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 + 546*d*n**
6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 32256*a**3*sin(c + d*x
)**5*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n
**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 21504*a**3*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**8 + 36*d*n**7 +
546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 10080*a**3*s
in(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 +
67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 60480*a**3*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x
)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n
+ 40320*d) + 40320*a**3*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 45
36*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 53760*a**3*sin(c + d*x)**3*s
in(c + d*x)**n*cos(c + d*x)**2/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 +
118124*d*n**2 + 109584*d*n + 40320*d) + 60480*a**3*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 3
6*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 40
320*a**3*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**8 + 36*d*n**7 + 546*d*n**6 + 4536*d*n**5 + 22449*d
*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d), True))
Maxima [A] (verification not implemented)
none
Time = 0.22 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.89
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\frac {a^{3} \sin \left (d x + c\right )^{n + 8}}{n + 8} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 7}}{n + 7} + \frac {a^{3} \sin \left (d x + c\right )^{n + 6}}{n + 6} - \frac {5 \, a^{3} \sin \left (d x + c\right )^{n + 5}}{n + 5} - \frac {5 \, a^{3} \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {a^{3} \sin \left (d x + c\right )^{n + 3}}{n + 3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a^{3} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d}
\]
[In]
integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="maxima")
[Out]
(a^3*sin(d*x + c)^(n + 8)/(n + 8) + 3*a^3*sin(d*x + c)^(n + 7)/(n + 7) + a^3*sin(d*x + c)^(n + 6)/(n + 6) - 5*
a^3*sin(d*x + c)^(n + 5)/(n + 5) - 5*a^3*sin(d*x + c)^(n + 4)/(n + 4) + a^3*sin(d*x + c)^(n + 3)/(n + 3) + 3*a
^3*sin(d*x + c)^(n + 2)/(n + 2) + a^3*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. 769 vs. \(2 (181) = 362\).
Time = 0.64 (sec) , antiderivative size = 769, normalized size of antiderivative = 4.25
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\frac {{\left (n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{8} + 10 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{8} - 2 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 24 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{8} - 24 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} - 64 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 14 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 48 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4}\right )} a^{3}}{n^{3} + 18 \, n^{2} + 104 \, n + 192} + \frac {3 \, {\left (n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7} + 8 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7} - 2 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 15 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7} - 20 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} - 42 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 12 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 35 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}\right )} a^{3}}{n^{3} + 15 \, n^{2} + 71 \, n + 105} + \frac {3 \, {\left (n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 6 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} - 2 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 8 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} - 16 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} - 24 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 10 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 24 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}\right )} a^{3}}{n^{3} + 12 \, n^{2} + 44 \, n + 48} + \frac {{\left (n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 4 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} - 2 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 3 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} - 12 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) - 10 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 8 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 15 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )\right )} a^{3}}{n^{3} + 9 \, n^{2} + 23 \, n + 15}}{d}
\]
[In]
integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="giac")
[Out]
((n^2*sin(d*x + c)^n*sin(d*x + c)^8 + 10*n*sin(d*x + c)^n*sin(d*x + c)^8 - 2*n^2*sin(d*x + c)^n*sin(d*x + c)^6
+ 24*sin(d*x + c)^n*sin(d*x + c)^8 - 24*n*sin(d*x + c)^n*sin(d*x + c)^6 + n^2*sin(d*x + c)^n*sin(d*x + c)^4 -
64*sin(d*x + c)^n*sin(d*x + c)^6 + 14*n*sin(d*x + c)^n*sin(d*x + c)^4 + 48*sin(d*x + c)^n*sin(d*x + c)^4)*a^3
/(n^3 + 18*n^2 + 104*n + 192) + 3*(n^2*sin(d*x + c)^n*sin(d*x + c)^7 + 8*n*sin(d*x + c)^n*sin(d*x + c)^7 - 2*n
^2*sin(d*x + c)^n*sin(d*x + c)^5 + 15*sin(d*x + c)^n*sin(d*x + c)^7 - 20*n*sin(d*x + c)^n*sin(d*x + c)^5 + n^2
*sin(d*x + c)^n*sin(d*x + c)^3 - 42*sin(d*x + c)^n*sin(d*x + c)^5 + 12*n*sin(d*x + c)^n*sin(d*x + c)^3 + 35*si
n(d*x + c)^n*sin(d*x + c)^3)*a^3/(n^3 + 15*n^2 + 71*n + 105) + 3*(n^2*sin(d*x + c)^n*sin(d*x + c)^6 + 6*n*sin(
d*x + c)^n*sin(d*x + c)^6 - 2*n^2*sin(d*x + c)^n*sin(d*x + c)^4 + 8*sin(d*x + c)^n*sin(d*x + c)^6 - 16*n*sin(d
*x + c)^n*sin(d*x + c)^4 + n^2*sin(d*x + c)^n*sin(d*x + c)^2 - 24*sin(d*x + c)^n*sin(d*x + c)^4 + 10*n*sin(d*x
+ c)^n*sin(d*x + c)^2 + 24*sin(d*x + c)^n*sin(d*x + c)^2)*a^3/(n^3 + 12*n^2 + 44*n + 48) + (n^2*sin(d*x + c)^
n*sin(d*x + c)^5 + 4*n*sin(d*x + c)^n*sin(d*x + c)^5 - 2*n^2*sin(d*x + c)^n*sin(d*x + c)^3 + 3*sin(d*x + c)^n*
sin(d*x + c)^5 - 12*n*sin(d*x + c)^n*sin(d*x + c)^3 + n^2*sin(d*x + c)^n*sin(d*x + c) - 10*sin(d*x + c)^n*sin(
d*x + c)^3 + 8*n*sin(d*x + c)^n*sin(d*x + c) + 15*sin(d*x + c)^n*sin(d*x + c))*a^3/(n^3 + 9*n^2 + 23*n + 15))/
d
Mupad [B] (verification not implemented)
Time = 18.90 (sec) , antiderivative size = 923, normalized size of antiderivative = 5.10
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\sin \left (c+d\,x\right )}^n\,\left (27\,n^7+1028\,n^6+17366\,n^5+162200\,n^4+870443\,n^3+2585492\,n^2+3757604\,n+1896720\right )}{128\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {a^3\,{\sin \left (c+d\,x\right )}^n\,\cos \left (8\,c+8\,d\,x\right )\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}{128\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a^3\,\sin \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^n\,\left (n^7\,17{}\mathrm {i}+n^6\,669{}\mathrm {i}+n^5\,11975{}\mathrm {i}+n^4\,118935{}\mathrm {i}+n^3\,675728{}\mathrm {i}+n^2\,2140836{}\mathrm {i}+n\,3467760{}\mathrm {i}+2217600{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a^3\,{\sin \left (c+d\,x\right )}^n\,\cos \left (6\,c+6\,d\,x\right )\,\left (3\,n^7+86\,n^6+1010\,n^5+6260\,n^4+21947\,n^3+43094\,n^2+43280\,n+16800\right )}{32\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a^3\,{\sin \left (c+d\,x\right )}^n\,\cos \left (4\,c+4\,d\,x\right )\,\left (7\,n^7+264\,n^6+3910\,n^5+29520\,n^4+122023\,n^3+273336\,n^2+303180\,n+126000\right )}{32\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a^3\,{\sin \left (c+d\,x\right )}^n\,\cos \left (2\,c+2\,d\,x\right )\,\left (-3\,n^7-86\,n^6-498\,n^5+5260\,n^4+75333\,n^3+333226\,n^2+596208\,n+332640\right )}{32\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {a^3\,{\sin \left (c+d\,x\right )}^n\,\sin \left (7\,c+7\,d\,x\right )\,\left (n^7\,1{}\mathrm {i}+n^6\,29{}\mathrm {i}+n^5\,343{}\mathrm {i}+n^4\,2135{}\mathrm {i}+n^3\,7504{}\mathrm {i}+n^2\,14756{}\mathrm {i}+n\,14832{}\mathrm {i}+5760{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {a^3\,{\sin \left (c+d\,x\right )}^n\,\sin \left (5\,c+5\,d\,x\right )\,\left (-n^7\,1{}\mathrm {i}+n^6\,11{}\mathrm {i}+n^5\,617{}\mathrm {i}+n^4\,6785{}\mathrm {i}+n^3\,33296{}\mathrm {i}+n^2\,81404{}\mathrm {i}+n\,94608{}\mathrm {i}+40320{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a^3\,{\sin \left (c+d\,x\right )}^n\,\sin \left (3\,c+3\,d\,x\right )\,\left (n^7\,21{}\mathrm {i}+n^6\,745{}\mathrm {i}+n^5\,10339{}\mathrm {i}+n^4\,72475{}\mathrm {i}+n^3\,275824{}\mathrm {i}+n^2\,567700{}\mathrm {i}+n\,583216{}\mathrm {i}+228480{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}
\]
[In]
int(cos(c + d*x)^5*sin(c + d*x)^n*(a + a*sin(c + d*x))^3,x)
[Out]
(a^3*sin(c + d*x)^n*(3757604*n + 2585492*n^2 + 870443*n^3 + 162200*n^4 + 17366*n^5 + 1028*n^6 + 27*n^7 + 18967
20))/(128*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (a^
3*sin(c + d*x)^n*cos(8*c + 8*d*x)*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040))
/(128*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (a^3*si
n(c + d*x)*sin(c + d*x)^n*(n*3467760i + n^2*2140836i + n^3*675728i + n^4*118935i + n^5*11975i + n^6*669i + n^7
*17i + 2217600i)*1i)/(64*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8
+ 40320)) - (a^3*sin(c + d*x)^n*cos(6*c + 6*d*x)*(43280*n + 43094*n^2 + 21947*n^3 + 6260*n^4 + 1010*n^5 + 86*n
^6 + 3*n^7 + 16800))/(32*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8
+ 40320)) - (a^3*sin(c + d*x)^n*cos(4*c + 4*d*x)*(303180*n + 273336*n^2 + 122023*n^3 + 29520*n^4 + 3910*n^5 +
264*n^6 + 7*n^7 + 126000))/(32*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7
+ n^8 + 40320)) - (a^3*sin(c + d*x)^n*cos(2*c + 2*d*x)*(596208*n + 333226*n^2 + 75333*n^3 + 5260*n^4 - 498*n^5
- 86*n^6 - 3*n^7 + 332640))/(32*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^
7 + n^8 + 40320)) + (a^3*sin(c + d*x)^n*sin(7*c + 7*d*x)*(n*14832i + n^2*14756i + n^3*7504i + n^4*2135i + n^5*
343i + n^6*29i + n^7*1i + 5760i)*3i)/(64*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6
+ 36*n^7 + n^8 + 40320)) + (a^3*sin(c + d*x)^n*sin(5*c + 5*d*x)*(n*94608i + n^2*81404i + n^3*33296i + n^4*678
5i + n^5*617i + n^6*11i - n^7*1i + 40320i)*1i)/(64*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5
+ 546*n^6 + 36*n^7 + n^8 + 40320)) - (a^3*sin(c + d*x)^n*sin(3*c + 3*d*x)*(n*583216i + n^2*567700i + n^3*2758
24i + n^4*72475i + n^5*10339i + n^6*745i + n^7*21i + 228480i)*1i)/(64*d*(109584*n + 118124*n^2 + 67284*n^3 + 2
2449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320))