\(\int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx\) [985]

Optimal result

Integrand size = 31, antiderivative size = 179 \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2 (9 A-2 B) \sec ^9(c+d x)}{99 d}+\frac {(A+B) \sec ^{11}(c+d x) (a+a \sin (c+d x))^2}{11 d}+\frac {a^2 (9 A-2 B) \tan (c+d x)}{11 d}+\frac {4 a^2 (9 A-2 B) \tan ^3(c+d x)}{33 d}+\frac {6 a^2 (9 A-2 B) \tan ^5(c+d x)}{55 d}+\frac {4 a^2 (9 A-2 B) \tan ^7(c+d x)}{77 d}+\frac {a^2 (9 A-2 B) \tan ^9(c+d x)}{99 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2934, 2748, 3852} \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2 (9 A-2 B) \tan ^9(c+d x)}{99 d}+\frac {4 a^2 (9 A-2 B) \tan ^7(c+d x)}{77 d}+\frac {6 a^2 (9 A-2 B) \tan ^5(c+d x)}{55 d}+\frac {4 a^2 (9 A-2 B) \tan ^3(c+d x)}{33 d}+\frac {a^2 (9 A-2 B) \tan (c+d x)}{11 d}+\frac {a^2 (9 A-2 B) \sec ^9(c+d x)}{99 d}+\frac {(A+B) \sec ^{11}(c+d x) (a \sin (c+d x)+a)^2}{11 d} \]

[In]

Int[Sec[c + d*x]^12*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]

[Out]

(a^2*(9*A - 2*B)*Sec[c + d*x]^9)/(99*d) + ((A + B)*Sec[c + d*x]^11*(a + a*Sin[c + d*x])^2)/(11*d) + (a^2*(9*A
- 2*B)*Tan[c + d*x])/(11*d) + (4*a^2*(9*A - 2*B)*Tan[c + d*x]^3)/(33*d) + (6*a^2*(9*A - 2*B)*Tan[c + d*x]^5)/(
55*d) + (4*a^2*(9*A - 2*B)*Tan[c + d*x]^7)/(77*d) + (a^2*(9*A - 2*B)*Tan[c + d*x]^9)/(99*d)

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[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])

Rule 2934

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 3852

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*} \text {integral}& = \frac {(A+B) \sec ^{11}(c+d x) (a+a \sin (c+d x))^2}{11 d}+\frac {1}{11} (a (9 A-2 B)) \int \sec ^{10}(c+d x) (a+a \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 {1}{11} \left (a^2 (9 A-2 B)\right ) \int \sec ^{10}(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 {\left (a^2 (9 A-2 B)\right ) \text {Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,-\tan (c+d x)\right )}{11 d} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.01 \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2 \left (35 (18 A+7 B) \sec ^{11}(c+d x)+3465 A \sec ^{10}(c+d x) \tan (c+d x)+385 B \sec ^9(c+d x) \tan ^2(c+d x)-1155 (9 A-2 B) \sec ^8(c+d x) \tan ^3(c+d x)+1848 (9 A-2 B) \sec ^6(c+d x) \tan ^5(c+d x)-1584 (9 A-2 B) \sec ^4(c+d x) \tan ^7(c+d x)+704 (9 A-2 B) \sec ^2(c+d x) \tan ^9(c+d x)+128 (-9 A+2 B) \tan ^{11}(c+d x)\right )}{3465 d} \]

[In]

Integrate[Sec[c + d*x]^12*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]

[Out]

(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*Tan[c + d*x]^9 + 128*(-9*A + 2*B)*Ta
n[c + d*x]^11))/(3465*d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.23 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.74

method	result	size
risch	\(-\frac {256 \left (-48 B \,a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-8 B \,a^{2} {\mathrm e}^{i \left (d x +c \right )}-9 i A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+504 A \,a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+504 A \,a^{2} {\mathrm e}^{7 i \left (d x +c \right )}-112 B \,a^{2} {\mathrm e}^{7 i \left (d x +c \right )}-112 B \,a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+180 i A \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+1386 i A \,a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-40 i B \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+2 i B \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+756 i A \,a^{2} {\mathrm e}^{6 i \left (d x +c \right )}-9 i A \,a^{2}-308 i B \,a^{2} {\mathrm e}^{8 i \left (d x +c \right )}+2 i B \,a^{2}-168 i B \,a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+36 A \,a^{2} {\mathrm e}^{i \left (d x +c \right )}+216 A \,a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+770 B \,a^{2} {\mathrm e}^{9 i \left (d x +c \right )}\right )}{3465 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{11} d}\)	\(312\)
parallelrisch	\(-\frac {2 \left (A \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (B -2 A \right ) \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4 B \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+4 \left (2 A +\frac {B}{3}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 \left (\frac {11 B}{3}+A \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {32 \left (\frac {2 B}{3}-3 A \right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {4 \left (188 A -69 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35}+\frac {4 \left (158 A +101 B \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35}+\frac {142 \left (-A +\frac {2 B}{9}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {2 \left (-174 A +\frac {103 B}{9}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35}+\frac {4 \left (3048 A -\frac {3401 B}{9}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{385}+\frac {4 \left (-54 A +\frac {421 B}{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{55}+\frac {4 \left (-129 A -\frac {127 B}{9}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{55}+\frac {8 \left (6 A -\frac {B}{9}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}+\frac {4 \left (12 A +B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}+\frac {4 \left (-6 A +\frac {23 B}{9}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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 ^{3}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{231 \cos \left (d x +c \right )^{7}}+\frac {64 \left (\sin ^{3}\left (d x +c \right )\right )}{1155 \cos \left (d x +c \right )^{5}}+\frac {128 \left (\sin ^{3}\left (d x +c \right )\right )}{3465 \cos \left (d x +c \right )^{3}}\right )+B \,a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {7 \left (\sin ^{4}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{33 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{99 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{99 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{99}\right )+\frac {2 A \,a^{2}}{11 \cos \left (d x +c \right )^{11}}+2 B \,a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{231 \cos \left (d x +c \right )^{7}}+\frac {64 \left (\sin ^{3}\left (d x +c \right )\right )}{1155 \cos \left (d x +c \right )^{5}}+\frac {128 \left (\sin ^{3}\left (d x +c \right )\right )}{3465 \cos \left (d x +c \right )^{3}}\right )-A \,a^{2} \left (-\frac {256}{693}-\frac {\left (\sec ^{10}\left (d x +c \right )\right )}{11}-\frac {10 \left (\sec ^{8}\left (d x +c \right )\right )}{99}-\frac {80 \left (\sec ^{6}\left (d x +c \right )\right )}{693}-\frac {32 \left (\sec ^{4}\left (d x +c \right )\right )}{231}-\frac {128 \left (\sec ^{2}\left (d x +c \right )\right )}{693}\right ) \tan \left (d x +c \right )+\frac {B \,a^{2}}{11 \cos \left (d x +c \right )^{11}}}{d}\)	\(423\)
default	\(\frac {A \,a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{231 \cos \left (d x +c \right )^{7}}+\frac {64 \left (\sin ^{3}\left (d x +c \right )\right )}{1155 \cos \left (d x +c \right )^{5}}+\frac {128 \left (\sin ^{3}\left (d x +c \right )\right )}{3465 \cos \left (d x +c \right )^{3}}\right )+B \,a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {7 \left (\sin ^{4}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{33 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{99 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{99 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{99}\right )+\frac {2 A \,a^{2}}{11 \cos \left (d x +c \right )^{11}}+2 B \,a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{231 \cos \left (d x +c \right )^{7}}+\frac {64 \left (\sin ^{3}\left (d x +c \right )\right )}{1155 \cos \left (d x +c \right )^{5}}+\frac {128 \left (\sin ^{3}\left (d x +c \right )\right )}{3465 \cos \left (d x +c \right )^{3}}\right )-A \,a^{2} \left (-\frac {256}{693}-\frac {\left (\sec ^{10}\left (d x +c \right )\right )}{11}-\frac {10 \left (\sec ^{8}\left (d x +c \right )\right )}{99}-\frac {80 \left (\sec ^{6}\left (d x +c \right )\right )}{693}-\frac {32 \left (\sec ^{4}\left (d x +c \right )\right )}{231}-\frac {128 \left (\sec ^{2}\left (d x +c \right )\right )}{693}\right ) \tan \left (d x +c \right )+\frac {B \,a^{2}}{11 \cos \left (d x +c \right )^{11}}}{d}\)	\(423\)

[In]

int(sec(d*x+c)^12*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-256/3465*(-48*B*a^2*exp(3*I*(d*x+c))-8*B*a^2*exp(I*(d*x+c))-9*I*A*a^2*exp(2*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))+180*I*A*a^2*exp(4*I*(d*x
+c))+1386*I*A*a^2*exp(8*I*(d*x+c))-40*I*B*a^2*exp(4*I*(d*x+c))+2*I*B*a^2*exp(2*I*(d*x+c))+756*I*A*a^2*exp(6*I*
(d*x+c))-9*I*A*a^2-308*I*B*a^2*exp(8*I*(d*x+c))+2*I*B*a^2-168*I*B*a^2*exp(6*I*(d*x+c))+36*A*a^2*exp(I*(d*x+c))
+216*A*a^2*exp(3*I*(d*x+c))+770*B*a^2*exp(9*I*(d*x+c)))/(exp(I*(d*x+c))+I)^7/(exp(I*(d*x+c))-I)^11/d

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.32 \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {256 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{8} - 128 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} - 32 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} - 16 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} - 45 \, {\left (2 \, A - 9 \, B\right )} a^{2} - {\left (128 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{8} - 192 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} - 80 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} - 56 \, {\left (9 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} - 45 \, {\left (9 \, A - 2 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right )^{9} + 2 \, d \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{7}\right )}} \]

[In]

integrate(sec(d*x+c)^12*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

-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)

Sympy [F(-1)]

Timed out. \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**12*(a+a*sin(d*x+c))**2*(A+B*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.33 \[ \int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {{\left (315 \, \tan \left (d x + c\right )^{11} + 1540 \, \tan \left (d x + c\right )^{9} + 2970 \, \tan \left (d x + c\right )^{7} + 2772 \, \tan \left (d x + c\right )^{5} + 1155 \, \tan \left (d x + c\right )^{3}\right )} A a^{2} + 5 \, {\left (63 \, \tan \left (d x + c\right )^{11} + 385 \, \tan \left (d x + c\right )^{9} + 990 \, \tan \left (d x + c\right )^{7} + 1386 \, \tan \left (d x + c\right )^{5} + 1155 \, \tan \left (d x + c\right )^{3} + 693 \, \tan \left (d x + c\right )\right )} A a^{2} + 2 \, {\left (315 \, \tan \left (d x + c\right )^{11} + 1540 \, \tan \left (d x + c\right )^{9} + 2970 \, \tan \left (d x + c\right )^{7} + 2772 \, \tan \left (d x + c\right )^{5} + 1155 \, \tan \left (d x + c\right )^{3}\right )} B a^{2} - \frac {35 \, {\left (11 \, \cos \left (d x + c\right )^{2} - 9\right )} B a^{2}}{\cos \left (d x + c\right )^{11}} + \frac {630 \, A a^{2}}{\cos \left (d x + c\right )^{11}} + \frac {315 \, B a^{2}}{\cos \left (d x + c\right )^{11}}}{3465 \, d} \]

[In]

integrate(sec(d*x+c)^12*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

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*t
an(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/cos(d*x + c)^11 + 315*B*a^2/cos(d*x + c)^11)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (165) = 330\).

Time = 0.62 (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=-\frac {\frac {33 \, {\left (6825 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2940 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 34965 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 13755 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 79800 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 30065 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 100170 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36470 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 73017 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 26166 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 29169 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 10367 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5142 \, A a^{2} - 1901 \, B a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}} + \frac {661815 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 97020 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 5083155 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 405405 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 19355490 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 952875 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 45446940 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1122660 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72295146 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 557172 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 80611146 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 563178 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 63771840 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1126950 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35253900 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 955020 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13119975 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 406120 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2978811 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 97163 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 330966 \, A a^{2} - 13 \, B a^{2}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{11}}}{443520 \, d} \]

[In]

integrate(sec(d*x+c)^12*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

-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 - 5083155*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 + 19355490*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 + 955
020*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

Mupad [B] (verification not implemented)

Time = 14.95 (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=-\frac {a^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {8127\,A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}-\frac {24255\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}-\frac {21357\,A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{64}+\frac {5229\,A\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{64}-\frac {8379\,A\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{64}+\frac {1467\,A\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{64}-\frac {2619\,A\,\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{128}+\frac {315\,A\,\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{128}-385\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {30415\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{128}-\frac {23247\,B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{128}+\frac {12957\,B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{128}-\frac {5789\,B\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{128}+\frac {3339\,B\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{128}-\frac {267\,B\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{128}+\frac {779\,B\,\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{256}+\frac {245\,B\,\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{256}-\frac {47889\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{32}+\frac {25713\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{32}-\frac {21303\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}+\frac {9207\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}-\frac {4797\,A\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}+\frac {1917\,A\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}-\frac {27\,A\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{32}+\frac {171\,A\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{32}+\frac {9\,A\,\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{2}+\frac {7809\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {2047\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}+\frac {1383\,B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}+\frac {3993\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{64}-\frac {563\,B\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{64}+\frac {1843\,B\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{64}-\frac {373\,B\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{64}+\frac {309\,B\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{64}-B\,\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )\right )}{887040\,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 )}^{11}} \]

[In]

int(((A + B*sin(c + d*x))*(a + a*sin(c + d*x))^2)/cos(c + d*x)^12,x)

[Out]

-(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 - (2135
7*A*cos((7*c)/2 + (7*d*x)/2))/64 + (5229*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
+ (245*B*cos((17*c)/2 + (17*d*x)/2))/256 - (47889*A*sin(c/2 + (d*x)/2))/32 + (25713*A*sin((3*c)/2 + (3*d*x)/2
))/32 - (21303*A*sin((5*c)/2 + (5*d*x)/2))/32 + (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)