\(\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx\) [71]

Optimal result

Integrand size = 31, antiderivative size = 201 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {1}{16} a^3 (23 A+26 B) x+\frac {a^3 (17 A+19 B) \sin (c+d x)}{5 d}+\frac {a^3 (23 A+26 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (21 A+22 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A+3 B) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (17 A+19 B) \sin ^3(c+d x)}{15 d} \]

[Out]

1/16*a^3*(23*A+26*B)*x+1/5*a^3*(17*A+19*B)*sin(d*x+c)/d+1/16*a^3*(23*A+26*B)*cos(d*x+c)*sin(d*x+c)/d+1/40*a^3*
(21*A+22*B)*cos(d*x+c)^3*sin(d*x+c)/d+1/6*a*A*cos(d*x+c)^5*(a+a*sec(d*x+c))^2*sin(d*x+c)/d+1/15*(4*A+3*B)*cos(
d*x+c)^4*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d-1/15*a^3*(17*A+19*B)*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4102, 4081, 3872, 2713, 2715, 8} \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=-\frac {a^3 (17 A+19 B) \sin ^3(c+d x)}{15 d}+\frac {a^3 (17 A+19 B) \sin (c+d x)}{5 d}+\frac {a^3 (21 A+22 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {a^3 (23 A+26 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {(4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac {1}{16} a^3 x (23 A+26 B)+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d} \]

[In]

Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]

[Out]

(a^3*(23*A + 26*B)*x)/16 + (a^3*(17*A + 19*B)*Sin[c + d*x])/(5*d) + (a^3*(23*A + 26*B)*Cos[c + d*x]*Sin[c + d*
x])/(16*d) + (a^3*(21*A + 22*B)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])
^2*Sin[c + d*x])/(6*d) + ((4*A + 3*B)*Cos[c + d*x]^4*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(17*
A + 19*B)*Sin[c + d*x]^3)/(15*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 ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (2 a (4 A+3 B)+3 a (A+2 B) \sec (c+d x)) \, dx \\ & = \frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A+3 B) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{30} \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (3 a^2 (21 A+22 B)+3 a^2 (13 A+16 B) \sec (c+d x)\right ) \, dx \\ & = \frac {a^3 (21 A+22 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A+3 B) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {1}{120} \int \cos ^3(c+d x) \left (-24 a^3 (17 A+19 B)-15 a^3 (23 A+26 B) \sec (c+d x)\right ) \, dx \\ & = \frac {a^3 (21 A+22 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A+3 B) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{5} \left (a^3 (17 A+19 B)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{8} \left (a^3 (23 A+26 B)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {a^3 (23 A+26 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (21 A+22 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A+3 B) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{16} \left (a^3 (23 A+26 B)\right ) \int 1 \, dx-\frac {\left (a^3 (17 A+19 B)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {1}{16} a^3 (23 A+26 B) x+\frac {a^3 (17 A+19 B) \sin (c+d x)}{5 d}+\frac {a^3 (23 A+26 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (21 A+22 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A+3 B) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (17 A+19 B) \sin ^3(c+d x)}{15 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.67 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 (1380 A c+1380 A d x+1560 B d x+120 (21 A+23 B) \sin (c+d x)+15 (63 A+64 B) \sin (2 (c+d x))+380 A \sin (3 (c+d x))+340 B \sin (3 (c+d x))+135 A \sin (4 (c+d x))+90 B \sin (4 (c+d x))+36 A \sin (5 (c+d x))+12 B \sin (5 (c+d x))+5 A \sin (6 (c+d x)))}{960 d} \]

[In]

Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]

[Out]

(a^3*(1380*A*c + 1380*A*d*x + 1560*B*d*x + 120*(21*A + 23*B)*Sin[c + d*x] + 15*(63*A + 64*B)*Sin[2*(c + d*x)]
+ 380*A*Sin[3*(c + d*x)] + 340*B*Sin[3*(c + d*x)] + 135*A*Sin[4*(c + d*x)] + 90*B*Sin[4*(c + d*x)] + 36*A*Sin[
5*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)

Maple [A] (verified)

Time = 3.74 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.56

method	result	size
parallelrisch	\(\frac {9 \left (\left (7 A +\frac {64 B}{9}\right ) \sin \left (2 d x +2 c \right )+\frac {4 \left (19 A +17 B \right ) \sin \left (3 d x +3 c \right )}{27}+\left (A +\frac {2 B}{3}\right ) \sin \left (4 d x +4 c \right )+\frac {4 \left (A +\frac {B}{3}\right ) \sin \left (5 d x +5 c \right )}{15}+\frac {A \sin \left (6 d x +6 c \right )}{27}+\frac {8 \left (7 A +\frac {23 B}{3}\right ) \sin \left (d x +c \right )}{3}+\frac {92 \left (A +\frac {26 B}{23}\right ) d x}{9}\right ) a^{3}}{64 d}\)	\(112\)
risch	\(\frac {23 a^{3} A x}{16}+\frac {13 a^{3} x B}{8}+\frac {21 a^{3} A \sin \left (d x +c \right )}{8 d}+\frac {23 a^{3} B \sin \left (d x +c \right )}{8 d}+\frac {a^{3} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 a^{3} A \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{3}}{80 d}+\frac {9 a^{3} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {3 \sin \left (4 d x +4 c \right ) B \,a^{3}}{32 d}+\frac {19 a^{3} A \sin \left (3 d x +3 c \right )}{48 d}+\frac {17 \sin \left (3 d x +3 c \right ) B \,a^{3}}{48 d}+\frac {63 \sin \left (2 d x +2 c \right ) a^{3} A}{64 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{d}\)	\(207\)
derivativedivides	\(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {3 a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\)	\(266\)
default	\(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {3 a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\)	\(266\)

[In]

int(cos(d*x+c)^6*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

9/64*((7*A+64/9*B)*sin(2*d*x+2*c)+4/27*(19*A+17*B)*sin(3*d*x+3*c)+(A+2/3*B)*sin(4*d*x+4*c)+4/15*(A+1/3*B)*sin(
5*d*x+5*c)+1/27*A*sin(6*d*x+6*c)+8/3*(7*A+23/3*B)*sin(d*x+c)+92/9*(A+26/23*B)*d*x)*a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.65 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (23 \, A + 26 \, B\right )} a^{3} d x + {\left (40 \, A a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{4} + 10 \, {\left (23 \, A + 18 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (17 \, A + 19 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (23 \, A + 26 \, B\right )} a^{3} \cos \left (d x + c\right ) + 32 \, {\left (17 \, A + 19 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{240 \, d} \]

[In]

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/240*(15*(23*A + 26*B)*a^3*d*x + (40*A*a^3*cos(d*x + c)^5 + 48*(3*A + B)*a^3*cos(d*x + c)^4 + 10*(23*A + 18*B
)*a^3*cos(d*x + c)^3 + 16*(17*A + 19*B)*a^3*cos(d*x + c)^2 + 15*(23*A + 26*B)*a^3*cos(d*x + c) + 32*(17*A + 19
*B)*a^3)*sin(d*x + c))/d

Sympy [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*(a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.30 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3}}{960 \, d} \]

[In]

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)),x, algorithm="maxima")

[Out]

1/960*(192*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^3 - 5*(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^3 - 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 + 90*(12
*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 + 64*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin
(d*x + c))*B*a^3 - 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3 + 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(
2*d*x + 2*c))*B*a^3 + 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.21 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (23 \, A a^{3} + 26 \, B a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (345 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 390 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1955 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2210 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4554 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5148 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5814 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5988 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3165 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4190 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1530 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]

[In]

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)),x, algorithm="giac")

[Out]

1/240*(15*(23*A*a^3 + 26*B*a^3)*(d*x + c) + 2*(345*A*a^3*tan(1/2*d*x + 1/2*c)^11 + 390*B*a^3*tan(1/2*d*x + 1/2
*c)^11 + 1955*A*a^3*tan(1/2*d*x + 1/2*c)^9 + 2210*B*a^3*tan(1/2*d*x + 1/2*c)^9 + 4554*A*a^3*tan(1/2*d*x + 1/2*
c)^7 + 5148*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 5814*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 5988*B*a^3*tan(1/2*d*x + 1/2*c)
^5 + 3165*A*a^3*tan(1/2*d*x + 1/2*c)^3 + 4190*B*a^3*tan(1/2*d*x + 1/2*c)^3 + 1575*A*a^3*tan(1/2*d*x + 1/2*c) +
1530*B*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d

Mupad [B] (verification not implemented)

Time = 16.33 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.42 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {\left (\frac {23\,A\,a^3}{8}+\frac {13\,B\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {391\,A\,a^3}{24}+\frac {221\,B\,a^3}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {759\,A\,a^3}{20}+\frac {429\,B\,a^3}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {969\,A\,a^3}{20}+\frac {499\,B\,a^3}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {211\,A\,a^3}{8}+\frac {419\,B\,a^3}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {105\,A\,a^3}{8}+\frac {51\,B\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (23\,A+26\,B\right )}{8\,\left (\frac {23\,A\,a^3}{8}+\frac {13\,B\,a^3}{4}\right )}\right )\,\left (23\,A+26\,B\right )}{8\,d} \]

[In]

int(cos(c + d*x)^6*(A + B/cos(c + d*x))*(a + a/cos(c + d*x))^3,x)

[Out]

(tan(c/2 + (d*x)/2)*((105*A*a^3)/8 + (51*B*a^3)/4) + tan(c/2 + (d*x)/2)^11*((23*A*a^3)/8 + (13*B*a^3)/4) + tan
(c/2 + (d*x)/2)^3*((211*A*a^3)/8 + (419*B*a^3)/12) + tan(c/2 + (d*x)/2)^9*((391*A*a^3)/24 + (221*B*a^3)/12) +
tan(c/2 + (d*x)/2)^7*((759*A*a^3)/20 + (429*B*a^3)/10) + tan(c/2 + (d*x)/2)^5*((969*A*a^3)/20 + (499*B*a^3)/10
))/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 +
6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(23*A + 26*B))/(8*((
23*A*a^3)/8 + (13*B*a^3)/4)))*(23*A + 26*B))/(8*d)