\(\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx\) [699]
Optimal result
Integrand size = 27, antiderivative size = 167 \[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {a \sin ^{2+n}(c+d x)}{d (2+n)}-\frac {3 a \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {3 a \sin ^{4+n}(c+d x)}{d (4+n)}+\frac {3 a \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {3 a \sin ^{6+n}(c+d x)}{d (6+n)}-\frac {a \sin ^{7+n}(c+d x)}{d (7+n)}-\frac {a \sin ^{8+n}(c+d x)}{d (8+n)}
\]
[Out]
a*sin(d*x+c)^(1+n)/d/(1+n)+a*sin(d*x+c)^(2+n)/d/(2+n)-3*a*sin(d*x+c)^(3+n)/d/(3+n)-3*a*sin(d*x+c)^(4+n)/d/(4+n
)+3*a*sin(d*x+c)^(5+n)/d/(5+n)+3*a*sin(d*x+c)^(6+n)/d/(6+n)-a*sin(d*x+c)^(7+n)/d/(7+n)-a*sin(d*x+c)^(8+n)/d/(8
+n)
Rubi [A] (verified)
Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2915, 90}
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac {3 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {3 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac {3 a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {3 a \sin ^{n+6}(c+d x)}{d (n+6)}-\frac {a \sin ^{n+7}(c+d x)}{d (n+7)}-\frac {a \sin ^{n+8}(c+d x)}{d (n+8)}
\]
[In]
Int[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]
[Out]
(a*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (a*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (3*a*Sin[c + d*x]^(3 + n))/(d*(3
+ n)) - (3*a*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (3*a*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (3*a*Sin[c + d*x]^(
6 + n))/(d*(6 + n)) - (a*Sin[c + d*x]^(7 + n))/(d*(7 + n)) - (a*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)^3 \left (\frac {x}{a}\right )^n (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \left (a^7 \left (\frac {x}{a}\right )^n+a^7 \left (\frac {x}{a}\right )^{1+n}-3 a^7 \left (\frac {x}{a}\right )^{2+n}-3 a^7 \left (\frac {x}{a}\right )^{3+n}+3 a^7 \left (\frac {x}{a}\right )^{4+n}+3 a^7 \left (\frac {x}{a}\right )^{5+n}-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^7 d} \\ & = \frac {a \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {a \sin ^{2+n}(c+d x)}{d (2+n)}-\frac {3 a \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {3 a \sin ^{4+n}(c+d x)}{d (4+n)}+\frac {3 a \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {3 a \sin ^{6+n}(c+d x)}{d (6+n)}-\frac {a \sin ^{7+n}(c+d x)}{d (7+n)}-\frac {a \sin ^{8+n}(c+d x)}{d (8+n)} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.29 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.92
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^{1+n}(c+d x) (1+\sin (c+d x)) \left (\frac {1}{1+n}+\frac {\sin (c+d x)}{2+n}-\frac {3 \sin ^2(c+d x)}{3+n}-\frac {3 \sin ^3(c+d x)}{4+n}+\frac {3 \sin ^4(c+d x)}{5+n}+\frac {3 \sin ^5(c+d x)}{6+n}-\frac {\sin ^6(c+d x)}{7+n}-\frac {\sin ^7(c+d x)}{8+n}\right )}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}
\]
[In]
Integrate[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]
[Out]
(a*Sin[c + d*x]^(1 + n)*(1 + Sin[c + d*x])*((1 + n)^(-1) + Sin[c + d*x]/(2 + n) - (3*Sin[c + d*x]^2)/(3 + n) -
(3*Sin[c + d*x]^3)/(4 + n) + (3*Sin[c + d*x]^4)/(5 + n) + (3*Sin[c + d*x]^5)/(6 + n) - Sin[c + d*x]^6/(7 + n)
- Sin[c + d*x]^7/(8 + n)))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)
Maple [A] (verified)
Time = 9.14 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.38
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {a \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}-\frac {3 a \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}-\frac {3 a \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}+\frac {3 a \left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {3 a \left (\sin ^{6}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (6+n \right )}-\frac {a \left (\sin ^{7}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (7+n \right )}-\frac {a \left (\sin ^{8}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (8+n \right )}\) |
\(230\) |
| default |
\(\frac {a \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}-\frac {3 a \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}-\frac {3 a \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}+\frac {3 a \left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {3 a \left (\sin ^{6}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (6+n \right )}-\frac {a \left (\sin ^{7}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (7+n \right )}-\frac {a \left (\sin ^{8}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (8+n \right )}\) |
\(230\) |
| parallelrisch |
\(\frac {5 \left (\sin ^{n}\left (d x +c \right )\right ) \left (\frac {2 \left (n -4\right ) \left (7+n \right ) \left (5+n \right ) \left (3+n \right ) \left (1+n \right ) \left (n^{2}+22 n +168\right ) \cos \left (2 d x +2 c \right )}{5}-\frac {2 \left (7+n \right ) \left (5+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \left (n^{2}+22 n +168\right ) \cos \left (4 d x +4 c \right )}{5}-\frac {2 \left (n +12\right ) \left (7+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (6 d x +6 c \right )}{5}-\frac {\left (7+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (8 d x +8 c \right )}{10}+\frac {9 \left (n^{2}+16 n +\frac {245}{3}\right ) \left (4+n \right ) \left (8+n \right ) \left (6+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (3 d x +3 c \right )}{5}+\left (4+n \right ) \left (n +\frac {49}{5}\right ) \left (3+n \right ) \left (8+n \right ) \left (6+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (5 d x +5 c \right )+\frac {\left (8+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (7 d x +7 c \right )}{5}+\left (4+n \right ) \left (n^{3}+\frac {93}{5} n^{2}+\frac {691}{5} n +735\right ) \left (8+n \right ) \left (6+n \right ) \left (2+n \right ) \sin \left (d x +c \right )+\frac {\left (3+n \right ) \left (7+n \right ) \left (5+n \right ) \left (1+n \right ) \left (n^{3}+\frac {108}{5} n^{2}+\frac {892}{5} n +\frac {4464}{5}\right )}{2}\right ) a}{64 \left (4+n \right ) \left (2+n \right ) \left (3+n \right ) \left (1+n \right ) \left (7+n \right ) \left (5+n \right ) \left (8+n \right ) d \left (6+n \right )}\) |
\(344\) |
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[In]
int(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
a/d/(1+n)*sin(d*x+c)*exp(n*ln(sin(d*x+c)))+a/d/(2+n)*sin(d*x+c)^2*exp(n*ln(sin(d*x+c)))-3*a/d/(3+n)*sin(d*x+c)
^3*exp(n*ln(sin(d*x+c)))-3*a/d/(4+n)*sin(d*x+c)^4*exp(n*ln(sin(d*x+c)))+3*a/d/(5+n)*sin(d*x+c)^5*exp(n*ln(sin(
d*x+c)))+3*a/d/(6+n)*sin(d*x+c)^6*exp(n*ln(sin(d*x+c)))-a/d/(7+n)*sin(d*x+c)^7*exp(n*ln(sin(d*x+c)))-a/d/(8+n)
*sin(d*x+c)^8*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. 445 vs. \(2 (167) = 334\).
Time = 0.32 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.66
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {{\left ({\left (a n^{7} + 28 \, a n^{6} + 322 \, a n^{5} + 1960 \, a n^{4} + 6769 \, a n^{3} + 13132 \, a n^{2} + 13068 \, a n + 5040 \, a\right )} \cos \left (d x + c\right )^{8} - {\left (a n^{7} + 22 \, a n^{6} + 190 \, a n^{5} + 820 \, a n^{4} + 1849 \, a n^{3} + 2038 \, a n^{2} + 840 \, a n\right )} \cos \left (d x + c\right )^{6} - 48 \, a n^{4} - 6 \, {\left (a n^{6} + 18 \, a n^{5} + 118 \, a n^{4} + 348 \, a n^{3} + 457 \, a n^{2} + 210 \, a n\right )} \cos \left (d x + c\right )^{4} - 768 \, a n^{3} - 4128 \, a n^{2} - 24 \, {\left (a n^{5} + 16 \, a n^{4} + 86 \, a n^{3} + 176 \, a n^{2} + 105 \, a n\right )} \cos \left (d x + c\right )^{2} - 8448 \, a n - {\left ({\left (a n^{7} + 29 \, a n^{6} + 343 \, a n^{5} + 2135 \, a n^{4} + 7504 \, a n^{3} + 14756 \, a n^{2} + 14832 \, a n + 5760 \, a\right )} \cos \left (d x + c\right )^{6} + 48 \, a n^{4} + 6 \, {\left (a n^{6} + 24 \, a n^{5} + 223 \, a n^{4} + 1020 \, a n^{3} + 2404 \, a n^{2} + 2736 \, a n + 1152 \, a\right )} \cos \left (d x + c\right )^{4} + 960 \, a n^{3} + 6720 \, a n^{2} + 24 \, {\left (a n^{5} + 21 \, a n^{4} + 160 \, a n^{3} + 540 \, a n^{2} + 784 \, a n + 384 \, a\right )} \cos \left (d x + c\right )^{2} + 19200 \, a n + 18432 \, a\right )} \sin \left (d x + c\right ) - 5040 \, a\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 36 \, d n^{7} + 546 \, d n^{6} + 4536 \, d n^{5} + 22449 \, d n^{4} + 67284 \, d n^{3} + 118124 \, d n^{2} + 109584 \, d n + 40320 \, d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
-((a*n^7 + 28*a*n^6 + 322*a*n^5 + 1960*a*n^4 + 6769*a*n^3 + 13132*a*n^2 + 13068*a*n + 5040*a)*cos(d*x + c)^8 -
(a*n^7 + 22*a*n^6 + 190*a*n^5 + 820*a*n^4 + 1849*a*n^3 + 2038*a*n^2 + 840*a*n)*cos(d*x + c)^6 - 48*a*n^4 - 6*
(a*n^6 + 18*a*n^5 + 118*a*n^4 + 348*a*n^3 + 457*a*n^2 + 210*a*n)*cos(d*x + c)^4 - 768*a*n^3 - 4128*a*n^2 - 24*
(a*n^5 + 16*a*n^4 + 86*a*n^3 + 176*a*n^2 + 105*a*n)*cos(d*x + c)^2 - 8448*a*n - ((a*n^7 + 29*a*n^6 + 343*a*n^5
+ 2135*a*n^4 + 7504*a*n^3 + 14756*a*n^2 + 14832*a*n + 5760*a)*cos(d*x + c)^6 + 48*a*n^4 + 6*(a*n^6 + 24*a*n^5
+ 223*a*n^4 + 1020*a*n^3 + 2404*a*n^2 + 2736*a*n + 1152*a)*cos(d*x + c)^4 + 960*a*n^3 + 6720*a*n^2 + 24*(a*n^
5 + 21*a*n^4 + 160*a*n^3 + 540*a*n^2 + 784*a*n + 384*a)*cos(d*x + c)^2 + 19200*a*n + 18432*a)*sin(d*x + c) - 5
040*a)*sin(d*x + c)^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)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 19968 vs. \(2 (141) = 282\).
Time = 27.65 (sec) , antiderivative size = 19968, normalized size of antiderivative = 119.57
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)**7*sin(d*x+c)**n*(a+a*sin(d*x+c)),x)
[Out]
Piecewise((x*(a*sin(c) + a)*sin(c)**n*cos(c)**7, Eq(d, 0)), (-a*log(sin(c + d*x))/d + 16*a/(35*d*sin(c + d*x))
- a*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) - 8*a*cos(c + d*x)**2/(35*d*sin(c + d*x)**3) + a*cos(c + d*x)**4/(4
*d*sin(c + d*x)**4) + 6*a*cos(c + d*x)**4/(35*d*sin(c + d*x)**5) - a*cos(c + d*x)**6/(6*d*sin(c + d*x)**6) - a
*cos(c + d*x)**6/(7*d*sin(c + d*x)**7), Eq(n, -8)), (-a*log(sin(c + d*x))/d - 16*a*sin(c + d*x)/(5*d) - 8*a*co
s(c + d*x)**2/(5*d*sin(c + d*x)) - a*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) + 2*a*cos(c + d*x)**4/(5*d*sin(c +
d*x)**3) + a*cos(c + d*x)**4/(4*d*sin(c + d*x)**4) - a*cos(c + d*x)**6/(5*d*sin(c + d*x)**5) - a*cos(c + d*x)*
*6/(6*d*sin(c + d*x)**6), Eq(n, -7)), (-960*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**9/(320*d*tan(c/2
+ d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) - 1920*a*log(tan(c/2 + d*x/2)**2 + 1)*tan
(c/2 + d*x/2)**7/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) - 960*a*l
og(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**5/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d
*tan(c/2 + d*x/2)**5) + 960*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**9/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan
(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) + 1920*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**7/(320*d*tan(c/
2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) + 960*a*log(tan(c/2 + d*x/2))*tan(c/2 +
d*x/2)**5/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) - 2*a*tan(c/2 +
d*x/2)**14/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) - 5*a*tan(c/2
+ d*x/2)**13/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) + 26*a*tan(c/
2 + d*x/2)**12/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) + 90*a*tan(
c/2 + d*x/2)**11/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) - 322*a*t
an(c/2 + d*x/2)**10/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) - 1750
*a*tan(c/2 + d*x/2)**8/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) - 8
30*a*tan(c/2 + d*x/2)**7/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5) -
1750*a*tan(c/2 + d*x/2)**6/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5
) - 322*a*tan(c/2 + d*x/2)**4/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)*
*5) + 90*a*tan(c/2 + d*x/2)**3/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)
**5) + 26*a*tan(c/2 + d*x/2)**2/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2
)**5) - 5*a*tan(c/2 + d*x/2)/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**
5) - 2*a/(320*d*tan(c/2 + d*x/2)**9 + 640*d*tan(c/2 + d*x/2)**7 + 320*d*tan(c/2 + d*x/2)**5), Eq(n, -6)), (-57
6*a*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) - 1728*a*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) - 1728*a*log(tan(c/2 + d*x/2)**2 + 1)*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) - 576*a*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*ta
n(c/2 + d*x/2)**4) + 576*a*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) + 1728*a*log(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) + 1728*a*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) + 576*a*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*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2
+ d*x/2)**4) - 3*a*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) - 8*a*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) + 51*a*tan(c/2 + d*x/2)**12/(192*d*t
an(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) + 240
*a*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) + 2184*a*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) - 660*a*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) + 3360*a*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) - 660*a*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) + 2184*a*tan(c/2 + d*x/2)**5/(192*d*tan(c/2 + d*x/2)**10 + 576*d*ta
n(c/2 + d*x/2)**8 + 576*d*tan(c/2 + d*x/2)**6 + 192*d*tan(c/2 + d*x/2)**4) + 240*a*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) + 51
*a*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 + 1
92*d*tan(c/2 + d*x/2)**4) - 8*a*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/(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)), (72*a*log(tan(c/2 + d*x/2)**2 + 1)*t
an(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*t
an(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 288*a*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) + 432*a*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) + 288
*a*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 + 14
4*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) + 72*a*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 + 9
6*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 72*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**11/(24*d*ta
n(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*ta
n(c/2 + d*x/2)**3) - 288*a*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) - 432*a*log(ta
n(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) - 288*a*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) - 72*a*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*
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) - 3*a*tan(c/2 + d*x/2)**13/(24*d*tan(c/2 + d*x/2)**11 + 96*d*t
an(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) + 29*a*t
an(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*t
an(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 303*a*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) + 171*a
*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) + 693*a*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) + 240*a
*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) + 693*a*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) + 171*a
*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) + 303*a*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) + 29*a*
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 + 144*d*tan(c/2 + d*x/2)**7 + 96*d*t
an(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 3*a*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/(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*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**12/(40*d*tan(c/2 + d*x/2)
**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*tan(c/2 + d*
x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) + 600*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**10/(40*d*tan(c/2 +
d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*tan(c/
2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) + 1200*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**8/(40*d*tan(
c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*
tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) + 1200*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**6/(40*
d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 +
200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) + 600*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**4
/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)*
*6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) + 120*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/
2)**2/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*
x/2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) - 120*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)
**12/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x
/2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) - 600*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)*
*10/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/
2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) - 1200*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)*
*8/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2
)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) - 1200*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**
6/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)
**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) - 600*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**4/
(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**
6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) - 120*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(4
0*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6
+ 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) - 5*a*tan(c/2 + d*x/2)**14/(40*d*tan(c/2 + d*x/2)**12
+ 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*tan(c/2 + d*x/2)*
*4 + 40*d*tan(c/2 + d*x/2)**2) - 20*a*tan(c/2 + d*x/2)**13/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)
**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/
2)**2) - 360*a*tan(c/2 + d*x/2)**11/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 +
d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) + 310*a*tan(c/2
+ d*x/2)**10/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c
/2 + d*x/2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) - 940*a*tan(c/2 + d*x/2)**9/(40*d*tan(c
/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*t
an(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) + 735*a*tan(c/2 + d*x/2)**8/(40*d*tan(c/2 + d*x/2)**12 + 200*d*
tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*
d*tan(c/2 + d*x/2)**2) - 1456*a*tan(c/2 + d*x/2)**7/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 +
400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2)
+ 735*a*tan(c/2 + d*x/2)**6/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**
8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) - 940*a*tan(c/2 + d*x/2)
**5/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/
2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) + 310*a*tan(c/2 + d*x/2)**4/(40*d*tan(c/2 + d*x/
2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*tan(c/2 +
d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) - 360*a*tan(c/2 + d*x/2)**3/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 +
d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2
+ d*x/2)**2) - 20*a*tan(c/2 + d*x/2)/(40*d*tan(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2
+ d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2) - 5*a/(40*d*ta
n(c/2 + d*x/2)**12 + 200*d*tan(c/2 + d*x/2)**10 + 400*d*tan(c/2 + d*x/2)**8 + 400*d*tan(c/2 + d*x/2)**6 + 200*
d*tan(c/2 + d*x/2)**4 + 40*d*tan(c/2 + d*x/2)**2), Eq(n, -3)), (-30*a*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)) - 180*a*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)) - 450*a*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*t
an(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 600*a*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)) - 450*a*log(tan(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*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)) - 180*
a*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 + 4
50*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)) - 30*a*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 + 1
80*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) + 30*a*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 + 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)) + 180*a*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 + 30*d*tan(c/2 + d*x/2)) + 450*a*log(t
an(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 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2
+ d*x/2)) + 600*a*log(tan(c/2 + d*x/2))*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)) + 450*a*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)) + 180*a*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)) + 30*a*log(tan(c/2 + d*x/2))*ta
n(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*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*ta
n(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)) - 285*a*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 + 180*d*tan(c/2 + d*x/2)**3 + 30*d*tan(c/2 + d*x/2)) - 180*a*t
an(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)) - 975*a
*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 + 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)) - 360
*a*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 + 6
00*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)) - 17
97*a*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)) -
680*a*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)) -
1797*a*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 + 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))
- 360*a*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)
) - 975*a*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
)) - 180*a*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)) - 285*a*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/(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)), Eq(n, -2)), (-
105*a*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*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*log(t
an(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*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*t
an(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*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*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*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 + 3
675*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*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 + 220
5*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*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*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*log(tan(c/2 + d*x/2))*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*t
an(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) + 3675*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**8/(10
5*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*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*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*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*ta
n(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*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) + 210*a*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) - 630*a*tan(c/2 + d*x/2)**12/(105*d*tan(c/2 + d*x/2)**14 + 735*d*t
an(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) + 420*a*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*ta
n(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) - 1890*a*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) + 180
6*a*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) - 3640*a*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) + 1272*a*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*ta
n(c/2 + d*x/2)**4 + 735*d*tan(c/2 + d*x/2)**2 + 105*d) - 3640*a*tan(c/2 + d*x/2)**6/(105*d*tan(c/2 + d*x/2)**1
4 + 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) + 1806*a*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 + 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) - 1890*a*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 + 10
5*d) + 420*a*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) - 630*a*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) + 210*a*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*n**7*sin(c + d*x)**2*sin(c + d*x)**
n*cos(c + d*x)**6/(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*n**7*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(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) + 6*a*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 + 22449*d*n**4 + 67284*d*n**
3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 6*a*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) +
34*a*n**6*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**6/(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) + 35*a*n**6*sin(c + d*x)*sin(c + d*x)**n*cos(
c + d*x)**6/(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) + 24*a*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*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) + 180*a*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)
+ 192*a*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) + 478*a*n**5*sin(c + d*x)**2*sin(c + d*x)
**n*cos(c + d*x)**6/(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*n**5*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(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) + 48*a*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 + 11812
4*d*n**2 + 109584*d*n + 40320*d) + 48*a*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) + 576*a*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) + 648*a*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)
+ 2148*a*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) + 2490*a*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) + 3580*a*n**4*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**6/(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) + 4025*a*n**4
*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(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) + 768*a*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 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 960
*a*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) + 5136*a*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 + 403
20*d) + 6720*a*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) + 12888*a*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 + 118
124*d*n**2 + 109584*d*n + 40320*d) + 16680*a*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) + 152
89*a*n**3*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**6/(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) + 18424*a*n**3*sin(c + d*x)*sin(c + d*x)**n*c
os(c + d*x)**6/(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) + 4128*a*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) + 6720*a*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 + 1095
84*d*n + 40320*d) + 20736*a*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 + 40320*d) + 33120*a*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) + 40182*a*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 + 4
0320*d) + 60504*a*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) + 36706*a*n**2*sin(c + d*x)**2*s
in(c + d*x)**n*cos(c + d*x)**6/(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*n**2*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(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) + 844
8*a*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) + 19200*a*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) + 36312*a*n*si
n(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) + 76416*a*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 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n +
40320*d) + 59508*a*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 + 22449*d*n**4 + 67284*d*n**3 + 118124*d*n**2 + 109584*d*n + 40320*d) + 111648*a*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 + 118
124*d*n**2 + 109584*d*n + 40320*d) + 44712*a*n*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**6/(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) + 69264*
a*n*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(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) + 5040*a*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) + 1843
2*a*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) + 20160*a*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) +
64512*a*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) + 30240*a*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 + 1095
84*d*n + 40320*d) + 80640*a*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) + 20160*a*sin(c + d*x)**2*s
in(c + d*x)**n*cos(c + d*x)**6/(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*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**6/(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 = 0.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {\frac {a \sin \left (d x + c\right )^{n + 8}}{n + 8} + \frac {a \sin \left (d x + c\right )^{n + 7}}{n + 7} - \frac {3 \, a \sin \left (d x + c\right )^{n + 6}}{n + 6} - \frac {3 \, a \sin \left (d x + c\right )^{n + 5}}{n + 5} + \frac {3 \, a \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {3 \, a \sin \left (d x + c\right )^{n + 3}}{n + 3} - \frac {a \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
-(a*sin(d*x + c)^(n + 8)/(n + 8) + a*sin(d*x + c)^(n + 7)/(n + 7) - 3*a*sin(d*x + c)^(n + 6)/(n + 6) - 3*a*sin
(d*x + c)^(n + 5)/(n + 5) + 3*a*sin(d*x + c)^(n + 4)/(n + 4) + 3*a*sin(d*x + c)^(n + 3)/(n + 3) - a*sin(d*x +
c)^(n + 2)/(n + 2) - a*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. 674 vs. \(2 (167) = 334\).
Time = 0.36 (sec) , antiderivative size = 674, normalized size of antiderivative = 4.04
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {\frac {{\left (n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{8} + 12 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{8} - 3 \, n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 44 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{8} - 42 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 48 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{8} + 3 \, n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} - 168 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 48 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} - 192 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} - n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 228 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} - 18 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 288 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} - 104 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} - 192 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}\right )} a}{n^{4} + 20 \, n^{3} + 140 \, n^{2} + 400 \, n + 384} + \frac {{\left (n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7} + 9 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7} - 3 \, n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 23 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7} - 33 \, 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} + 3 \, n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} - 93 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 39 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} - 63 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} - n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 141 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} - 15 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 105 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} - 71 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) - 105 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )\right )} a}{n^{4} + 16 \, n^{3} + 86 \, n^{2} + 176 \, n + 105}}{d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
-((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*sin(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 - 192*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*si
n(d*x + c)^2 - 192*sin(d*x + c)^n*sin(d*x + c)^2)*a/(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*sin(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/(n^4 + 16*n^3 + 86*n^2 + 176*n + 105))/d
Mupad [B] (verification not implemented)
Time = 17.20 (sec) , antiderivative size = 901, normalized size of antiderivative = 5.40
\[
\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,{\sin \left (c+d\,x\right )}^n\,\left (5\,n^7+188\,n^6+3050\,n^5+28904\,n^4+167669\,n^3+552236\,n^2+879324\,n+468720\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\,{\sin \left (c+d\,x\right )}^n\,\cos \left (2\,c+2\,d\,x\right )\,\left (-n^7-34\,n^6-454\,n^5-2332\,n^4+599\,n^3+41822\,n^2+109872\,n+70560\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\,{\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 )\,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\,{\sin \left (c+d\,x\right )}^n\,\sin \left (5\,c+5\,d\,x\right )\,\left (n^7\,5{}\mathrm {i}+n^6\,169{}\mathrm {i}+n^5\,2291{}\mathrm {i}+n^4\,16027{}\mathrm {i}+n^3\,62000{}\mathrm {i}+n^2\,131476{}\mathrm {i}+n\,139824{}\mathrm {i}+56448{}\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\,{\sin \left (c+d\,x\right )}^n\,\sin \left (3\,c+3\,d\,x\right )\,\left (n^7\,3{}\mathrm {i}+n^6\,111{}\mathrm {i}+n^5\,1733{}\mathrm {i}+n^4\,14445{}\mathrm {i}+n^3\,67472{}\mathrm {i}+n^2\,171084{}\mathrm {i}+n\,210512{}\mathrm {i}+94080{}\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\,{\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\,{\sin \left (c+d\,x\right )}^n\,\cos \left (6\,c+6\,d\,x\right )\,\left (n^7+34\,n^6+454\,n^5+3100\,n^4+11689\,n^3+24226\,n^2+25296\,n+10080\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\,{\sin \left (c+d\,x\right )}^n\,\cos \left (4\,c+4\,d\,x\right )\,\left (n^7+40\,n^6+682\,n^5+5968\,n^4+27937\,n^3+68728\,n^2+81396\,n+35280\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\,\sin \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^n\,\left (n^7\,5{}\mathrm {i}+n^6\,193{}\mathrm {i}+n^5\,3251{}\mathrm {i}+n^4\,32515{}\mathrm {i}+n^3\,209360{}\mathrm {i}+n^2\,826612{}\mathrm {i}+n\,1735344{}\mathrm {i}+1411200{}\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)^7*sin(c + d*x)^n*(a + a*sin(c + d*x)),x)
[Out]
(a*sin(c + d*x)^n*(879324*n + 552236*n^2 + 167669*n^3 + 28904*n^4 + 3050*n^5 + 188*n^6 + 5*n^7 + 468720))/(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*sin(c + d
*x)^n*cos(2*c + 2*d*x)*(109872*n + 41822*n^2 + 599*n^3 - 2332*n^4 - 454*n^5 - 34*n^6 - n^7 + 70560))/(32*d*(10
9584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (a*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)*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*sin(c + d
*x)^n*sin(5*c + 5*d*x)*(n*139824i + n^2*131476i + n^3*62000i + n^4*16027i + n^5*2291i + n^6*169i + n^7*5i + 56
448i)*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*sin(c + d*x)^n*sin(3*c + 3*d*x)*(n*210512i + n^2*171084i + n^3*67472i + n^4*14445i + n^5*1733i + n^6*111i
+ n^7*3i + 94080i)*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*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 + 40
320)) - (a*sin(c + d*x)^n*cos(6*c + 6*d*x)*(25296*n + 24226*n^2 + 11689*n^3 + 3100*n^4 + 454*n^5 + 34*n^6 + n^
7 + 10080))/(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*sin(c + d*x)^n*cos(4*c + 4*d*x)*(81396*n + 68728*n^2 + 27937*n^3 + 5968*n^4 + 682*n^5 + 40*n^6 + n^7 + 3
5280))/(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
*sin(c + d*x)*sin(c + d*x)^n*(n*1735344i + n^2*826612i + n^3*209360i + n^4*32515i + n^5*3251i + n^6*193i + n^7
*5i + 1411200i)*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))