\(\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 (A+C \sec ^2(c+d x)) \, dx\) [119]

Optimal result

Integrand size = 33, antiderivative size = 192 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {7}{16} a^4 (7 A+10 C) x+\frac {4 a^4 (7 A+10 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (7 A+10 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (7 A+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {2 A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac {2 a^4 (7 A+10 C) \sin ^3(c+d x)}{15 d} \]

[Out]

7/16*a^4*(7*A+10*C)*x+4/5*a^4*(7*A+10*C)*sin(d*x+c)/d+27/80*a^4*(7*A+10*C)*cos(d*x+c)*sin(d*x+c)/d+1/40*a^4*(7
*A+10*C)*cos(d*x+c)^3*sin(d*x+c)/d+2/15*A*cos(d*x+c)^4*(a+a*sec(d*x+c))^4*sin(d*x+c)/d+1/6*A*cos(d*x+c)^5*(a+a
*sec(d*x+c))^4*sin(d*x+c)/d-2/15*a^4*(7*A+10*C)*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4172, 4098, 3876, 2717, 2715, 8, 2713} \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 a^4 (7 A+10 C) \sin ^3(c+d x)}{15 d}+\frac {4 a^4 (7 A+10 C) \sin (c+d x)}{5 d}+\frac {a^4 (7 A+10 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {27 a^4 (7 A+10 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac {7}{16} a^4 x (7 A+10 C)+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d}+\frac {2 A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{15 d} \]

[In]

Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]

[Out]

(7*a^4*(7*A + 10*C)*x)/16 + (4*a^4*(7*A + 10*C)*Sin[c + d*x])/(5*d) + (27*a^4*(7*A + 10*C)*Cos[c + d*x]*Sin[c
+ d*x])/(80*d) + (a^4*(7*A + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (2*A*Cos[c + d*x]^4*(a + a*Sec[c + d*
x])^4*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(6*d) - (2*a^4*(7*A + 10*C
)*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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3876

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[Expand
Trig[(a + b*csc[e + f*x])^m*(d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0]
&& IGtQ[m, 0] && RationalQ[n]

Rule 4098

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

Rule 4172

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b
_.) + (a_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Dis
t[1/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*(A*(m + n + 1) + C*n)*Csc[e +
f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -2
^(-1)] || EqQ[m + n + 1, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 (4 a A+a (A+6 C) \sec (c+d x)) \, dx}{6 a} \\ & = \frac {2 A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {1}{10} (7 A+10 C) \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \, dx \\ & = \frac {2 A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {1}{10} (7 A+10 C) \int \left (a^4+4 a^4 \cos (c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^4(c+d x)\right ) \, dx \\ & = \frac {1}{10} a^4 (7 A+10 C) x+\frac {2 A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {1}{10} \left (a^4 (7 A+10 C)\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (7 A+10 C)\right ) \int \cos (c+d x) \, dx+\frac {1}{5} \left (2 a^4 (7 A+10 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{5} \left (3 a^4 (7 A+10 C)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {1}{10} a^4 (7 A+10 C) x+\frac {2 a^4 (7 A+10 C) \sin (c+d x)}{5 d}+\frac {3 a^4 (7 A+10 C) \cos (c+d x) \sin (c+d x)}{10 d}+\frac {a^4 (7 A+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {2 A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {1}{40} \left (3 a^4 (7 A+10 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{10} \left (3 a^4 (7 A+10 C)\right ) \int 1 \, dx-\frac {\left (2 a^4 (7 A+10 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {2}{5} a^4 (7 A+10 C) x+\frac {4 a^4 (7 A+10 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (7 A+10 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (7 A+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {2 A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac {2 a^4 (7 A+10 C) \sin ^3(c+d x)}{15 d}+\frac {1}{80} \left (3 a^4 (7 A+10 C)\right ) \int 1 \, dx \\ & = \frac {7}{16} a^4 (7 A+10 C) x+\frac {4 a^4 (7 A+10 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (7 A+10 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (7 A+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {2 A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac {2 a^4 (7 A+10 C) \sin ^3(c+d x)}{15 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.62 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (2940 A d x+4200 C d x+480 (11 A+14 C) \sin (c+d x)+15 (127 A+112 C) \sin (2 (c+d x))+720 A \sin (3 (c+d x))+320 C \sin (3 (c+d x))+225 A \sin (4 (c+d x))+30 C \sin (4 (c+d x))+48 A \sin (5 (c+d x))+5 A \sin (6 (c+d x)))}{960 d} \]

[In]

Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]

[Out]

(a^4*(2940*A*d*x + 4200*C*d*x + 480*(11*A + 14*C)*Sin[c + d*x] + 15*(127*A + 112*C)*Sin[2*(c + d*x)] + 720*A*S
in[3*(c + d*x)] + 320*C*Sin[3*(c + d*x)] + 225*A*Sin[4*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 48*A*Sin[5*(c + d*
x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.56

method	result	size
parallelrisch	\(\frac {15 a^{4} \left (\frac {\left (127 A +112 C \right ) \sin \left (2 d x +2 c \right )}{15}+\frac {16 \left (A +\frac {4 C}{9}\right ) \sin \left (3 d x +3 c \right )}{5}+\left (A +\frac {2 C}{15}\right ) \sin \left (4 d x +4 c \right )+\frac {16 A \sin \left (5 d x +5 c \right )}{75}+\frac {A \sin \left (6 d x +6 c \right )}{45}+\frac {32 \left (11 A +14 C \right ) \sin \left (d x +c \right )}{15}+\frac {196 \left (A +\frac {10 C}{7}\right ) x d}{15}\right )}{64 d}\)	\(107\)
risch	\(\frac {49 a^{4} A x}{16}+\frac {35 a^{4} x C}{8}+\frac {11 \sin \left (d x +c \right ) a^{4} A}{2 d}+\frac {7 \sin \left (d x +c \right ) a^{4} C}{d}+\frac {a^{4} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{4} A \sin \left (5 d x +5 c \right )}{20 d}+\frac {15 a^{4} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} C}{32 d}+\frac {3 a^{4} A \sin \left (3 d x +3 c \right )}{4 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} C}{3 d}+\frac {127 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+\frac {7 \sin \left (2 d x +2 c \right ) a^{4} C}{4 d}\)	\(190\)
derivativedivides	\(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 a^{4} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \sin \left (d x +c \right )+a^{4} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} C \left (d x +c \right )}{d}\)	\(284\)
default	\(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 a^{4} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \sin \left (d x +c \right )+a^{4} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} C \left (d x +c \right )}{d}\)	\(284\)

[In]

int(cos(d*x+c)^6*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

15/64*a^4*(1/15*(127*A+112*C)*sin(2*d*x+2*c)+16/5*(A+4/9*C)*sin(3*d*x+3*c)+(A+2/15*C)*sin(4*d*x+4*c)+16/75*A*s
in(5*d*x+5*c)+1/45*A*sin(6*d*x+6*c)+32/15*(11*A+14*C)*sin(d*x+c)+196/15*(A+10/7*C)*x*d)/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.66 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (7 \, A + 10 \, C\right )} a^{4} d x + {\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 192 \, A a^{4} \cos \left (d x + c\right )^{4} + 10 \, {\left (41 \, A + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 64 \, {\left (9 \, A + 5 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (49 \, A + 54 \, C\right )} a^{4} \cos \left (d x + c\right ) + 64 \, {\left (18 \, A + 25 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \]

[In]

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/240*(105*(7*A + 10*C)*a^4*d*x + (40*A*a^4*cos(d*x + c)^5 + 192*A*a^4*cos(d*x + c)^4 + 10*(41*A + 6*C)*a^4*co
s(d*x + c)^3 + 64*(9*A + 5*C)*a^4*cos(d*x + c)^2 + 15*(49*A + 54*C)*a^4*cos(d*x + c) + 64*(18*A + 25*C)*a^4)*s
in(d*x + c))/d

Sympy [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*(a+a*sec(d*x+c))**4*(A+C*sec(d*x+c)**2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.42 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1440 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 960 \, {\left (d x + c\right )} C a^{4} + 3840 \, C a^{4} \sin \left (d x + c\right )}{960 \, d} \]

[In]

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/960*(256*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 - 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^4 - 1280*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 + 180*(
12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 - 12
80*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 + 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4
+ 1440*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^4 + 960*(d*x + c)*C*a^4 + 3840*C*a^4*sin(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.27 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (7 \, A a^{4} + 10 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (735 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1050 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 4165 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5950 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9702 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 13860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11802 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7355 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10690 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2790 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]

[In]

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/240*(105*(7*A*a^4 + 10*C*a^4)*(d*x + c) + 2*(735*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 1050*C*a^4*tan(1/2*d*x + 1/
2*c)^11 + 4165*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 5950*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 9702*A*a^4*tan(1/2*d*x + 1/2
*c)^7 + 13860*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 11802*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 16860*C*a^4*tan(1/2*d*x + 1/
2*c)^5 + 7355*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 10690*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 3105*A*a^4*tan(1/2*d*x + 1/2
*c) + 2790*C*a^4*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 = 18.47 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.49 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {49\,A\,a^4}{8}+\frac {35\,C\,a^4}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {833\,A\,a^4}{24}+\frac {595\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1617\,A\,a^4}{20}+\frac {231\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1967\,A\,a^4}{20}+\frac {281\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1471\,A\,a^4}{24}+\frac {1069\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+\frac {93\,C\,a^4}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,A+10\,C\right )}{8\,\left (\frac {49\,A\,a^4}{8}+\frac {35\,C\,a^4}{4}\right )}\right )\,\left (7\,A+10\,C\right )}{8\,d} \]

[In]

int(cos(c + d*x)^6*(A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^4,x)

[Out]

(tan(c/2 + (d*x)/2)*((207*A*a^4)/8 + (93*C*a^4)/4) + tan(c/2 + (d*x)/2)^11*((49*A*a^4)/8 + (35*C*a^4)/4) + tan
(c/2 + (d*x)/2)^9*((833*A*a^4)/24 + (595*C*a^4)/12) + tan(c/2 + (d*x)/2)^7*((1617*A*a^4)/20 + (231*C*a^4)/2) +
tan(c/2 + (d*x)/2)^5*((1967*A*a^4)/20 + (281*C*a^4)/2) + tan(c/2 + (d*x)/2)^3*((1471*A*a^4)/24 + (1069*C*a^4)
/12))/(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)) + (7*a^4*atan((7*a^4*tan(c/2 + (d*x)/2)*(7*A + 10*C))
/(8*((49*A*a^4)/8 + (35*C*a^4)/4)))*(7*A + 10*C))/(8*d)