\(\int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx\) [967]

Optimal result

Integrand size = 29, antiderivative size = 96 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))}{7 d}+\frac {a (6 A-B) \tan (c+d x)}{7 d}+\frac {2 a (6 A-B) \tan ^3(c+d x)}{21 d}+\frac {a (6 A-B) \tan ^5(c+d x)}{35 d} \]

[Out]

1/7*(A+B)*sec(d*x+c)^7*(a+a*sin(d*x+c))/d+1/7*a*(6*A-B)*tan(d*x+c)/d+2/21*a*(6*A-B)*tan(d*x+c)^3/d+1/35*a*(6*A
-B)*tan(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 96, 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 = {2934, 3852} \[ \int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a (6 A-B) \tan ^5(c+d x)}{35 d}+\frac {2 a (6 A-B) \tan ^3(c+d x)}{21 d}+\frac {a (6 A-B) \tan (c+d x)}{7 d}+\frac {(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)}{7 d} \]

[In]

Int[Sec[c + d*x]^8*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

((A + B)*Sec[c + d*x]^7*(a + a*Sin[c + d*x]))/(7*d) + (a*(6*A - B)*Tan[c + d*x])/(7*d) + (2*a*(6*A - B)*Tan[c
+ d*x]^3)/(21*d) + (a*(6*A - B)*Tan[c + d*x]^5)/(35*d)

Rule 2934

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c + a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(p +
1))), x] + Dist[b*((a*d*m + b*c*(m + p + 1))/(a*g^2*(p + 1))), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*
x])^(m - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, -1] && LtQ[p, -1]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))}{7 d}+\frac {1}{7} (a (6 A-B)) \int \sec ^6(c+d x) \, dx \\ & = \frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))}{7 d}-\frac {(a (6 A-B)) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{7 d} \\ & = \frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))}{7 d}+\frac {a (6 A-B) \tan (c+d x)}{7 d}+\frac {2 a (6 A-B) \tan ^3(c+d x)}{21 d}+\frac {a (6 A-B) \tan ^5(c+d x)}{35 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.94 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a \left (15 (A+B) \sec ^7(c+d x)+105 A \sec ^6(c+d x) \tan (c+d x)-35 (6 A-B) \sec ^4(c+d x) \tan ^3(c+d x)+28 (6 A-B) \sec ^2(c+d x) \tan ^5(c+d x)+8 (-6 A+B) \tan ^7(c+d x)\right )}{105 d} \]

[In]

Integrate[Sec[c + d*x]^8*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

(a*(15*(A + B)*Sec[c + d*x]^7 + 105*A*Sec[c + d*x]^6*Tan[c + d*x] - 35*(6*A - B)*Sec[c + d*x]^4*Tan[c + d*x]^3
+ 28*(6*A - B)*Sec[c + d*x]^2*Tan[c + d*x]^5 + 8*(-6*A + B)*Tan[c + d*x]^7))/(105*d)

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.35

method	result	size
derivativedivides	\(\frac {\frac {a A}{7 \cos \left (d x +c \right )^{7}}+B a \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )-a A \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {B a}{7 \cos \left (d x +c \right )^{7}}}{d}\)	\(130\)
default	\(\frac {\frac {a A}{7 \cos \left (d x +c \right )^{7}}+B a \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )-a A \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {B a}{7 \cos \left (d x +c \right )^{7}}}{d}\)	\(130\)
risch	\(-\frac {16 i a \left (120 i A \,{\mathrm e}^{5 i \left (d x +c \right )}-20 i B \,{\mathrm e}^{5 i \left (d x +c \right )}+70 B \,{\mathrm e}^{6 i \left (d x +c \right )}+60 i A \,{\mathrm e}^{3 i \left (d x +c \right )}+30 A \,{\mathrm e}^{4 i \left (d x +c \right )}-10 i B \,{\mathrm e}^{3 i \left (d x +c \right )}-5 B \,{\mathrm e}^{4 i \left (d x +c \right )}+12 i A \,{\mathrm e}^{i \left (d x +c \right )}+24 A \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i B \,{\mathrm e}^{i \left (d x +c \right )}-4 B \,{\mathrm e}^{2 i \left (d x +c \right )}+6 A -B \right )}{105 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{7} d}\)	\(181\)
parallelrisch	\(\frac {13 \left (\left (A -\frac {94 B}{39}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {376 A}{13}+\frac {188 B}{39}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {298 A}{13}-\frac {674 B}{39}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {480 A}{13}-\frac {80 B}{13}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {985 A}{13}-\frac {820 B}{39}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1520 A}{13}+\frac {760 B}{39}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (140 A -\frac {2660 B}{39}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-105 A -\frac {210 B}{13}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {280 A}{13}+\frac {140 B}{39}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {602 A}{13}-\frac {826 B}{39}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {224 A}{13}+\frac {112 B}{39}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {63 A}{13}-\frac {56 B}{39}\right ) a}{175 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{7}}\)	\(245\)

[In]

int(sec(d*x+c)^8*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/7*a*A/cos(d*x+c)^7+B*a*(1/7*sin(d*x+c)^3/cos(d*x+c)^7+4/35*sin(d*x+c)^3/cos(d*x+c)^5+8/105*sin(d*x+c)^3
/cos(d*x+c)^3)-a*A*(-16/35-1/7*sec(d*x+c)^6-6/35*sec(d*x+c)^4-8/35*sec(d*x+c)^2)*tan(d*x+c)+1/7*B*a/cos(d*x+c)
^7)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.55 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {8 \, {\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{6} - 4 \, {\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{4} - {\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{2} - 3 \, {\left (A - 6 \, B\right )} a + {\left (8 \, {\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{4} + 4 \, {\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{2} + 3 \, {\left (6 \, A - B\right )} a\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{5}\right )}} \]

[In]

integrate(sec(d*x+c)^8*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/105*(8*(6*A - B)*a*cos(d*x + c)^6 - 4*(6*A - B)*a*cos(d*x + c)^4 - (6*A - B)*a*cos(d*x + c)^2 - 3*(A - 6*B)
*a + (8*(6*A - B)*a*cos(d*x + c)^4 + 4*(6*A - B)*a*cos(d*x + c)^2 + 3*(6*A - B)*a)*sin(d*x + c))/(d*cos(d*x +
c)^5*sin(d*x + c) - d*cos(d*x + c)^5)

Sympy [F(-1)]

Timed out. \[ \int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**8*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.11 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {3 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} A a + {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} B a + \frac {15 \, A a}{\cos \left (d x + c\right )^{7}} + \frac {15 \, B a}{\cos \left (d x + c\right )^{7}}}{105 \, d} \]

[In]

integrate(sec(d*x+c)^8*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/105*(3*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*tan(d*x + c))*A*a + (15*tan(d*x + c)^7
+ 42*tan(d*x + c)^5 + 35*tan(d*x + c)^3)*B*a + 15*A*a/cos(d*x + c)^7 + 15*B*a/cos(d*x + c)^7)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (88) = 176\).

Time = 0.49 (sec) , antiderivative size = 345, normalized size of antiderivative = 3.59 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {\frac {7 \, {\left (165 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 75 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 540 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 210 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 750 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 280 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 480 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 170 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 129 \, A a - 49 \, B a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}} + \frac {2205 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 525 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 10080 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1470 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21945 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2555 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 26460 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2240 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18963 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1407 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7476 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 434 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1383 \, A a + 137 \, B a}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{7}}}{1680 \, d} \]

[In]

integrate(sec(d*x+c)^8*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/1680*(7*(165*A*a*tan(1/2*d*x + 1/2*c)^4 - 75*B*a*tan(1/2*d*x + 1/2*c)^4 + 540*A*a*tan(1/2*d*x + 1/2*c)^3 -
210*B*a*tan(1/2*d*x + 1/2*c)^3 + 750*A*a*tan(1/2*d*x + 1/2*c)^2 - 280*B*a*tan(1/2*d*x + 1/2*c)^2 + 480*A*a*tan
(1/2*d*x + 1/2*c) - 170*B*a*tan(1/2*d*x + 1/2*c) + 129*A*a - 49*B*a)/(tan(1/2*d*x + 1/2*c) + 1)^5 + (2205*A*a*
tan(1/2*d*x + 1/2*c)^6 + 525*B*a*tan(1/2*d*x + 1/2*c)^6 - 10080*A*a*tan(1/2*d*x + 1/2*c)^5 - 1470*B*a*tan(1/2*
d*x + 1/2*c)^5 + 21945*A*a*tan(1/2*d*x + 1/2*c)^4 + 2555*B*a*tan(1/2*d*x + 1/2*c)^4 - 26460*A*a*tan(1/2*d*x +
1/2*c)^3 - 2240*B*a*tan(1/2*d*x + 1/2*c)^3 + 18963*A*a*tan(1/2*d*x + 1/2*c)^2 + 1407*B*a*tan(1/2*d*x + 1/2*c)^
2 - 7476*A*a*tan(1/2*d*x + 1/2*c) - 434*B*a*tan(1/2*d*x + 1/2*c) + 1383*A*a + 137*B*a)/(tan(1/2*d*x + 1/2*c) -
1)^7)/d

Mupad [B] (verification not implemented)

Time = 14.12 (sec) , antiderivative size = 320, normalized size of antiderivative = 3.33 \[ \int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {15\,A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}-\frac {75\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {105\,A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {9\,A\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}-\frac {3\,A\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{2}-\frac {35\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {65\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {55\,B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {35\,B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}-\frac {19\,B\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}+\frac {B\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{4}-\frac {843\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {363\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}-\frac {651\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}+\frac {171\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}-\frac {111\,A\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}+\frac {15\,A\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}+\frac {53\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {27\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}+\frac {21\,B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}+\frac {59\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}+\frac {B\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}+\frac {15\,B\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}\right )}{3360\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^5\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^7} \]

[In]

int(((A + B*sin(c + d*x))*(a + a*sin(c + d*x)))/cos(c + d*x)^8,x)

[Out]

-(a*cos(c/2 + (d*x)/2)*((15*A*cos((5*c)/2 + (5*d*x)/2))/8 - (75*A*cos((3*c)/2 + (3*d*x)/2))/8 - (105*A*cos((7*
c)/2 + (7*d*x)/2))/16 + (9*A*cos((9*c)/2 + (9*d*x)/2))/16 - (3*A*cos((11*c)/2 + (11*d*x)/2))/2 - (35*B*cos(c/2
+ (d*x)/2))/2 + (65*B*cos((3*c)/2 + (3*d*x)/2))/8 - (55*B*cos((5*c)/2 + (5*d*x)/2))/8 + (35*B*cos((7*c)/2 + (
7*d*x)/2))/16 - (19*B*cos((9*c)/2 + (9*d*x)/2))/16 + (B*cos((11*c)/2 + (11*d*x)/2))/4 - (843*A*sin(c/2 + (d*x)
/2))/16 + (363*A*sin((3*c)/2 + (3*d*x)/2))/16 - (651*A*sin((5*c)/2 + (5*d*x)/2))/32 + (171*A*sin((7*c)/2 + (7*
d*x)/2))/32 - (111*A*sin((9*c)/2 + (9*d*x)/2))/32 + (15*A*sin((11*c)/2 + (11*d*x)/2))/32 + (53*B*sin(c/2 + (d*
x)/2))/16 + (27*B*sin((3*c)/2 + (3*d*x)/2))/16 + (21*B*sin((5*c)/2 + (5*d*x)/2))/32 + (59*B*sin((7*c)/2 + (7*d
*x)/2))/32 + (B*sin((9*c)/2 + (9*d*x)/2))/32 + (15*B*sin((11*c)/2 + (11*d*x)/2))/32))/(3360*d*cos(c/2 - pi/4 +
(d*x)/2)^5*cos(c/2 + pi/4 + (d*x)/2)^7)