\(\int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx\) [968]

Optimal result

Integrand size = 29, antiderivative size = 119 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))}{9 d}+\frac {a (8 A-B) \tan (c+d x)}{9 d}+\frac {a (8 A-B) \tan ^3(c+d x)}{9 d}+\frac {a (8 A-B) \tan ^5(c+d x)}{15 d}+\frac {a (8 A-B) \tan ^7(c+d x)}{63 d} \]

[Out]

1/9*(A+B)*sec(d*x+c)^9*(a+a*sin(d*x+c))/d+1/9*a*(8*A-B)*tan(d*x+c)/d+1/9*a*(8*A-B)*tan(d*x+c)^3/d+1/15*a*(8*A-
B)*tan(d*x+c)^5/d+1/63*a*(8*A-B)*tan(d*x+c)^7/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 119, 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} \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a (8 A-B) \tan ^7(c+d x)}{63 d}+\frac {a (8 A-B) \tan ^5(c+d x)}{15 d}+\frac {a (8 A-B) \tan ^3(c+d x)}{9 d}+\frac {a (8 A-B) \tan (c+d x)}{9 d}+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)}{9 d} \]

[In]

Int[Sec[c + d*x]^10*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x]))/(9*d) + (a*(8*A - B)*Tan[c + d*x])/(9*d) + (a*(8*A - B)*Tan[c +
d*x]^3)/(9*d) + (a*(8*A - B)*Tan[c + d*x]^5)/(15*d) + (a*(8*A - B)*Tan[c + d*x]^7)/(63*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 ^9(c+d x) (a+a \sin (c+d x))}{9 d}+\frac {1}{9} (a (8 A-B)) \int \sec ^8(c+d x) \, dx \\ & = \frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))}{9 d}-\frac {(a (8 A-B)) \text {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (c+d x)\right )}{9 d} \\ & = \frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))}{9 d}+\frac {a (8 A-B) \tan (c+d x)}{9 d}+\frac {a (8 A-B) \tan ^3(c+d x)}{9 d}+\frac {a (8 A-B) \tan ^5(c+d x)}{15 d}+\frac {a (8 A-B) \tan ^7(c+d x)}{63 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.31 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.10 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a \left (35 (A+B) \sec ^9(c+d x)+315 A \sec ^8(c+d x) \tan (c+d x)-105 (8 A-B) \sec ^6(c+d x) \tan ^3(c+d x)+126 (8 A-B) \sec ^4(c+d x) \tan ^5(c+d x)-72 (8 A-B) \sec ^2(c+d x) \tan ^7(c+d x)+16 (8 A-B) \tan ^9(c+d x)\right )}{315 d} \]

[In]

Integrate[Sec[c + d*x]^10*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

(a*(35*(A + B)*Sec[c + d*x]^9 + 315*A*Sec[c + d*x]^8*Tan[c + d*x] - 105*(8*A - B)*Sec[c + d*x]^6*Tan[c + d*x]^
3 + 126*(8*A - B)*Sec[c + d*x]^4*Tan[c + d*x]^5 - 72*(8*A - B)*Sec[c + d*x]^2*Tan[c + d*x]^7 + 16*(8*A - B)*Ta
n[c + d*x]^9))/(315*d)

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.33

method	result	size
derivativedivides	\(\frac {\frac {a A}{9 \cos \left (d x +c \right )^{9}}+B a \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )-a A \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (d x +c \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (d x +c \right )\right )}{315}\right ) \tan \left (d x +c \right )+\frac {B a}{9 \cos \left (d x +c \right )^{9}}}{d}\)	\(158\)
default	\(\frac {\frac {a A}{9 \cos \left (d x +c \right )^{9}}+B a \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )-a A \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (d x +c \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (d x +c \right )\right )}{315}\right ) \tan \left (d x +c \right )+\frac {B a}{9 \cos \left (d x +c \right )^{9}}}{d}\)	\(158\)
risch	\(-\frac {32 i a \left (336 i A \,{\mathrm e}^{5 i \left (d x +c \right )}-14 i B \,{\mathrm e}^{3 i \left (d x +c \right )}+315 B \,{\mathrm e}^{8 i \left (d x +c \right )}+560 i A \,{\mathrm e}^{7 i \left (d x +c \right )}+112 A \,{\mathrm e}^{6 i \left (d x +c \right )}-42 i B \,{\mathrm e}^{5 i \left (d x +c \right )}-14 B \,{\mathrm e}^{6 i \left (d x +c \right )}+112 i A \,{\mathrm e}^{3 i \left (d x +c \right )}+112 A \,{\mathrm e}^{4 i \left (d x +c \right )}-70 i B \,{\mathrm e}^{7 i \left (d x +c \right )}-14 B \,{\mathrm e}^{4 i \left (d x +c \right )}+16 i A \,{\mathrm e}^{i \left (d x +c \right )}+48 A \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i B \,{\mathrm e}^{i \left (d x +c \right )}-6 B \,{\mathrm e}^{2 i \left (d x +c \right )}+8 A -B \right )}{315 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{9} d}\)	\(231\)
parallelrisch	\(-\frac {2 \left (A \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-A +B \right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (-5 A -2 B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {\left (B +13 A \right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {\left (41 A +8 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (-57 A +29 B \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (-57 A -146 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35}+\frac {\left (513 A +89 B \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35}+\frac {\left (1937 A +1136 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315}+\frac {\left (-583 A +191 B \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{45}+\frac {\left (53 A -46 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {\left (53 A +17 B \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {\left (-A +8 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {\left (B -5 A \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {\left (7 A -2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{9}+\frac {A}{9}+\frac {B}{9}\right ) a}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{9}}\)	\(324\)

[In]

int(sec(d*x+c)^10*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/9*a*A/cos(d*x+c)^9+B*a*(1/9*sin(d*x+c)^3/cos(d*x+c)^9+2/21*sin(d*x+c)^3/cos(d*x+c)^7+8/105*sin(d*x+c)^3
/cos(d*x+c)^5+16/315*sin(d*x+c)^3/cos(d*x+c)^3)-a*A*(-128/315-1/9*sec(d*x+c)^8-8/63*sec(d*x+c)^6-16/105*sec(d*
x+c)^4-64/315*sec(d*x+c)^2)*tan(d*x+c)+1/9*B*a/cos(d*x+c)^9)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.55 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {16 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{8} - 8 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{6} - 2 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{4} - {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{2} - 5 \, {\left (A - 8 \, B\right )} a + {\left (16 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{6} + 8 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{4} + 6 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{2} + 5 \, {\left (8 \, A - B\right )} a\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{7}\right )}} \]

[In]

integrate(sec(d*x+c)^10*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/315*(16*(8*A - B)*a*cos(d*x + c)^8 - 8*(8*A - B)*a*cos(d*x + c)^6 - 2*(8*A - B)*a*cos(d*x + c)^4 - (8*A - B
)*a*cos(d*x + c)^2 - 5*(A - 8*B)*a + (16*(8*A - B)*a*cos(d*x + c)^6 + 8*(8*A - B)*a*cos(d*x + c)^4 + 6*(8*A -
B)*a*cos(d*x + c)^2 + 5*(8*A - B)*a)*sin(d*x + c))/(d*cos(d*x + c)^7*sin(d*x + c) - d*cos(d*x + c)^7)

Sympy [F(-1)]

Timed out. \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**10*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.06 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {{\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} A a + {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} B a + \frac {35 \, A a}{\cos \left (d x + c\right )^{9}} + \frac {35 \, B a}{\cos \left (d x + c\right )^{9}}}{315 \, d} \]

[In]

integrate(sec(d*x+c)^10*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/315*((35*tan(d*x + c)^9 + 180*tan(d*x + c)^7 + 378*tan(d*x + c)^5 + 420*tan(d*x + c)^3 + 315*tan(d*x + c))*A
*a + (35*tan(d*x + c)^9 + 135*tan(d*x + c)^7 + 189*tan(d*x + c)^5 + 105*tan(d*x + c)^3)*B*a + 35*A*a/cos(d*x +
c)^9 + 35*B*a/cos(d*x + c)^9)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (109) = 218\).

Time = 0.43 (sec) , antiderivative size = 465, normalized size of antiderivative = 3.91 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {\frac {3 \, {\left (9765 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3675 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 48720 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15960 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 109865 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 33775 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 136640 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 39760 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 99183 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 28161 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 39536 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 11032 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7043 \, A a - 2101 \, B a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}} + \frac {51345 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 11025 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 322560 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 47880 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 976500 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 117180 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1753920 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 168840 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2037294 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 165942 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1550976 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 106008 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 760644 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 47772 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 219456 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12888 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 30089 \, A a + 2657 \, B a}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{9}}}{40320 \, d} \]

[In]

integrate(sec(d*x+c)^10*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/40320*(3*(9765*A*a*tan(1/2*d*x + 1/2*c)^6 - 3675*B*a*tan(1/2*d*x + 1/2*c)^6 + 48720*A*a*tan(1/2*d*x + 1/2*c
)^5 - 15960*B*a*tan(1/2*d*x + 1/2*c)^5 + 109865*A*a*tan(1/2*d*x + 1/2*c)^4 - 33775*B*a*tan(1/2*d*x + 1/2*c)^4
+ 136640*A*a*tan(1/2*d*x + 1/2*c)^3 - 39760*B*a*tan(1/2*d*x + 1/2*c)^3 + 99183*A*a*tan(1/2*d*x + 1/2*c)^2 - 28
161*B*a*tan(1/2*d*x + 1/2*c)^2 + 39536*A*a*tan(1/2*d*x + 1/2*c) - 11032*B*a*tan(1/2*d*x + 1/2*c) + 7043*A*a -
2101*B*a)/(tan(1/2*d*x + 1/2*c) + 1)^7 + (51345*A*a*tan(1/2*d*x + 1/2*c)^8 + 11025*B*a*tan(1/2*d*x + 1/2*c)^8
- 322560*A*a*tan(1/2*d*x + 1/2*c)^7 - 47880*B*a*tan(1/2*d*x + 1/2*c)^7 + 976500*A*a*tan(1/2*d*x + 1/2*c)^6 + 1
17180*B*a*tan(1/2*d*x + 1/2*c)^6 - 1753920*A*a*tan(1/2*d*x + 1/2*c)^5 - 168840*B*a*tan(1/2*d*x + 1/2*c)^5 + 20
37294*A*a*tan(1/2*d*x + 1/2*c)^4 + 165942*B*a*tan(1/2*d*x + 1/2*c)^4 - 1550976*A*a*tan(1/2*d*x + 1/2*c)^3 - 10
6008*B*a*tan(1/2*d*x + 1/2*c)^3 + 760644*A*a*tan(1/2*d*x + 1/2*c)^2 + 47772*B*a*tan(1/2*d*x + 1/2*c)^2 - 21945
6*A*a*tan(1/2*d*x + 1/2*c) - 12888*B*a*tan(1/2*d*x + 1/2*c) + 30089*A*a + 2657*B*a)/(tan(1/2*d*x + 1/2*c) - 1)
^9)/d

Mupad [B] (verification not implemented)

Time = 14.03 (sec) , antiderivative size = 416, normalized size of antiderivative = 3.50 \[ \int \sec ^{10}(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 {329\,A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}-\frac {1225\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}-\frac {133\,A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}+\frac {21\,A\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{8}-\frac {413\,A\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{64}+\frac {29\,A\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{64}-A\,\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )-\frac {315\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {1295\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}-\frac {1183\,B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}+7\,B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )-\frac {21\,B\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{4}+\frac {91\,B\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{64}-\frac {43\,B\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{64}+\frac {B\,\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{8}-\frac {17609\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {8649\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{128}-\frac {8159\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{128}+\frac {2783\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{128}-\frac {2293\,A\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{128}+\frac {501\,A\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{128}-\frac {291\,A\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{128}+\frac {35\,A\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{128}+\frac {823\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {297\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{128}+\frac {193\,B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{128}+\frac {479\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{128}+\frac {11\,B\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{128}+\frac {213\,B\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{128}-\frac {3\,B\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{128}+\frac {35\,B\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{128}\right )}{40320\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^7\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^9} \]

[In]

int(((A + B*sin(c + d*x))*(a + a*sin(c + d*x)))/cos(c + d*x)^10,x)

[Out]

-(a*cos(c/2 + (d*x)/2)*((329*A*cos((5*c)/2 + (5*d*x)/2))/64 - (1225*A*cos((3*c)/2 + (3*d*x)/2))/64 - (133*A*co
s((7*c)/2 + (7*d*x)/2))/8 + (21*A*cos((9*c)/2 + (9*d*x)/2))/8 - (413*A*cos((11*c)/2 + (11*d*x)/2))/64 + (29*A*
cos((13*c)/2 + (13*d*x)/2))/64 - A*cos((15*c)/2 + (15*d*x)/2) - (315*B*cos(c/2 + (d*x)/2))/8 + (1295*B*cos((3*
c)/2 + (3*d*x)/2))/64 - (1183*B*cos((5*c)/2 + (5*d*x)/2))/64 + 7*B*cos((7*c)/2 + (7*d*x)/2) - (21*B*cos((9*c)/
2 + (9*d*x)/2))/4 + (91*B*cos((11*c)/2 + (11*d*x)/2))/64 - (43*B*cos((13*c)/2 + (13*d*x)/2))/64 + (B*cos((15*c
)/2 + (15*d*x)/2))/8 - (17609*A*sin(c/2 + (d*x)/2))/128 + (8649*A*sin((3*c)/2 + (3*d*x)/2))/128 - (8159*A*sin(
(5*c)/2 + (5*d*x)/2))/128 + (2783*A*sin((7*c)/2 + (7*d*x)/2))/128 - (2293*A*sin((9*c)/2 + (9*d*x)/2))/128 + (5
01*A*sin((11*c)/2 + (11*d*x)/2))/128 - (291*A*sin((13*c)/2 + (13*d*x)/2))/128 + (35*A*sin((15*c)/2 + (15*d*x)/
2))/128 + (823*B*sin(c/2 + (d*x)/2))/128 + (297*B*sin((3*c)/2 + (3*d*x)/2))/128 + (193*B*sin((5*c)/2 + (5*d*x)
/2))/128 + (479*B*sin((7*c)/2 + (7*d*x)/2))/128 + (11*B*sin((9*c)/2 + (9*d*x)/2))/128 + (213*B*sin((11*c)/2 +
(11*d*x)/2))/128 - (3*B*sin((13*c)/2 + (13*d*x)/2))/128 + (35*B*sin((15*c)/2 + (15*d*x)/2))/128))/(40320*d*cos
(c/2 - pi/4 + (d*x)/2)^7*cos(c/2 + pi/4 + (d*x)/2)^9)