\(\int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx\) [965]

Optimal result

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

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
 

Mathematica [A] (verified)

Time = 5.49 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.10 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a \left (35 (A+B) \sec ^9(c+d x)+315 A \sec ^8(c+d x) \tan (c+d x)-105 (8 A-B) \sec ^6(c+d x) \tan ^3(c+d x)+126 (8 A-B) \sec ^4(c+d x) \tan ^5(c+d x)-72 (8 A-B) \sec ^2(c+d x) \tan ^7(c+d x)+16 (8 A-B) \tan ^9(c+d x)\right )}{315 d} \] Input:

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

Output:

(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)*Tan[ 
c + d*x]^9))/(315*d)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.73, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3334, 3042, 4254, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (c+d x)+a) (A+B \sin (c+d x))}{\cos (c+d x)^{10}}dx\)

\(\Big \downarrow \) 3334

\(\displaystyle \frac {1}{9} a (8 A-B) \int \sec ^8(c+d x)dx+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} a (8 A-B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^8dx+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)}{9 d}\)

\(\Big \downarrow \) 4254

\(\displaystyle \frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)}{9 d}-\frac {a (8 A-B) \int \left (\tan ^6(c+d x)+3 \tan ^4(c+d x)+3 \tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{9 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)}{9 d}-\frac {a (8 A-B) \left (-\frac {1}{7} \tan ^7(c+d x)-\frac {3}{5} \tan ^5(c+d x)-\tan ^3(c+d x)-\tan (c+d x)\right )}{9 d}\)

Input:

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

Output:

((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x]))/(9*d) - (a*(8*A - B)*(-Tan[c 
 + d*x] - Tan[c + d*x]^3 - (3*Tan[c + d*x]^5)/5 - Tan[c + d*x]^7/7))/(9*d)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3334

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] + Simp[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 4254

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

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.33

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

Input:

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

Output:

1/d*(1/9*a*A/cos(d*x+c)^9+a*B*(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*a*B/cos(d*x+c)^9)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.55 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {16 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{8} - 8 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{6} - 2 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{4} - {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{2} - 5 \, {\left (A - 8 \, B\right )} a + {\left (16 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{6} + 8 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{4} + 6 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{2} + 5 \, {\left (8 \, A - B\right )} a\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{7}\right )}} \] Input:

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

Output:

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.06 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {{\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} A a + {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} B a + \frac {35 \, A a}{\cos \left (d x + c\right )^{9}} + \frac {35 \, B a}{\cos \left (d x + c\right )^{9}}}{315 \, d} \] Input:

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

Output:

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 = 0.19 (sec) , antiderivative size = 465, normalized size of antiderivative = 3.91 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx =\text {Too large to display} \] Input:

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

Output:

-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 + 991 
83*A*a*tan(1/2*d*x + 1/2*c)^2 - 28161*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 + 1 
1025*B*a*tan(1/2*d*x + 1/2*c)^8 - 322560*A*a*tan(1/2*d*x + 1/2*c)^7 - 4788 
0*B*a*tan(1/2*d*x + 1/2*c)^7 + 976500*A*a*tan(1/2*d*x + 1/2*c)^6 + 117180* 
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 + 2037294*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 - 106008*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*ta 
n(1/2*d*x + 1/2*c)^2 - 219456*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 = 39.67 (sec) , antiderivative size = 416, normalized size of antiderivative = 3.50 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx =\text {Too large to display} \] Input:

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

Output:

-(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*cos((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*co 
s((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 + (501*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))/12 
8 + (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)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 515, normalized size of antiderivative = 4.33 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a \left (35 a +35 b -280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a -35 \sin \left (d x +c \right ) b -16 \sin \left (d x +c \right )^{8} b -128 \sin \left (d x +c \right )^{7} a +16 \sin \left (d x +c \right )^{7} b -448 \sin \left (d x +c \right )^{6} a +56 \sin \left (d x +c \right )^{6} b +448 \sin \left (d x +c \right )^{5} a -56 \sin \left (d x +c \right )^{5} b +560 \sin \left (d x +c \right )^{4} a -70 \sin \left (d x +c \right )^{4} b -560 \sin \left (d x +c \right )^{3} a +70 \sin \left (d x +c \right )^{3} b -280 \sin \left (d x +c \right )^{2} a +35 \sin \left (d x +c \right )^{2} b +280 \sin \left (d x +c \right ) a +128 \sin \left (d x +c \right )^{8} a +35 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} b +280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a -35 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b +840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a -105 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b -840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a +105 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b -840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a +105 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a -105 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b +280 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a -35 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b -280 \cos \left (d x +c \right ) a +35 \cos \left (d x +c \right ) b \right )}{315 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{7}-\sin \left (d x +c \right )^{6}-3 \sin \left (d x +c \right )^{5}+3 \sin \left (d x +c \right )^{4}+3 \sin \left (d x +c \right )^{3}-3 \sin \left (d x +c \right )^{2}-\sin \left (d x +c \right )+1\right )} \] Input:

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

Output:

(a*( - 280*cos(c + d*x)*sin(c + d*x)**7*a + 35*cos(c + d*x)*sin(c + d*x)** 
7*b + 280*cos(c + d*x)*sin(c + d*x)**6*a - 35*cos(c + d*x)*sin(c + d*x)**6 
*b + 840*cos(c + d*x)*sin(c + d*x)**5*a - 105*cos(c + d*x)*sin(c + d*x)**5 
*b - 840*cos(c + d*x)*sin(c + d*x)**4*a + 105*cos(c + d*x)*sin(c + d*x)**4 
*b - 840*cos(c + d*x)*sin(c + d*x)**3*a + 105*cos(c + d*x)*sin(c + d*x)**3 
*b + 840*cos(c + d*x)*sin(c + d*x)**2*a - 105*cos(c + d*x)*sin(c + d*x)**2 
*b + 280*cos(c + d*x)*sin(c + d*x)*a - 35*cos(c + d*x)*sin(c + d*x)*b - 28 
0*cos(c + d*x)*a + 35*cos(c + d*x)*b + 128*sin(c + d*x)**8*a - 16*sin(c + 
d*x)**8*b - 128*sin(c + d*x)**7*a + 16*sin(c + d*x)**7*b - 448*sin(c + d*x 
)**6*a + 56*sin(c + d*x)**6*b + 448*sin(c + d*x)**5*a - 56*sin(c + d*x)**5 
*b + 560*sin(c + d*x)**4*a - 70*sin(c + d*x)**4*b - 560*sin(c + d*x)**3*a 
+ 70*sin(c + d*x)**3*b - 280*sin(c + d*x)**2*a + 35*sin(c + d*x)**2*b + 28 
0*sin(c + d*x)*a - 35*sin(c + d*x)*b + 35*a + 35*b))/(315*cos(c + d*x)*d*( 
sin(c + d*x)**7 - sin(c + d*x)**6 - 3*sin(c + d*x)**5 + 3*sin(c + d*x)**4 
+ 3*sin(c + d*x)**3 - 3*sin(c + d*x)**2 - sin(c + d*x) + 1))