Integrand size = 31, antiderivative size = 154 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2 (7 A-2 B) \sec ^7(c+d x)}{63 d}+\frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))^2}{9 d}+\frac {a^2 (7 A-2 B) \tan (c+d x)}{9 d}+\frac {a^2 (7 A-2 B) \tan ^3(c+d x)}{9 d}+\frac {a^2 (7 A-2 B) \tan ^5(c+d x)}{15 d}+\frac {a^2 (7 A-2 B) \tan ^7(c+d x)}{63 d} \] Output:
1/63*a^2*(7*A-2*B)*sec(d*x+c)^7/d+1/9*(A+B)*sec(d*x+c)^9*(a+a*sin(d*x+c))^ 2/d+1/9*a^2*(7*A-2*B)*tan(d*x+c)/d+1/9*a^2*(7*A-2*B)*tan(d*x+c)^3/d+1/15*a ^2*(7*A-2*B)*tan(d*x+c)^5/d+1/63*a^2*(7*A-2*B)*tan(d*x+c)^7/d
Time = 0.58 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.01 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2 \left (5 (14 A+5 B) \sec ^9(c+d x)+315 A \sec ^8(c+d x) \tan (c+d x)+45 B \sec ^7(c+d x) \tan ^2(c+d x)-105 (7 A-2 B) \sec ^6(c+d x) \tan ^3(c+d x)+126 (7 A-2 B) \sec ^4(c+d x) \tan ^5(c+d x)-72 (7 A-2 B) \sec ^2(c+d x) \tan ^7(c+d x)+16 (7 A-2 B) \tan ^9(c+d x)\right )}{315 d} \] Input:
Integrate[Sec[c + d*x]^10*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]
Output:
(a^2*(5*(14*A + 5*B)*Sec[c + d*x]^9 + 315*A*Sec[c + d*x]^8*Tan[c + d*x] + 45*B*Sec[c + d*x]^7*Tan[c + d*x]^2 - 105*(7*A - 2*B)*Sec[c + d*x]^6*Tan[c + d*x]^3 + 126*(7*A - 2*B)*Sec[c + d*x]^4*Tan[c + d*x]^5 - 72*(7*A - 2*B)* Sec[c + d*x]^2*Tan[c + d*x]^7 + 16*(7*A - 2*B)*Tan[c + d*x]^9))/(315*d)
Time = 0.48 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.71, 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, 3148, 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.
| \(\displaystyle \int \sec ^{10}(c+d x) (a \sin (c+d x)+a)^2 (A+B \sin (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^2 (A+B \sin (c+d x))}{\cos (c+d x)^{10}}dx\) |
| \(\Big \downarrow \) 3334 |
| \(\displaystyle \frac {1}{9} a (7 A-2 B) \int \sec ^8(c+d x) (\sin (c+d x) a+a)dx+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} a (7 A-2 B) \int \frac {\sin (c+d x) a+a}{\cos (c+d x)^8}dx+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{9} a (7 A-2 B) \left (a \int \sec ^8(c+d x)dx+\frac {a \sec ^7(c+d x)}{7 d}\right )+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} a (7 A-2 B) \left (a \int \csc \left (c+d x+\frac {\pi }{2}\right )^8dx+\frac {a \sec ^7(c+d x)}{7 d}\right )+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{9} a (7 A-2 B) \left (\frac {a \sec ^7(c+d x)}{7 d}-\frac {a \int \left (\tan ^6(c+d x)+3 \tan ^4(c+d x)+3 \tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{d}\right )+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^2}{9 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^2}{9 d}+\frac {1}{9} a (7 A-2 B) \left (\frac {a \sec ^7(c+d x)}{7 d}-\frac {a \left (-\frac {1}{7} \tan ^7(c+d x)-\frac {3}{5} \tan ^5(c+d x)-\tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )\) |
Input:
Int[Sec[c + d*x]^10*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]
Output:
((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x])^2)/(9*d) + (a*(7*A - 2*B)*((a *Sec[c + d*x]^7)/(7*d) - (a*(-Tan[c + d*x] - Tan[c + d*x]^3 - (3*Tan[c + d *x]^5)/5 - Tan[c + d*x]^7/7))/d))/9
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
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 = 1.66 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.62
| method | result | size |
| risch | \(-\frac {32 \left (133 i A \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+7 i A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-38 i B \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-2 i B \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+315 i A \,a^{2} {\mathrm e}^{6 i \left (d x +c \right )}-90 i B \,a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+140 A \,a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+2 i B \,a^{2}+112 A \,a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-40 B \,a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-32 B \,a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-7 i A \,a^{2}+28 A \,a^{2} {\mathrm e}^{i \left (d x +c \right )}+180 B \,a^{2} {\mathrm e}^{7 i \left (d x +c \right )}-8 B \,a^{2} {\mathrm e}^{i \left (d x +c \right )}\right )}{315 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{9} d}\) | \(250\) |
| parallelrisch | \(-\frac {2 \left (A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+\left (B -2 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\frac {2 \left (A -2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3}+\frac {4 \left (4 A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3}+\frac {\left (-23 A +28 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{15}+\frac {\left (-118 A +23 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{15}+\frac {172 \left (-\frac {2 B}{7}+A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{15}+\frac {8 \left (4 A +\frac {67 B}{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{15}+\frac {\left (-269 A +\frac {88 B}{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{45}+\frac {\left (10 A -\frac {11 B}{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{9}+\frac {2 \left (19 A -\frac {2 B}{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{9}+\frac {4 \left (-4 A +\frac {17 B}{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{9}+\frac {\left (A -\frac {20 B}{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{9}+\frac {2 A}{9}+\frac {5 B}{63}\right ) a^{2}}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{9}}\) | \(282\) |
| derivativedivides | \(\frac {A \,a^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )+a^{2} B \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )+\frac {2 A \,a^{2}}{9 \cos \left (d x +c \right )^{9}}+2 a^{2} B \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )-A \,a^{2} \left (-\frac {128}{315}-\frac {\sec \left (d x +c \right )^{8}}{9}-\frac {8 \sec \left (d x +c \right )^{6}}{63}-\frac {16 \sec \left (d x +c \right )^{4}}{105}-\frac {64 \sec \left (d x +c \right )^{2}}{315}\right ) \tan \left (d x +c \right )+\frac {a^{2} B}{9 \cos \left (d x +c \right )^{9}}}{d}\) | \(359\) |
| default | \(\frac {A \,a^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )+a^{2} B \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )+\frac {2 A \,a^{2}}{9 \cos \left (d x +c \right )^{9}}+2 a^{2} B \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )-A \,a^{2} \left (-\frac {128}{315}-\frac {\sec \left (d x +c \right )^{8}}{9}-\frac {8 \sec \left (d x +c \right )^{6}}{63}-\frac {16 \sec \left (d x +c \right )^{4}}{105}-\frac {64 \sec \left (d x +c \right )^{2}}{315}\right ) \tan \left (d x +c \right )+\frac {a^{2} B}{9 \cos \left (d x +c \right )^{9}}}{d}\) | \(359\) |
Input:
int(sec(d*x+c)^10*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x,method=_RETURNVERB OSE)
Output:
-32/315*(133*I*A*a^2*exp(4*I*(d*x+c))+7*I*A*a^2*exp(2*I*(d*x+c))-38*I*B*a^ 2*exp(4*I*(d*x+c))-2*I*B*a^2*exp(2*I*(d*x+c))+315*I*A*a^2*exp(6*I*(d*x+c)) -90*I*B*a^2*exp(6*I*(d*x+c))+140*A*a^2*exp(5*I*(d*x+c))+2*I*B*a^2+112*A*a^ 2*exp(3*I*(d*x+c))-40*B*a^2*exp(5*I*(d*x+c))-32*B*a^2*exp(3*I*(d*x+c))-7*I *A*a^2+28*A*a^2*exp(I*(d*x+c))+180*B*a^2*exp(7*I*(d*x+c))-8*B*a^2*exp(I*(d *x+c)))/(exp(I*(d*x+c))+I)^5/(exp(I*(d*x+c))-I)^9/d
Time = 0.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.28 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {32 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} - 16 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} - 4 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} - 7 \, {\left (2 \, A - 7 \, B\right )} a^{2} - {\left (16 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} - 24 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} - 10 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} - 7 \, {\left (7 \, A - 2 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{7} + 2 \, d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{5}\right )}} \] Input:
integrate(sec(d*x+c)^10*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm=" fricas")
Output:
-1/315*(32*(7*A - 2*B)*a^2*cos(d*x + c)^6 - 16*(7*A - 2*B)*a^2*cos(d*x + c )^4 - 4*(7*A - 2*B)*a^2*cos(d*x + c)^2 - 7*(2*A - 7*B)*a^2 - (16*(7*A - 2* B)*a^2*cos(d*x + c)^6 - 24*(7*A - 2*B)*a^2*cos(d*x + c)^4 - 10*(7*A - 2*B) *a^2*cos(d*x + c)^2 - 7*(7*A - 2*B)*a^2)*sin(d*x + c))/(d*cos(d*x + c)^7 + 2*d*cos(d*x + c)^5*sin(d*x + c) - 2*d*cos(d*x + c)^5)
Timed out. \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**10*(a+a*sin(d*x+c))**2*(A+B*sin(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.34 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {{\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} A a^{2} + {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} A a^{2} + 2 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} B a^{2} - \frac {5 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} B a^{2}}{\cos \left (d x + c\right )^{9}} + \frac {70 \, A a^{2}}{\cos \left (d x + c\right )^{9}} + \frac {35 \, B a^{2}}{\cos \left (d x + c\right )^{9}}}{315 \, d} \] Input:
integrate(sec(d*x+c)^10*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm=" maxima")
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^2 + (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^2 + 2*(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 ^2 - 5*(9*cos(d*x + c)^2 - 7)*B*a^2/cos(d*x + c)^9 + 70*A*a^2/cos(d*x + c) ^9 + 35*B*a^2/cos(d*x + c)^9)/d
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (142) = 284\).
Time = 0.20 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.99 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)^10*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm=" giac")
Output:
-1/20160*(21*(435*A*a^2*tan(1/2*d*x + 1/2*c)^4 - 225*B*a^2*tan(1/2*d*x + 1 /2*c)^4 + 1470*A*a^2*tan(1/2*d*x + 1/2*c)^3 - 690*B*a^2*tan(1/2*d*x + 1/2* c)^3 + 2060*A*a^2*tan(1/2*d*x + 1/2*c)^2 - 940*B*a^2*tan(1/2*d*x + 1/2*c)^ 2 + 1330*A*a^2*tan(1/2*d*x + 1/2*c) - 590*B*a^2*tan(1/2*d*x + 1/2*c) + 353 *A*a^2 - 163*B*a^2)/(tan(1/2*d*x + 1/2*c) + 1)^5 + (31185*A*a^2*tan(1/2*d* x + 1/2*c)^8 + 4725*B*a^2*tan(1/2*d*x + 1/2*c)^8 - 185220*A*a^2*tan(1/2*d* x + 1/2*c)^7 - 11340*B*a^2*tan(1/2*d*x + 1/2*c)^7 + 546840*A*a^2*tan(1/2*d *x + 1/2*c)^6 + 15120*B*a^2*tan(1/2*d*x + 1/2*c)^6 - 961380*A*a^2*tan(1/2* d*x + 1/2*c)^5 + 3780*B*a^2*tan(1/2*d*x + 1/2*c)^5 + 1101618*A*a^2*tan(1/2 *d*x + 1/2*c)^4 - 24318*B*a^2*tan(1/2*d*x + 1/2*c)^4 - 828492*A*a^2*tan(1/ 2*d*x + 1/2*c)^3 + 33852*B*a^2*tan(1/2*d*x + 1/2*c)^3 + 404208*A*a^2*tan(1 /2*d*x + 1/2*c)^2 - 19368*B*a^2*tan(1/2*d*x + 1/2*c)^2 - 116172*A*a^2*tan( 1/2*d*x + 1/2*c) + 6732*B*a^2*tan(1/2*d*x + 1/2*c) + 16373*A*a^2 - 223*B*a ^2)/(tan(1/2*d*x + 1/2*c) - 1)^9)/d
Time = 38.92 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.40 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {a^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {455\,A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}-\frac {1575\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{32}-35\,A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )+7\,A\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )-\frac {259\,A\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}+\frac {35\,A\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{32}-45\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1755\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}-\frac {1115\,B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}+10\,B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )-2\,B\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )+\frac {103\,B\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{64}+\frac {25\,B\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{64}-\frac {623\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+77\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )-\frac {441\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {175\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}-\frac {35\,A\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{8}+\frac {21\,A\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{8}+\frac {7\,A\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{4}+\frac {131\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {49\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}+\frac {27\,B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{16}+\frac {125\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}-\frac {25\,B\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}+\frac {33\,B\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{16}-\frac {B\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{2}\right )}{20160\,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 )}^9} \] Input:
int(((A + B*sin(c + d*x))*(a + a*sin(c + d*x))^2)/cos(c + d*x)^10,x)
Output:
-(a^2*cos(c/2 + (d*x)/2)*((455*A*cos((5*c)/2 + (5*d*x)/2))/32 - (1575*A*co s((3*c)/2 + (3*d*x)/2))/32 - 35*A*cos((7*c)/2 + (7*d*x)/2) + 7*A*cos((9*c) /2 + (9*d*x)/2) - (259*A*cos((11*c)/2 + (11*d*x)/2))/32 + (35*A*cos((13*c) /2 + (13*d*x)/2))/32 - 45*B*cos(c/2 + (d*x)/2) + (1755*B*cos((3*c)/2 + (3* d*x)/2))/64 - (1115*B*cos((5*c)/2 + (5*d*x)/2))/64 + 10*B*cos((7*c)/2 + (7 *d*x)/2) - 2*B*cos((9*c)/2 + (9*d*x)/2) + (103*B*cos((11*c)/2 + (11*d*x)/2 ))/64 + (25*B*cos((13*c)/2 + (13*d*x)/2))/64 - (623*A*sin(c/2 + (d*x)/2))/ 4 + 77*A*sin((3*c)/2 + (3*d*x)/2) - (441*A*sin((5*c)/2 + (5*d*x)/2))/8 + ( 175*A*sin((7*c)/2 + (7*d*x)/2))/8 - (35*A*sin((9*c)/2 + (9*d*x)/2))/8 + (2 1*A*sin((11*c)/2 + (11*d*x)/2))/8 + (7*A*sin((13*c)/2 + (13*d*x)/2))/4 + ( 131*B*sin(c/2 + (d*x)/2))/8 + (49*B*sin((3*c)/2 + (3*d*x)/2))/8 + (27*B*si n((5*c)/2 + (5*d*x)/2))/16 + (125*B*sin((7*c)/2 + (7*d*x)/2))/16 - (25*B*s in((9*c)/2 + (9*d*x)/2))/16 + (33*B*sin((11*c)/2 + (11*d*x)/2))/16 - (B*si n((13*c)/2 + (13*d*x)/2))/2))/(20160*d*cos(c/2 - pi/4 + (d*x)/2)^5*cos(c/2 + pi/4 + (d*x)/2)^9)
Time = 0.18 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.93 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^{2} \left (140 a +50 b -100 \sin \left (d x +c \right ) b +224 \sin \left (d x +c \right )^{7} a -64 \sin \left (d x +c \right )^{7} b -448 \sin \left (d x +c \right )^{6} a +128 \sin \left (d x +c \right )^{6} b -336 \sin \left (d x +c \right )^{5} a +96 \sin \left (d x +c \right )^{5} b +1120 \sin \left (d x +c \right )^{4} a -320 \sin \left (d x +c \right )^{4} b -140 \sin \left (d x +c \right )^{3} a +40 \sin \left (d x +c \right )^{3} b -840 \sin \left (d x +c \right )^{2} a +240 \sin \left (d x +c \right )^{2} b +350 \sin \left (d x +c \right ) a -175 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a +50 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b +350 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a -100 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b +175 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a -50 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b -700 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a +200 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +175 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a -50 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b +350 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a -100 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b -175 \cos \left (d x +c \right ) a +50 \cos \left (d x +c \right ) b \right )}{630 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{6}-2 \sin \left (d x +c \right )^{5}-\sin \left (d x +c \right )^{4}+4 \sin \left (d x +c \right )^{3}-\sin \left (d x +c \right )^{2}-2 \sin \left (d x +c \right )+1\right )} \] Input:
int(sec(d*x+c)^10*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x)
Output:
(a**2*( - 175*cos(c + d*x)*sin(c + d*x)**6*a + 50*cos(c + d*x)*sin(c + d*x )**6*b + 350*cos(c + d*x)*sin(c + d*x)**5*a - 100*cos(c + d*x)*sin(c + d*x )**5*b + 175*cos(c + d*x)*sin(c + d*x)**4*a - 50*cos(c + d*x)*sin(c + d*x) **4*b - 700*cos(c + d*x)*sin(c + d*x)**3*a + 200*cos(c + d*x)*sin(c + d*x) **3*b + 175*cos(c + d*x)*sin(c + d*x)**2*a - 50*cos(c + d*x)*sin(c + d*x)* *2*b + 350*cos(c + d*x)*sin(c + d*x)*a - 100*cos(c + d*x)*sin(c + d*x)*b - 175*cos(c + d*x)*a + 50*cos(c + d*x)*b + 224*sin(c + d*x)**7*a - 64*sin(c + d*x)**7*b - 448*sin(c + d*x)**6*a + 128*sin(c + d*x)**6*b - 336*sin(c + d*x)**5*a + 96*sin(c + d*x)**5*b + 1120*sin(c + d*x)**4*a - 320*sin(c + d *x)**4*b - 140*sin(c + d*x)**3*a + 40*sin(c + d*x)**3*b - 840*sin(c + d*x) **2*a + 240*sin(c + d*x)**2*b + 350*sin(c + d*x)*a - 100*sin(c + d*x)*b + 140*a + 50*b))/(630*cos(c + d*x)*d*(sin(c + d*x)**6 - 2*sin(c + d*x)**5 - sin(c + d*x)**4 + 4*sin(c + d*x)**3 - sin(c + d*x)**2 - 2*sin(c + d*x) + 1 ))