Integrand size = 31, antiderivative size = 179 \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2 (9 A-2 B) \sec ^9(c+d x)}{99 d}+\frac {(A+B) \sec ^{11}(c+d x) (a+a \sin (c+d x))^2}{11 d}+\frac {a^2 (9 A-2 B) \tan (c+d x)}{11 d}+\frac {4 a^2 (9 A-2 B) \tan ^3(c+d x)}{33 d}+\frac {6 a^2 (9 A-2 B) \tan ^5(c+d x)}{55 d}+\frac {4 a^2 (9 A-2 B) \tan ^7(c+d x)}{77 d}+\frac {a^2 (9 A-2 B) \tan ^9(c+d x)}{99 d} \] Output:
1/99*a^2*(9*A-2*B)*sec(d*x+c)^9/d+1/11*(A+B)*sec(d*x+c)^11*(a+a*sin(d*x+c) )^2/d+1/11*a^2*(9*A-2*B)*tan(d*x+c)/d+4/33*a^2*(9*A-2*B)*tan(d*x+c)^3/d+6/ 55*a^2*(9*A-2*B)*tan(d*x+c)^5/d+4/77*a^2*(9*A-2*B)*tan(d*x+c)^7/d+1/99*a^2 *(9*A-2*B)*tan(d*x+c)^9/d
Time = 1.18 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.01 \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2 \left (35 (18 A+7 B) \sec ^{11}(c+d x)+3465 A \sec ^{10}(c+d x) \tan (c+d x)+385 B \sec ^9(c+d x) \tan ^2(c+d x)-1155 (9 A-2 B) \sec ^8(c+d x) \tan ^3(c+d x)+1848 (9 A-2 B) \sec ^6(c+d x) \tan ^5(c+d x)-1584 (9 A-2 B) \sec ^4(c+d x) \tan ^7(c+d x)+704 (9 A-2 B) \sec ^2(c+d x) \tan ^9(c+d x)+128 (-9 A+2 B) \tan ^{11}(c+d x)\right )}{3465 d} \] Input:
Integrate[Sec[c + d*x]^12*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]
Output:
(a^2*(35*(18*A + 7*B)*Sec[c + d*x]^11 + 3465*A*Sec[c + d*x]^10*Tan[c + d*x ] + 385*B*Sec[c + d*x]^9*Tan[c + d*x]^2 - 1155*(9*A - 2*B)*Sec[c + d*x]^8* Tan[c + d*x]^3 + 1848*(9*A - 2*B)*Sec[c + d*x]^6*Tan[c + d*x]^5 - 1584*(9* A - 2*B)*Sec[c + d*x]^4*Tan[c + d*x]^7 + 704*(9*A - 2*B)*Sec[c + d*x]^2*Ta n[c + d*x]^9 + 128*(-9*A + 2*B)*Tan[c + d*x]^11))/(3465*d)
Time = 0.49 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.69, 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.
| \(\displaystyle \int \sec ^{12}(c+d x) (a \sin (c+d x)+a)^2 (A+B \sin (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^2 (A+B \sin (c+d x))}{\cos (c+d x)^{12}}dx\) |
| \(\Big \downarrow \) 3334 |
| \(\displaystyle \frac {1}{11} a (9 A-2 B) \int \sec ^{10}(c+d x) (\sin (c+d x) a+a)dx+\frac {(A+B) \sec ^{11}(c+d x) (a \sin (c+d x)+a)^2}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} a (9 A-2 B) \int \frac {\sin (c+d x) a+a}{\cos (c+d x)^{10}}dx+\frac {(A+B) \sec ^{11}(c+d x) (a \sin (c+d x)+a)^2}{11 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{11} a (9 A-2 B) \left (a \int \sec ^{10}(c+d x)dx+\frac {a \sec ^9(c+d x)}{9 d}\right )+\frac {(A+B) \sec ^{11}(c+d x) (a \sin (c+d x)+a)^2}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} a (9 A-2 B) \left (a \int \csc \left (c+d x+\frac {\pi }{2}\right )^{10}dx+\frac {a \sec ^9(c+d x)}{9 d}\right )+\frac {(A+B) \sec ^{11}(c+d x) (a \sin (c+d x)+a)^2}{11 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{11} a (9 A-2 B) \left (\frac {a \sec ^9(c+d x)}{9 d}-\frac {a \int \left (\tan ^8(c+d x)+4 \tan ^6(c+d x)+6 \tan ^4(c+d x)+4 \tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{d}\right )+\frac {(A+B) \sec ^{11}(c+d x) (a \sin (c+d x)+a)^2}{11 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(A+B) \sec ^{11}(c+d x) (a \sin (c+d x)+a)^2}{11 d}+\frac {1}{11} a (9 A-2 B) \left (\frac {a \sec ^9(c+d x)}{9 d}-\frac {a \left (-\frac {1}{9} \tan ^9(c+d x)-\frac {4}{7} \tan ^7(c+d x)-\frac {6}{5} \tan ^5(c+d x)-\frac {4}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )\) |
Input:
Int[Sec[c + d*x]^12*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]
Output:
((A + B)*Sec[c + d*x]^11*(a + a*Sin[c + d*x])^2)/(11*d) + (a*(9*A - 2*B)*( (a*Sec[c + d*x]^9)/(9*d) - (a*(-Tan[c + d*x] - (4*Tan[c + d*x]^3)/3 - (6*T an[c + d*x]^5)/5 - (4*Tan[c + d*x]^7)/7 - Tan[c + d*x]^9/9))/d))/11
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 = 2.07 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.74
| method | result | size |
| risch | \(-\frac {256 \left (-9 i A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-168 i B \,a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+756 i A \,a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+2 i B \,a^{2}-308 i B \,a^{2} {\mathrm e}^{8 i \left (d x +c \right )}+1386 i A \,a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-9 i A \,a^{2}+2 i B \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-40 i B \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+770 B \,a^{2} {\mathrm e}^{9 i \left (d x +c \right )}+36 A \,a^{2} {\mathrm e}^{i \left (d x +c \right )}+180 i A \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+504 A \,a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+504 A \,a^{2} {\mathrm e}^{7 i \left (d x +c \right )}-112 B \,a^{2} {\mathrm e}^{7 i \left (d x +c \right )}-112 B \,a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-48 B \,a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-8 B \,a^{2} {\mathrm e}^{i \left (d x +c \right )}+216 A \,a^{2} {\mathrm e}^{3 i \left (d x +c \right )}\right )}{3465 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{11} d}\) | \(312\) |
| parallelrisch | \(-\frac {2 \left (A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}+\left (B -2 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}-\frac {4 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{3}+4 \left (2 A +\frac {B}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\frac {4 \left (A +\frac {11 B}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{5}+\frac {32 \left (-3 A +\frac {2 B}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{5}+\frac {4 \left (188 A -69 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{35}+\frac {4 \left (158 A +101 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{35}+\frac {142 \left (-A +\frac {2 B}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{7}+\frac {2 \left (-174 A +\frac {103 B}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{35}+\frac {4 \left (-\frac {3401 B}{9}+3048 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{385}+\frac {4 \left (-54 A +\frac {421 B}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{55}+\frac {4 \left (-129 A -\frac {127 B}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{55}+\frac {8 \left (6 A -\frac {B}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{11}+\frac {4 \left (12 A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{11}+\frac {4 \left (-6 A +\frac {23 B}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{11}+\frac {\left (3 A -\frac {28 B}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{11}+\frac {2 A}{11}+\frac {7 B}{99}\right ) a^{2}}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{11}}\) | \(360\) |
| derivativedivides | \(\frac {A \,a^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{11 \cos \left (d x +c \right )^{11}}+\frac {8 \sin \left (d x +c \right )^{3}}{99 \cos \left (d x +c \right )^{9}}+\frac {16 \sin \left (d x +c \right )^{3}}{231 \cos \left (d x +c \right )^{7}}+\frac {64 \sin \left (d x +c \right )^{3}}{1155 \cos \left (d x +c \right )^{5}}+\frac {128 \sin \left (d x +c \right )^{3}}{3465 \cos \left (d x +c \right )^{3}}\right )+a^{2} B \left (\frac {\sin \left (d x +c \right )^{4}}{11 \cos \left (d x +c \right )^{11}}+\frac {7 \sin \left (d x +c \right )^{4}}{99 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{99 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{33 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{99 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{99 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{99}\right )+\frac {2 A \,a^{2}}{11 \cos \left (d x +c \right )^{11}}+2 a^{2} B \left (\frac {\sin \left (d x +c \right )^{3}}{11 \cos \left (d x +c \right )^{11}}+\frac {8 \sin \left (d x +c \right )^{3}}{99 \cos \left (d x +c \right )^{9}}+\frac {16 \sin \left (d x +c \right )^{3}}{231 \cos \left (d x +c \right )^{7}}+\frac {64 \sin \left (d x +c \right )^{3}}{1155 \cos \left (d x +c \right )^{5}}+\frac {128 \sin \left (d x +c \right )^{3}}{3465 \cos \left (d x +c \right )^{3}}\right )-A \,a^{2} \left (-\frac {256}{693}-\frac {\sec \left (d x +c \right )^{10}}{11}-\frac {10 \sec \left (d x +c \right )^{8}}{99}-\frac {80 \sec \left (d x +c \right )^{6}}{693}-\frac {32 \sec \left (d x +c \right )^{4}}{231}-\frac {128 \sec \left (d x +c \right )^{2}}{693}\right ) \tan \left (d x +c \right )+\frac {a^{2} B}{11 \cos \left (d x +c \right )^{11}}}{d}\) | \(423\) |
| default | \(\frac {A \,a^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{11 \cos \left (d x +c \right )^{11}}+\frac {8 \sin \left (d x +c \right )^{3}}{99 \cos \left (d x +c \right )^{9}}+\frac {16 \sin \left (d x +c \right )^{3}}{231 \cos \left (d x +c \right )^{7}}+\frac {64 \sin \left (d x +c \right )^{3}}{1155 \cos \left (d x +c \right )^{5}}+\frac {128 \sin \left (d x +c \right )^{3}}{3465 \cos \left (d x +c \right )^{3}}\right )+a^{2} B \left (\frac {\sin \left (d x +c \right )^{4}}{11 \cos \left (d x +c \right )^{11}}+\frac {7 \sin \left (d x +c \right )^{4}}{99 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{99 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{33 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{99 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{99 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{99}\right )+\frac {2 A \,a^{2}}{11 \cos \left (d x +c \right )^{11}}+2 a^{2} B \left (\frac {\sin \left (d x +c \right )^{3}}{11 \cos \left (d x +c \right )^{11}}+\frac {8 \sin \left (d x +c \right )^{3}}{99 \cos \left (d x +c \right )^{9}}+\frac {16 \sin \left (d x +c \right )^{3}}{231 \cos \left (d x +c \right )^{7}}+\frac {64 \sin \left (d x +c \right )^{3}}{1155 \cos \left (d x +c \right )^{5}}+\frac {128 \sin \left (d x +c \right )^{3}}{3465 \cos \left (d x +c \right )^{3}}\right )-A \,a^{2} \left (-\frac {256}{693}-\frac {\sec \left (d x +c \right )^{10}}{11}-\frac {10 \sec \left (d x +c \right )^{8}}{99}-\frac {80 \sec \left (d x +c \right )^{6}}{693}-\frac {32 \sec \left (d x +c \right )^{4}}{231}-\frac {128 \sec \left (d x +c \right )^{2}}{693}\right ) \tan \left (d x +c \right )+\frac {a^{2} B}{11 \cos \left (d x +c \right )^{11}}}{d}\) | \(423\) |
Input:
int(sec(d*x+c)^12*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x,method=_RETURNVERB OSE)
Output:
-256/3465*(-9*I*A*a^2*exp(2*I*(d*x+c))-168*I*B*a^2*exp(6*I*(d*x+c))+756*I* A*a^2*exp(6*I*(d*x+c))+2*I*B*a^2-308*I*B*a^2*exp(8*I*(d*x+c))+1386*I*A*a^2 *exp(8*I*(d*x+c))-9*I*A*a^2+2*I*B*a^2*exp(2*I*(d*x+c))-40*I*B*a^2*exp(4*I* (d*x+c))+770*B*a^2*exp(9*I*(d*x+c))+36*A*a^2*exp(I*(d*x+c))+180*I*A*a^2*ex p(4*I*(d*x+c))+504*A*a^2*exp(5*I*(d*x+c))+504*A*a^2*exp(7*I*(d*x+c))-112*B *a^2*exp(7*I*(d*x+c))-112*B*a^2*exp(5*I*(d*x+c))-48*B*a^2*exp(3*I*(d*x+c)) -8*B*a^2*exp(I*(d*x+c))+216*A*a^2*exp(3*I*(d*x+c)))/(exp(I*(d*x+c))+I)^7/( exp(I*(d*x+c))-I)^11/d
Time = 0.10 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.32 \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {256 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{8} - 128 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} - 32 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} - 16 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} - 45 \, {\left (2 \, A - 9 \, B\right )} a^{2} - {\left (128 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{8} - 192 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} - 80 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} - 56 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} - 45 \, {\left (9 \, A - 2 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right )^{9} + 2 \, d \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{7}\right )}} \] Input:
integrate(sec(d*x+c)^12*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm=" fricas")
Output:
-1/3465*(256*(9*A - 2*B)*a^2*cos(d*x + c)^8 - 128*(9*A - 2*B)*a^2*cos(d*x + c)^6 - 32*(9*A - 2*B)*a^2*cos(d*x + c)^4 - 16*(9*A - 2*B)*a^2*cos(d*x + c)^2 - 45*(2*A - 9*B)*a^2 - (128*(9*A - 2*B)*a^2*cos(d*x + c)^8 - 192*(9*A - 2*B)*a^2*cos(d*x + c)^6 - 80*(9*A - 2*B)*a^2*cos(d*x + c)^4 - 56*(9*A - 2*B)*a^2*cos(d*x + c)^2 - 45*(9*A - 2*B)*a^2)*sin(d*x + c))/(d*cos(d*x + c)^9 + 2*d*cos(d*x + c)^7*sin(d*x + c) - 2*d*cos(d*x + c)^7)
Timed out. \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**12*(a+a*sin(d*x+c))**2*(A+B*sin(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.33 \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {{\left (315 \, \tan \left (d x + c\right )^{11} + 1540 \, \tan \left (d x + c\right )^{9} + 2970 \, \tan \left (d x + c\right )^{7} + 2772 \, \tan \left (d x + c\right )^{5} + 1155 \, \tan \left (d x + c\right )^{3}\right )} A a^{2} + 5 \, {\left (63 \, \tan \left (d x + c\right )^{11} + 385 \, \tan \left (d x + c\right )^{9} + 990 \, \tan \left (d x + c\right )^{7} + 1386 \, \tan \left (d x + c\right )^{5} + 1155 \, \tan \left (d x + c\right )^{3} + 693 \, \tan \left (d x + c\right )\right )} A a^{2} + 2 \, {\left (315 \, \tan \left (d x + c\right )^{11} + 1540 \, \tan \left (d x + c\right )^{9} + 2970 \, \tan \left (d x + c\right )^{7} + 2772 \, \tan \left (d x + c\right )^{5} + 1155 \, \tan \left (d x + c\right )^{3}\right )} B a^{2} - \frac {35 \, {\left (11 \, \cos \left (d x + c\right )^{2} - 9\right )} B a^{2}}{\cos \left (d x + c\right )^{11}} + \frac {630 \, A a^{2}}{\cos \left (d x + c\right )^{11}} + \frac {315 \, B a^{2}}{\cos \left (d x + c\right )^{11}}}{3465 \, d} \] Input:
integrate(sec(d*x+c)^12*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm=" maxima")
Output:
1/3465*((315*tan(d*x + c)^11 + 1540*tan(d*x + c)^9 + 2970*tan(d*x + c)^7 + 2772*tan(d*x + c)^5 + 1155*tan(d*x + c)^3)*A*a^2 + 5*(63*tan(d*x + c)^11 + 385*tan(d*x + c)^9 + 990*tan(d*x + c)^7 + 1386*tan(d*x + c)^5 + 1155*tan (d*x + c)^3 + 693*tan(d*x + c))*A*a^2 + 2*(315*tan(d*x + c)^11 + 1540*tan( d*x + c)^9 + 2970*tan(d*x + c)^7 + 2772*tan(d*x + c)^5 + 1155*tan(d*x + c) ^3)*B*a^2 - 35*(11*cos(d*x + c)^2 - 9)*B*a^2/cos(d*x + c)^11 + 630*A*a^2/c os(d*x + c)^11 + 315*B*a^2/cos(d*x + c)^11)/d
Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (165) = 330\).
Time = 0.22 (sec) , antiderivative size = 597, normalized size of antiderivative = 3.34 \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)^12*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm=" giac")
Output:
-1/443520*(33*(6825*A*a^2*tan(1/2*d*x + 1/2*c)^6 - 2940*B*a^2*tan(1/2*d*x + 1/2*c)^6 + 34965*A*a^2*tan(1/2*d*x + 1/2*c)^5 - 13755*B*a^2*tan(1/2*d*x + 1/2*c)^5 + 79800*A*a^2*tan(1/2*d*x + 1/2*c)^4 - 30065*B*a^2*tan(1/2*d*x + 1/2*c)^4 + 100170*A*a^2*tan(1/2*d*x + 1/2*c)^3 - 36470*B*a^2*tan(1/2*d*x + 1/2*c)^3 + 73017*A*a^2*tan(1/2*d*x + 1/2*c)^2 - 26166*B*a^2*tan(1/2*d*x + 1/2*c)^2 + 29169*A*a^2*tan(1/2*d*x + 1/2*c) - 10367*B*a^2*tan(1/2*d*x + 1/2*c) + 5142*A*a^2 - 1901*B*a^2)/(tan(1/2*d*x + 1/2*c) + 1)^7 + (661815* A*a^2*tan(1/2*d*x + 1/2*c)^10 + 97020*B*a^2*tan(1/2*d*x + 1/2*c)^10 - 5083 155*A*a^2*tan(1/2*d*x + 1/2*c)^9 - 405405*B*a^2*tan(1/2*d*x + 1/2*c)^9 + 1 9355490*A*a^2*tan(1/2*d*x + 1/2*c)^8 + 952875*B*a^2*tan(1/2*d*x + 1/2*c)^8 - 45446940*A*a^2*tan(1/2*d*x + 1/2*c)^7 - 1122660*B*a^2*tan(1/2*d*x + 1/2 *c)^7 + 72295146*A*a^2*tan(1/2*d*x + 1/2*c)^6 + 557172*B*a^2*tan(1/2*d*x + 1/2*c)^6 - 80611146*A*a^2*tan(1/2*d*x + 1/2*c)^5 + 563178*B*a^2*tan(1/2*d *x + 1/2*c)^5 + 63771840*A*a^2*tan(1/2*d*x + 1/2*c)^4 - 1126950*B*a^2*tan( 1/2*d*x + 1/2*c)^4 - 35253900*A*a^2*tan(1/2*d*x + 1/2*c)^3 + 955020*B*a^2* tan(1/2*d*x + 1/2*c)^3 + 13119975*A*a^2*tan(1/2*d*x + 1/2*c)^2 - 406120*B* a^2*tan(1/2*d*x + 1/2*c)^2 - 2978811*A*a^2*tan(1/2*d*x + 1/2*c) + 97163*B* a^2*tan(1/2*d*x + 1/2*c) + 330966*A*a^2 - 13*B*a^2)/(tan(1/2*d*x + 1/2*c) - 1)^11)/d
Time = 40.20 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.60 \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx =\text {Too large to display} \] Input:
int(((A + B*sin(c + d*x))*(a + a*sin(c + d*x))^2)/cos(c + d*x)^12,x)
Output:
-(a^2*cos(c/2 + (d*x)/2)*((8127*A*cos((5*c)/2 + (5*d*x)/2))/64 - (24255*A* cos((3*c)/2 + (3*d*x)/2))/64 - (21357*A*cos((7*c)/2 + (7*d*x)/2))/64 + (52 29*A*cos((9*c)/2 + (9*d*x)/2))/64 - (8379*A*cos((11*c)/2 + (11*d*x)/2))/64 + (1467*A*cos((13*c)/2 + (13*d*x)/2))/64 - (2619*A*cos((15*c)/2 + (15*d*x )/2))/128 + (315*A*cos((17*c)/2 + (17*d*x)/2))/128 - 385*B*cos(c/2 + (d*x) /2) + (30415*B*cos((3*c)/2 + (3*d*x)/2))/128 - (23247*B*cos((5*c)/2 + (5*d *x)/2))/128 + (12957*B*cos((7*c)/2 + (7*d*x)/2))/128 - (5789*B*cos((9*c)/2 + (9*d*x)/2))/128 + (3339*B*cos((11*c)/2 + (11*d*x)/2))/128 - (267*B*cos( (13*c)/2 + (13*d*x)/2))/128 + (779*B*cos((15*c)/2 + (15*d*x)/2))/256 + (24 5*B*cos((17*c)/2 + (17*d*x)/2))/256 - (47889*A*sin(c/2 + (d*x)/2))/32 + (2 5713*A*sin((3*c)/2 + (3*d*x)/2))/32 - (21303*A*sin((5*c)/2 + (5*d*x)/2))/3 2 + (9207*A*sin((7*c)/2 + (7*d*x)/2))/32 - (4797*A*sin((9*c)/2 + (9*d*x)/2 ))/32 + (1917*A*sin((11*c)/2 + (11*d*x)/2))/32 - (27*A*sin((13*c)/2 + (13* d*x)/2))/32 + (171*A*sin((15*c)/2 + (15*d*x)/2))/32 + (9*A*sin((17*c)/2 + (17*d*x)/2))/2 + (7809*B*sin(c/2 + (d*x)/2))/64 + (2047*B*sin((3*c)/2 + (3 *d*x)/2))/64 + (1383*B*sin((5*c)/2 + (5*d*x)/2))/64 + (3993*B*sin((7*c)/2 + (7*d*x)/2))/64 - (563*B*sin((9*c)/2 + (9*d*x)/2))/64 + (1843*B*sin((11*c )/2 + (11*d*x)/2))/64 - (373*B*sin((13*c)/2 + (13*d*x)/2))/64 + (309*B*sin ((15*c)/2 + (15*d*x)/2))/64 - B*sin((17*c)/2 + (17*d*x)/2)))/(887040*d*cos (c/2 - pi/4 + (d*x)/2)^7*cos(c/2 + pi/4 + (d*x)/2)^11)
Time = 0.18 (sec) , antiderivative size = 539, normalized size of antiderivative = 3.01 \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^{2} \left (-1260 a -490 b -2205 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} a +4410 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a +980 \sin \left (d x +c \right ) b -512 \sin \left (d x +c \right )^{9} b +1024 \sin \left (d x +c \right )^{8} b -5760 \sin \left (d x +c \right )^{7} a +1280 \sin \left (d x +c \right )^{7} b +16128 \sin \left (d x +c \right )^{6} a -3584 \sin \left (d x +c \right )^{6} b +2016 \sin \left (d x +c \right )^{5} a -448 \sin \left (d x +c \right )^{5} b -20160 \sin \left (d x +c \right )^{4} a +4480 \sin \left (d x +c \right )^{4} b +5040 \sin \left (d x +c \right )^{3} a -1120 \sin \left (d x +c \right )^{3} b +10080 \sin \left (d x +c \right )^{2} a -2240 \sin \left (d x +c \right )^{2} b +490 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} b -4410 \sin \left (d x +c \right ) a +2304 \sin \left (d x +c \right )^{9} a -4608 \sin \left (d x +c \right )^{8} a -980 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} b +4410 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a -980 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b -13230 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a +2940 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b +13230 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -2940 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b -4410 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a +980 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b -4410 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +980 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +2205 \cos \left (d x +c \right ) a -490 \cos \left (d x +c \right ) b \right )}{6930 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{8}-2 \sin \left (d x +c \right )^{7}-2 \sin \left (d x +c \right )^{6}+6 \sin \left (d x +c \right )^{5}-6 \sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )-1\right )} \] Input:
int(sec(d*x+c)^12*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x)
Output:
(a**2*( - 2205*cos(c + d*x)*sin(c + d*x)**8*a + 490*cos(c + d*x)*sin(c + d *x)**8*b + 4410*cos(c + d*x)*sin(c + d*x)**7*a - 980*cos(c + d*x)*sin(c + d*x)**7*b + 4410*cos(c + d*x)*sin(c + d*x)**6*a - 980*cos(c + d*x)*sin(c + d*x)**6*b - 13230*cos(c + d*x)*sin(c + d*x)**5*a + 2940*cos(c + d*x)*sin( c + d*x)**5*b + 13230*cos(c + d*x)*sin(c + d*x)**3*a - 2940*cos(c + d*x)*s in(c + d*x)**3*b - 4410*cos(c + d*x)*sin(c + d*x)**2*a + 980*cos(c + d*x)* sin(c + d*x)**2*b - 4410*cos(c + d*x)*sin(c + d*x)*a + 980*cos(c + d*x)*si n(c + d*x)*b + 2205*cos(c + d*x)*a - 490*cos(c + d*x)*b + 2304*sin(c + d*x )**9*a - 512*sin(c + d*x)**9*b - 4608*sin(c + d*x)**8*a + 1024*sin(c + d*x )**8*b - 5760*sin(c + d*x)**7*a + 1280*sin(c + d*x)**7*b + 16128*sin(c + d *x)**6*a - 3584*sin(c + d*x)**6*b + 2016*sin(c + d*x)**5*a - 448*sin(c + d *x)**5*b - 20160*sin(c + d*x)**4*a + 4480*sin(c + d*x)**4*b + 5040*sin(c + d*x)**3*a - 1120*sin(c + d*x)**3*b + 10080*sin(c + d*x)**2*a - 2240*sin(c + d*x)**2*b - 4410*sin(c + d*x)*a + 980*sin(c + d*x)*b - 1260*a - 490*b)) /(6930*cos(c + d*x)*d*(sin(c + d*x)**8 - 2*sin(c + d*x)**7 - 2*sin(c + d*x )**6 + 6*sin(c + d*x)**5 - 6*sin(c + d*x)**3 + 2*sin(c + d*x)**2 + 2*sin(c + d*x) - 1))