\(\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx\) [81]

Optimal result

Integrand size = 31, antiderivative size = 241 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {1}{16} a^4 (44 A+49 B) x+\frac {a^4 (227 A+252 B) \sin (c+d x)}{35 d}+\frac {a^4 (44 A+49 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (276 A+301 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^4 (227 A+252 B) \sin ^3(c+d x)}{105 d} \]

[Out]

1/16*a^4*(44*A+49*B)*x+1/35*a^4*(227*A+252*B)*sin(d*x+c)/d+1/16*a^4*(44*A+49*B)*cos(d*x+c)*sin(d*x+c)/d+1/280*
a^4*(276*A+301*B)*cos(d*x+c)^3*sin(d*x+c)/d+1/7*a*A*cos(d*x+c)^6*(a+a*sec(d*x+c))^3*sin(d*x+c)/d+1/42*(10*A+7*
B)*cos(d*x+c)^5*(a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/d+7/15*(A+B)*cos(d*x+c)^4*(a^4+a^4*sec(d*x+c))*sin(d*x+c)/d-
1/105*a^4*(227*A+252*B)*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4102, 4081, 3872, 2713, 2715, 8} \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=-\frac {a^4 (227 A+252 B) \sin ^3(c+d x)}{105 d}+\frac {a^4 (227 A+252 B) \sin (c+d x)}{35 d}+\frac {a^4 (276 A+301 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {a^4 (44 A+49 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {7 (A+B) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{15 d}+\frac {1}{16} a^4 x (44 A+49 B)+\frac {(10 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{42 d}+\frac {a A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d} \]

[In]

Int[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]

[Out]

(a^4*(44*A + 49*B)*x)/16 + (a^4*(227*A + 252*B)*Sin[c + d*x])/(35*d) + (a^4*(44*A + 49*B)*Cos[c + d*x]*Sin[c +
d*x])/(16*d) + (a^4*(276*A + 301*B)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (a*A*Cos[c + d*x]^6*(a + a*Sec[c +
d*x])^3*Sin[c + d*x])/(7*d) + ((10*A + 7*B)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(42*d) +
(7*(A + B)*Cos[c + d*x]^4*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^4*(227*A + 252*B)*Sin[c + d*x]^3)
/(105*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 4081

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.)
+ (A_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Dist[1/(d*n), Int[(d*Csc[e + f*x
])^(n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B},
x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]

Rule 4102

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x]
- Dist[b/(a*d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*(m - n - 1) - b*B*n - (a*
B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2
- b^2, 0] && GtQ[m, 1/2] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {1}{7} \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (a (10 A+7 B)+a (3 A+7 B) \sec (c+d x)) \, dx \\ & = \frac {a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {1}{42} \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (98 a^2 (A+B)+3 a^2 (16 A+21 B) \sec (c+d x)\right ) \, dx \\ & = \frac {a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{210} \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (3 a^3 (276 A+301 B)+3 a^3 (178 A+203 B) \sec (c+d x)\right ) \, dx \\ & = \frac {a^4 (276 A+301 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {1}{840} \int \cos ^3(c+d x) \left (-24 a^4 (227 A+252 B)-105 a^4 (44 A+49 B) \sec (c+d x)\right ) \, dx \\ & = \frac {a^4 (276 A+301 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{8} \left (a^4 (44 A+49 B)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{35} \left (a^4 (227 A+252 B)\right ) \int \cos ^3(c+d x) \, dx \\ & = \frac {a^4 (44 A+49 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (276 A+301 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{16} \left (a^4 (44 A+49 B)\right ) \int 1 \, dx-\frac {\left (a^4 (227 A+252 B)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d} \\ & = \frac {1}{16} a^4 (44 A+49 B) x+\frac {a^4 (227 A+252 B) \sin (c+d x)}{35 d}+\frac {a^4 (44 A+49 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (276 A+301 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^4 (227 A+252 B) \sin ^3(c+d x)}{105 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.65 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {a^4 (18480 A c+18480 A d x+20580 B d x+105 (323 A+352 B) \sin (c+d x)+105 (124 A+127 B) \sin (2 (c+d x))+5495 A \sin (3 (c+d x))+5040 B \sin (3 (c+d x))+2100 A \sin (4 (c+d x))+1575 B \sin (4 (c+d x))+651 A \sin (5 (c+d x))+336 B \sin (5 (c+d x))+140 A \sin (6 (c+d x))+35 B \sin (6 (c+d x))+15 A \sin (7 (c+d x)))}{6720 d} \]

[In]

Integrate[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]

[Out]

(a^4*(18480*A*c + 18480*A*d*x + 20580*B*d*x + 105*(323*A + 352*B)*Sin[c + d*x] + 105*(124*A + 127*B)*Sin[2*(c
+ d*x)] + 5495*A*Sin[3*(c + d*x)] + 5040*B*Sin[3*(c + d*x)] + 2100*A*Sin[4*(c + d*x)] + 1575*B*Sin[4*(c + d*x)
] + 651*A*Sin[5*(c + d*x)] + 336*B*Sin[5*(c + d*x)] + 140*A*Sin[6*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 15*A*Si
n[7*(c + d*x)]))/(6720*d)

Maple [A] (verified)

Time = 4.61 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.53

method	result	size
parallelrisch	\(\frac {5 \left (\left (\frac {31 A}{5}+\frac {127 B}{20}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {157 A}{60}+\frac {12 B}{5}\right ) \sin \left (3 d x +3 c \right )+\left (A +\frac {3 B}{4}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {31 A}{100}+\frac {4 B}{25}\right ) \sin \left (5 d x +5 c \right )+\left (\frac {A}{15}+\frac {B}{60}\right ) \sin \left (6 d x +6 c \right )+\frac {A \sin \left (7 d x +7 c \right )}{140}+\left (\frac {323 A}{20}+\frac {88 B}{5}\right ) \sin \left (d x +c \right )+\frac {44 d \left (A +\frac {49 B}{44}\right ) x}{5}\right ) a^{4}}{16 d}\)	\(128\)
risch	\(\frac {11 a^{4} A x}{4}+\frac {49 a^{4} x B}{16}+\frac {323 \sin \left (d x +c \right ) a^{4} A}{64 d}+\frac {11 \sin \left (d x +c \right ) B \,a^{4}}{2 d}+\frac {a^{4} A \sin \left (7 d x +7 c \right )}{448 d}+\frac {a^{4} A \sin \left (6 d x +6 c \right )}{48 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{4}}{192 d}+\frac {31 a^{4} A \sin \left (5 d x +5 c \right )}{320 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{20 d}+\frac {5 a^{4} A \sin \left (4 d x +4 c \right )}{16 d}+\frac {15 \sin \left (4 d x +4 c \right ) B \,a^{4}}{64 d}+\frac {157 a^{4} A \sin \left (3 d x +3 c \right )}{192 d}+\frac {3 \sin \left (3 d x +3 c \right ) B \,a^{4}}{4 d}+\frac {31 \sin \left (2 d x +2 c \right ) a^{4} A}{16 d}+\frac {127 \sin \left (2 d x +2 c \right ) B \,a^{4}}{64 d}\)	\(244\)
derivativedivides	\(\frac {\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {6 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {a^{4} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(358\)
default	\(\frac {\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {6 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {a^{4} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(358\)

[In]

int(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

5/16*((31/5*A+127/20*B)*sin(2*d*x+2*c)+(157/60*A+12/5*B)*sin(3*d*x+3*c)+(A+3/4*B)*sin(4*d*x+4*c)+(31/100*A+4/2
5*B)*sin(5*d*x+5*c)+(1/15*A+1/60*B)*sin(6*d*x+6*c)+1/140*A*sin(7*d*x+7*c)+(323/20*A+88/5*B)*sin(d*x+c)+44/5*d*
(A+49/44*B)*x)*a^4/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.62 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {105 \, {\left (44 \, A + 49 \, B\right )} a^{4} d x + {\left (240 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{5} + 192 \, {\left (12 \, A + 7 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (44 \, A + 41 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \, {\left (227 \, A + 252 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (44 \, A + 49 \, B\right )} a^{4} \cos \left (d x + c\right ) + 32 \, {\left (227 \, A + 252 \, B\right )} a^{4}\right )} \sin \left (d x + c\right )}{1680 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/1680*(105*(44*A + 49*B)*a^4*d*x + (240*A*a^4*cos(d*x + c)^6 + 280*(4*A + B)*a^4*cos(d*x + c)^5 + 192*(12*A +
7*B)*a^4*cos(d*x + c)^4 + 70*(44*A + 41*B)*a^4*cos(d*x + c)^3 + 16*(227*A + 252*B)*a^4*cos(d*x + c)^2 + 105*(
44*A + 49*B)*a^4*cos(d*x + c) + 32*(227*A + 252*B)*a^4)*sin(d*x + c))/d

Sympy [F(-1)]

Timed out. \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7*(a+a*sec(d*x+c))**4*(A+B*sec(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.48 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=-\frac {192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} A a^{4} - 2688 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} + 140 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 840 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1792 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 8960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 1260 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4}}{6720 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)),x, algorithm="maxima")

[Out]

-1/6720*(192*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*A*a^4 - 2688*(3*sin(
d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 + 140*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*
d*x + 4*c) - 48*sin(2*d*x + 2*c))*A*a^4 + 2240*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 - 840*(12*d*x + 12*c +
sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 - 1792*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B
*a^4 + 35*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*B*a^4 + 8960*(sin(
d*x + c)^3 - 3*sin(d*x + c))*B*a^4 - 1260*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 - 1680
*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4)/d

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.15 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {105 \, {\left (44 \, A a^{4} + 49 \, B a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (4620 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5145 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 30800 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 34300 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 87164 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 97069 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 135168 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 150528 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 126084 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 134099 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 58800 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 73220 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 22260 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21735 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7}}}{1680 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)),x, algorithm="giac")

[Out]

1/1680*(105*(44*A*a^4 + 49*B*a^4)*(d*x + c) + 2*(4620*A*a^4*tan(1/2*d*x + 1/2*c)^13 + 5145*B*a^4*tan(1/2*d*x +
1/2*c)^13 + 30800*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 34300*B*a^4*tan(1/2*d*x + 1/2*c)^11 + 87164*A*a^4*tan(1/2*d
*x + 1/2*c)^9 + 97069*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 135168*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 150528*B*a^4*tan(1/
2*d*x + 1/2*c)^7 + 126084*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 134099*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 58800*A*a^4*tan
(1/2*d*x + 1/2*c)^3 + 73220*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 22260*A*a^4*tan(1/2*d*x + 1/2*c) + 21735*B*a^4*tan(
1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^7)/d

Mupad [B] (verification not implemented)

Time = 15.67 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.34 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {\left (\frac {11\,A\,a^4}{2}+\frac {49\,B\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {110\,A\,a^4}{3}+\frac {245\,B\,a^4}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3113\,A\,a^4}{30}+\frac {13867\,B\,a^4}{120}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {5632\,A\,a^4}{35}+\frac {896\,B\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1501\,A\,a^4}{10}+\frac {19157\,B\,a^4}{120}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (70\,A\,a^4+\frac {523\,B\,a^4}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {53\,A\,a^4}{2}+\frac {207\,B\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (44\,A+49\,B\right )}{8\,\left (\frac {11\,A\,a^4}{2}+\frac {49\,B\,a^4}{8}\right )}\right )\,\left (44\,A+49\,B\right )}{8\,d} \]

[In]

int(cos(c + d*x)^7*(A + B/cos(c + d*x))*(a + a/cos(c + d*x))^4,x)

[Out]

(tan(c/2 + (d*x)/2)*((53*A*a^4)/2 + (207*B*a^4)/8) + tan(c/2 + (d*x)/2)^13*((11*A*a^4)/2 + (49*B*a^4)/8) + tan
(c/2 + (d*x)/2)^11*((110*A*a^4)/3 + (245*B*a^4)/6) + tan(c/2 + (d*x)/2)^3*(70*A*a^4 + (523*B*a^4)/6) + tan(c/2
+ (d*x)/2)^7*((5632*A*a^4)/35 + (896*B*a^4)/5) + tan(c/2 + (d*x)/2)^9*((3113*A*a^4)/30 + (13867*B*a^4)/120) +
tan(c/2 + (d*x)/2)^5*((1501*A*a^4)/10 + (19157*B*a^4)/120))/(d*(7*tan(c/2 + (d*x)/2)^2 + 21*tan(c/2 + (d*x)/2
)^4 + 35*tan(c/2 + (d*x)/2)^6 + 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7*tan(c/2 + (d*x)/2)^12 +
tan(c/2 + (d*x)/2)^14 + 1)) + (a^4*atan((a^4*tan(c/2 + (d*x)/2)*(44*A + 49*B))/(8*((11*A*a^4)/2 + (49*B*a^4)/
8)))*(44*A + 49*B))/(8*d)