Integrand size = 31, antiderivative size = 140 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))^3}{9 d}+\frac {2 (2 A-B) \sec ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{21 d}+\frac {5 a^3 (2 A-B) \tan (c+d x)}{21 d}+\frac {10 a^3 (2 A-B) \tan ^3(c+d x)}{63 d}+\frac {a^3 (2 A-B) \tan ^5(c+d x)}{21 d} \]
1/9*(A+B)*sec(d*x+c)^9*(a+a*sin(d*x+c))^3/d+2/21*(2*A-B)*sec(d*x+c)^7*(a^3 +a^3*sin(d*x+c))/d+5/21*a^3*(2*A-B)*tan(d*x+c)/d+10/63*a^3*(2*A-B)*tan(d*x +c)^3/d+1/21*a^3*(2*A-B)*tan(d*x+c)^5/d
Time = 0.44 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.26 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3 (-42 B+27 (-2 A+B) \cos (2 (c+d x))+12 (-2 A+B) \cos (4 (c+d x))+2 A \cos (6 (c+d x))-B \cos (6 (c+d x))-72 A \sin (c+d x)+36 B \sin (c+d x)-4 A \sin (3 (c+d x))+2 B \sin (3 (c+d x))+12 A \sin (5 (c+d x))-6 B \sin (5 (c+d x)))}{252 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^9 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
-1/252*(a^3*(-42*B + 27*(-2*A + B)*Cos[2*(c + d*x)] + 12*(-2*A + B)*Cos[4* (c + d*x)] + 2*A*Cos[6*(c + d*x)] - B*Cos[6*(c + d*x)] - 72*A*Sin[c + d*x] + 36*B*Sin[c + d*x] - 4*A*Sin[3*(c + d*x)] + 2*B*Sin[3*(c + d*x)] + 12*A* Sin[5*(c + d*x)] - 6*B*Sin[5*(c + d*x)]))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^9*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)
Time = 0.48 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.83, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3042, 3334, 3042, 3155, 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.
| \(\displaystyle \int \sec ^{10}(c+d x) (a \sin (c+d x)+a)^3 (A+B \sin (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^3 (A+B \sin (c+d x))}{\cos (c+d x)^{10}}dx\) |
| \(\Big \downarrow \) 3334 |
| \(\displaystyle \frac {1}{3} a (2 A-B) \int \sec ^8(c+d x) (\sin (c+d x) a+a)^2dx+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} a (2 A-B) \int \frac {(\sin (c+d x) a+a)^2}{\cos (c+d x)^8}dx+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3155 |
| \(\displaystyle \frac {1}{3} a (2 A-B) \left (\frac {5}{7} a^2 \int \sec ^6(c+d x)dx+\frac {2 \sec ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{7 d}\right )+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} a (2 A-B) \left (\frac {5}{7} a^2 \int \csc \left (c+d x+\frac {\pi }{2}\right )^6dx+\frac {2 \sec ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{7 d}\right )+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{3} a (2 A-B) \left (\frac {2 \sec ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{7 d}-\frac {5 a^2 \int \left (\tan ^4(c+d x)+2 \tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{7 d}\right )+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} a (2 A-B) \left (\frac {2 \sec ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{7 d}-\frac {5 a^2 \left (-\frac {1}{5} \tan ^5(c+d x)-\frac {2}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{7 d}\right )+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^3}{9 d}\) |
((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x])^3)/(9*d) + (a*(2*A - B)*((2*S ec[c + d*x]^7*(a^2 + a^2*Sin[c + d*x]))/(7*d) - (5*a^2*(-Tan[c + d*x] - (2 *Tan[c + d*x]^3)/3 - Tan[c + d*x]^5/5))/(7*d)))/3
3.11.2.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[-2*b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(p + 1))), x] + Simp[b^2*((2*m + p - 1)/(g^2*(p + 1))) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2), x], x] /; FreeQ [{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && Int egersQ[2*m, 2*p]
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]
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]
Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.29
| method | result | size |
| risch | \(\frac {16 i a^{3} \left (72 i A \,{\mathrm e}^{5 i \left (d x +c \right )}-36 i B \,{\mathrm e}^{5 i \left (d x +c \right )}+42 B \,{\mathrm e}^{6 i \left (d x +c \right )}+4 i A \,{\mathrm e}^{3 i \left (d x +c \right )}+54 A \,{\mathrm e}^{4 i \left (d x +c \right )}-2 i B \,{\mathrm e}^{3 i \left (d x +c \right )}-27 B \,{\mathrm e}^{4 i \left (d x +c \right )}-12 i A \,{\mathrm e}^{i \left (d x +c \right )}+24 A \,{\mathrm e}^{2 i \left (d x +c \right )}+6 i B \,{\mathrm e}^{i \left (d x +c \right )}-12 B \,{\mathrm e}^{2 i \left (d x +c \right )}-2 A +B \right )}{63 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{9} d}\) | \(181\) |
| parallelrisch | \(-\frac {2 \left (A \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 A +B \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {13 A}{3}-2 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (A +3 B \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 A \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 \left (7 A -B \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {2 \left (17 A +2 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {2 \left (-25 A +9 B \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (235 A -128 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63}+\frac {\left (13 A +25 B \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21}+\frac {\left (-17 A -2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{21}+\frac {19 A}{63}+\frac {B}{63}\right ) a^{3}}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{9}}\) | \(240\) |
| derivativedivides | \(\frac {A \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )+B \,a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )+3 A \,a^{3} \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 )+3 B \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )+\frac {A \,a^{3}}{3 \cos \left (d x +c \right )^{9}}+3 B \,a^{3} \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^{3} \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^{3}}{9 \cos \left (d x +c \right )^{9}}}{d}\) | \(535\) |
| default | \(\frac {A \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )+B \,a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )+3 A \,a^{3} \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 )+3 B \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )+\frac {A \,a^{3}}{3 \cos \left (d x +c \right )^{9}}+3 B \,a^{3} \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^{3} \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^{3}}{9 \cos \left (d x +c \right )^{9}}}{d}\) | \(535\) |
16/63*I*a^3*(72*I*A*exp(5*I*(d*x+c))-36*I*B*exp(5*I*(d*x+c))+42*B*exp(6*I* (d*x+c))+4*I*A*exp(3*I*(d*x+c))+54*A*exp(4*I*(d*x+c))-2*I*B*exp(3*I*(d*x+c ))-27*B*exp(4*I*(d*x+c))-12*I*A*exp(I*(d*x+c))+24*A*exp(2*I*(d*x+c))+6*I*B *exp(I*(d*x+c))-12*B*exp(2*I*(d*x+c))-2*A+B)/(exp(I*(d*x+c))+I)^3/(exp(I*( d*x+c))-I)^9/d
Time = 0.29 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.34 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {8 \, {\left (2 \, A - B\right )} a^{3} \cos \left (d x + c\right )^{6} - 36 \, {\left (2 \, A - B\right )} a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (2 \, A - B\right )} a^{3} \cos \left (d x + c\right )^{2} + 7 \, {\left (A - 2 \, B\right )} a^{3} + {\left (24 \, {\left (2 \, A - B\right )} a^{3} \cos \left (d x + c\right )^{4} - 20 \, {\left (2 \, A - B\right )} a^{3} \cos \left (d x + c\right )^{2} - 7 \, {\left (2 \, A - B\right )} a^{3}\right )} \sin \left (d x + c\right )}{63 \, {\left (3 \, d \cos \left (d x + c\right )^{5} - 4 \, d \cos \left (d x + c\right )^{3} - {\left (d \cos \left (d x + c\right )^{5} - 4 \, d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
1/63*(8*(2*A - B)*a^3*cos(d*x + c)^6 - 36*(2*A - B)*a^3*cos(d*x + c)^4 + 1 5*(2*A - B)*a^3*cos(d*x + c)^2 + 7*(A - 2*B)*a^3 + (24*(2*A - B)*a^3*cos(d *x + c)^4 - 20*(2*A - B)*a^3*cos(d*x + c)^2 - 7*(2*A - B)*a^3)*sin(d*x + c ))/(3*d*cos(d*x + c)^5 - 4*d*cos(d*x + c)^3 - (d*cos(d*x + c)^5 - 4*d*cos( d*x + c)^3)*sin(d*x + c))
Timed out. \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (130) = 260\).
Time = 0.22 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.93 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {{\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} A a^{3} + 3 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} A a^{3} + 3 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} B a^{3} + {\left (35 \, \tan \left (d x + c\right )^{9} + 90 \, \tan \left (d x + c\right )^{7} + 63 \, \tan \left (d x + c\right )^{5}\right )} B a^{3} - \frac {5 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} A a^{3}}{\cos \left (d x + c\right )^{9}} - \frac {15 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} B a^{3}}{\cos \left (d x + c\right )^{9}} + \frac {105 \, A a^{3}}{\cos \left (d x + c\right )^{9}} + \frac {35 \, B a^{3}}{\cos \left (d x + c\right )^{9}}}{315 \, d} \]
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^3 + 3*(35*tan(d*x + c)^9 + 135*tan( d*x + c)^7 + 189*tan(d*x + c)^5 + 105*tan(d*x + c)^3)*A*a^3 + 3*(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^3 + (35*tan(d*x + c)^9 + 90*tan(d*x + c)^7 + 63*tan(d*x + c)^5)*B*a^3 - 5*(9*cos(d*x + c)^2 - 7)*A*a^3/cos(d*x + c)^9 - 15*(9*cos(d*x + c)^2 - 7) *B*a^3/cos(d*x + c)^9 + 105*A*a^3/cos(d*x + c)^9 + 35*B*a^3/cos(d*x + c)^9 )/d
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (130) = 260\).
Time = 0.37 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.81 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {\frac {21 \, {\left (21 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 19 \, A a^{3} - 13 \, B a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}} + \frac {3591 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 315 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 19656 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 756 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 56196 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4200 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 95760 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 11340 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 107730 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 14994 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 79464 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13356 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 38484 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6768 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10944 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2196 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1615 \, A a^{3} - 209 \, B a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{9}}}{2016 \, d} \]
-1/2016*(21*(21*A*a^3*tan(1/2*d*x + 1/2*c)^2 - 15*B*a^3*tan(1/2*d*x + 1/2* c)^2 + 36*A*a^3*tan(1/2*d*x + 1/2*c) - 24*B*a^3*tan(1/2*d*x + 1/2*c) + 19* A*a^3 - 13*B*a^3)/(tan(1/2*d*x + 1/2*c) + 1)^3 + (3591*A*a^3*tan(1/2*d*x + 1/2*c)^8 + 315*B*a^3*tan(1/2*d*x + 1/2*c)^8 - 19656*A*a^3*tan(1/2*d*x + 1 /2*c)^7 + 756*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 56196*A*a^3*tan(1/2*d*x + 1/2 *c)^6 - 4200*B*a^3*tan(1/2*d*x + 1/2*c)^6 - 95760*A*a^3*tan(1/2*d*x + 1/2* c)^5 + 11340*B*a^3*tan(1/2*d*x + 1/2*c)^5 + 107730*A*a^3*tan(1/2*d*x + 1/2 *c)^4 - 14994*B*a^3*tan(1/2*d*x + 1/2*c)^4 - 79464*A*a^3*tan(1/2*d*x + 1/2 *c)^3 + 13356*B*a^3*tan(1/2*d*x + 1/2*c)^3 + 38484*A*a^3*tan(1/2*d*x + 1/2 *c)^2 - 6768*B*a^3*tan(1/2*d*x + 1/2*c)^2 - 10944*A*a^3*tan(1/2*d*x + 1/2* c) + 2196*B*a^3*tan(1/2*d*x + 1/2*c) + 1615*A*a^3 - 209*B*a^3)/(tan(1/2*d* x + 1/2*c) - 1)^9)/d
Time = 13.84 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.30 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {63\,A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}-\frac {171\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {145\,A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {49\,A\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}+\frac {A\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{2}-\frac {21\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {75\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {21\,B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {41\,B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {7\,B\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}-\frac {B\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{4}-\frac {617\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {329\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}-\frac {145\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}+\frac {113\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}+\frac {115\,A\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}-\frac {19\,A\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}+\frac {109\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {35\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}-\frac {43\,B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}+\frac {59\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}-\frac {47\,B\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}-\frac {B\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}\right )}{2016\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^3\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^9} \]
-(a^3*cos(c/2 + (d*x)/2)*((63*A*cos((5*c)/2 + (5*d*x)/2))/8 - (171*A*cos(( 3*c)/2 + (3*d*x)/2))/8 - (145*A*cos((7*c)/2 + (7*d*x)/2))/16 + (49*A*cos(( 9*c)/2 + (9*d*x)/2))/16 + (A*cos((11*c)/2 + (11*d*x)/2))/2 - (21*B*cos(c/2 + (d*x)/2))/2 + (75*B*cos((3*c)/2 + (3*d*x)/2))/8 - (21*B*cos((5*c)/2 + ( 5*d*x)/2))/8 + (41*B*cos((7*c)/2 + (7*d*x)/2))/16 + (7*B*cos((9*c)/2 + (9* d*x)/2))/16 - (B*cos((11*c)/2 + (11*d*x)/2))/4 - (617*A*sin(c/2 + (d*x)/2) )/16 + (329*A*sin((3*c)/2 + (3*d*x)/2))/16 - (145*A*sin((5*c)/2 + (5*d*x)/ 2))/32 + (113*A*sin((7*c)/2 + (7*d*x)/2))/32 + (115*A*sin((9*c)/2 + (9*d*x )/2))/32 - (19*A*sin((11*c)/2 + (11*d*x)/2))/32 + (109*B*sin(c/2 + (d*x)/2 ))/16 + (35*B*sin((3*c)/2 + (3*d*x)/2))/16 - (43*B*sin((5*c)/2 + (5*d*x)/2 ))/32 + (59*B*sin((7*c)/2 + (7*d*x)/2))/32 - (47*B*sin((9*c)/2 + (9*d*x)/2 ))/32 - (B*sin((11*c)/2 + (11*d*x)/2))/32))/(2016*d*cos(c/2 - pi/4 + (d*x) /2)^3*cos(c/2 + pi/4 + (d*x)/2)^9)