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

Optimal. Leaf size=119 \[ \frac {a (8 A-B) \tan ^7(c+d x)}{63 d}+\frac {a (8 A-B) \tan ^5(c+d x)}{15 d}+\frac {a (8 A-B) \tan ^3(c+d x)}{9 d}+\frac {a (8 A-B) \tan (c+d x)}{9 d}+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)}{9 d} \]

[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

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Rubi [A]  time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2855, 3767} \[ \frac {a (8 A-B) \tan ^7(c+d x)}{63 d}+\frac {a (8 A-B) \tan ^5(c+d x)}{15 d}+\frac {a (8 A-B) \tan ^3(c+d x)}{9 d}+\frac {a (8 A-B) \tan (c+d x)}{9 d}+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)}{9 d} \]

Antiderivative was successfully verified.

[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 2855

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 3767

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*} \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 {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)) \operatorname {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*}

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Mathematica [B]  time = 4.34, size = 407, normalized size = 3.42 \[ \frac {a \sec (c) (-85750 (A+B) \cos (c+d x)+17150 A \sin (2 (c+d x))+17150 A \sin (4 (c+d x))+7350 A \sin (6 (c+d x))+1225 A \sin (8 (c+d x))+688128 A \sin (2 c+3 d x)+229376 A \sin (4 c+5 d x)+32768 A \sin (6 c+7 d x)-51450 A \cos (3 (c+d x))-17150 A \cos (5 (c+d x))-2450 A \cos (7 (c+d x))+229376 A \cos (c+2 d x)+229376 A \cos (3 c+4 d x)+98304 A \cos (5 c+6 d x)+16384 A \cos (7 c+8 d x)+1146880 A \sin (d x)+17150 B \sin (2 (c+d x))+17150 B \sin (4 (c+d x))+7350 B \sin (6 (c+d x))+1225 B \sin (8 (c+d x))-86016 B \sin (2 c+3 d x)-28672 B \sin (4 c+5 d x)-4096 B \sin (6 c+7 d x)-51450 B \cos (3 (c+d x))-17150 B \cos (5 (c+d x))-2450 B \cos (7 (c+d x))-28672 B \cos (c+2 d x)-28672 B \cos (3 c+4 d x)-12288 B \cos (5 c+6 d x)-2048 B \cos (7 c+8 d x)+645120 B \cos (c)-143360 B \sin (d x))}{5160960 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^9 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sec[c]*(645120*B*Cos[c] - 85750*(A + B)*Cos[c + d*x] - 51450*A*Cos[3*(c + d*x)] - 51450*B*Cos[3*(c + d*x)]
- 17150*A*Cos[5*(c + d*x)] - 17150*B*Cos[5*(c + d*x)] - 2450*A*Cos[7*(c + d*x)] - 2450*B*Cos[7*(c + d*x)] + 22
9376*A*Cos[c + 2*d*x] - 28672*B*Cos[c + 2*d*x] + 229376*A*Cos[3*c + 4*d*x] - 28672*B*Cos[3*c + 4*d*x] + 98304*
A*Cos[5*c + 6*d*x] - 12288*B*Cos[5*c + 6*d*x] + 16384*A*Cos[7*c + 8*d*x] - 2048*B*Cos[7*c + 8*d*x] + 1146880*A
*Sin[d*x] - 143360*B*Sin[d*x] + 17150*A*Sin[2*(c + d*x)] + 17150*B*Sin[2*(c + d*x)] + 17150*A*Sin[4*(c + d*x)]
+ 17150*B*Sin[4*(c + d*x)] + 7350*A*Sin[6*(c + d*x)] + 7350*B*Sin[6*(c + d*x)] + 1225*A*Sin[8*(c + d*x)] + 12
25*B*Sin[8*(c + d*x)] + 688128*A*Sin[2*c + 3*d*x] - 86016*B*Sin[2*c + 3*d*x] + 229376*A*Sin[4*c + 5*d*x] - 286
72*B*Sin[4*c + 5*d*x] + 32768*A*Sin[6*c + 7*d*x] - 4096*B*Sin[6*c + 7*d*x]))/(5160960*d*(Cos[(c + d*x)/2] - Si
n[(c + d*x)/2])^9*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)

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fricas [A]  time = 0.64, size = 185, normalized size = 1.55 \[ -\frac {16 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{8} - 8 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{6} - 2 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{4} - {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{2} - 5 \, {\left (A - 8 \, B\right )} a + {\left (16 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{6} + 8 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{4} + 6 \, {\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{2} + 5 \, {\left (8 \, A - B\right )} a\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.23, size = 465, normalized size = 3.91 \[ -\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} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.63, size = 158, normalized size = 1.33 \[ \frac {\frac {a A}{9 \cos \left (d x +c \right )^{9}}+a B \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )-a A \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (d x +c \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (d x +c \right )\right )}{315}\right ) \tan \left (d x +c \right )+\frac {a B}{9 \cos \left (d x +c \right )^{9}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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maxima [A]  time = 0.33, size = 126, normalized size = 1.06 \[ \frac {{\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} A a + {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} B a + \frac {35 \, A a}{\cos \left (d x + c\right )^{9}} + \frac {35 \, B a}{\cos \left (d x + c\right )^{9}}}{315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 13.30, size = 416, normalized size = 3.50 \[ -\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} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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