\(\int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+C \sec ^2(c+d x)) \, dx\) [671]

Optimal result

Integrand size = 33, antiderivative size = 298 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} \left (8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) x+\frac {4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \cos (c+d x) \sin (c+d x)}{240 d}+\frac {a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac {2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d} \]

[Out]

1/16*(8*b^4*(A+2*C)+12*a^2*b^2*(3*A+4*C)+a^4*(5*A+6*C))*x+4/15*a*b*(5*b^2*(2*A+3*C)+2*a^2*(4*A+5*C))*sin(d*x+c
)/d+1/240*(24*A*b^4+15*a^4*(5*A+6*C)+10*a^2*b^2*(49*A+66*C))*cos(d*x+c)*sin(d*x+c)/d+1/60*a*b*(4*A*b^2+a^2*(39
*A+50*C))*cos(d*x+c)^2*sin(d*x+c)/d+1/120*(12*A*b^2+5*a^2*(5*A+6*C))*cos(d*x+c)^3*(a+b*sec(d*x+c))^2*sin(d*x+c
)/d+2/15*A*b*cos(d*x+c)^4*(a+b*sec(d*x+c))^3*sin(d*x+c)/d+1/6*A*cos(d*x+c)^5*(a+b*sec(d*x+c))^4*sin(d*x+c)/d

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4180, 4179, 4159, 4132, 2717, 4130, 8} \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {4 a b \left (2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \sin (c+d x)}{15 d}+\frac {a b \left (a^2 (39 A+50 C)+4 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{60 d}+\frac {\left (5 a^2 (5 A+6 C)+12 A b^2\right ) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{120 d}+\frac {\left (15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)+24 A b^4\right ) \sin (c+d x) \cos (c+d x)}{240 d}+\frac {1}{16} x \left (a^4 (5 A+6 C)+12 a^2 b^2 (3 A+4 C)+8 b^4 (A+2 C)\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}+\frac {2 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{15 d} \]

[In]

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

[Out]

((8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*x)/16 + (4*a*b*(5*b^2*(2*A + 3*C) + 2*a^2*(4*A +
5*C))*Sin[c + d*x])/(15*d) + ((24*A*b^4 + 15*a^4*(5*A + 6*C) + 10*a^2*b^2*(49*A + 66*C))*Cos[c + d*x]*Sin[c +
d*x])/(240*d) + (a*b*(4*A*b^2 + a^2*(39*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + ((12*A*b^2 + 5*a^2*(
5*A + 6*C))*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(120*d) + (2*A*b*Cos[c + d*x]^4*(a + b*Sec[c +
d*x])^3*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(6*d)

Rule 8

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

Rule 2717

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

Rule 4130

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e
+ f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rule 4132

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rule 4159

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(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) + (n*(a*C + B*b) + A*a*(n + 1))*Csc[e + f*x]
+ b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]

Rule 4179

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + 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] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*
m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]

Rule 4180

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/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*(C*n + A*(n + 1))*Csc[e +
f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2,
0] && GtQ[m, 0] && LeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (4 A b+a (5 A+6 C) \sec (c+d x)+b (A+6 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {1}{30} \int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \left (12 A b^2+5 a^2 (5 A+6 C)+2 a b (23 A+30 C) \sec (c+d x)+3 b^2 (3 A+10 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac {2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {1}{120} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (6 b \left (4 A b^2+a^2 (39 A+50 C)\right )+a \left (15 a^2 (5 A+6 C)+8 b^2 (32 A+45 C)\right ) \sec (c+d x)+b \left (24 b^2 (2 A+5 C)+5 a^2 (5 A+6 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac {2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac {1}{360} \int \cos ^2(c+d x) \left (-3 \left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right )-96 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \sec (c+d x)-3 b^2 \left (24 b^2 (2 A+5 C)+5 a^2 (5 A+6 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac {2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac {1}{360} \int \cos ^2(c+d x) \left (-3 \left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right )-3 b^2 \left (24 b^2 (2 A+5 C)+5 a^2 (5 A+6 C)\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{15} \left (4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right )\right ) \int \cos (c+d x) \, dx \\ & = \frac {4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \cos (c+d x) \sin (c+d x)}{240 d}+\frac {a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac {2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac {1}{16} \left (-8 b^4 (A+2 C)-12 a^2 b^2 (3 A+4 C)-a^4 (5 A+6 C)\right ) \int 1 \, dx \\ & = \frac {1}{16} \left (8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) x+\frac {4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \cos (c+d x) \sin (c+d x)}{240 d}+\frac {a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac {2 A b \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.01 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {300 a^4 A c+2160 a^2 A b^2 c+480 A b^4 c+360 a^4 c C+2880 a^2 b^2 c C+960 b^4 c C+300 a^4 A d x+2160 a^2 A b^2 d x+480 A b^4 d x+360 a^4 C d x+2880 a^2 b^2 C d x+960 b^4 C d x+480 a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \sin (c+d x)+15 \left (16 A b^4+96 a^2 b^2 (A+C)+a^4 (15 A+16 C)\right ) \sin (2 (c+d x))+400 a^3 A b \sin (3 (c+d x))+320 a A b^3 \sin (3 (c+d x))+320 a^3 b C \sin (3 (c+d x))+45 a^4 A \sin (4 (c+d x))+180 a^2 A b^2 \sin (4 (c+d x))+30 a^4 C \sin (4 (c+d x))+48 a^3 A b \sin (5 (c+d x))+5 a^4 A \sin (6 (c+d x))}{960 d} \]

[In]

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

[Out]

(300*a^4*A*c + 2160*a^2*A*b^2*c + 480*A*b^4*c + 360*a^4*c*C + 2880*a^2*b^2*c*C + 960*b^4*c*C + 300*a^4*A*d*x +
2160*a^2*A*b^2*d*x + 480*A*b^4*d*x + 360*a^4*C*d*x + 2880*a^2*b^2*C*d*x + 960*b^4*C*d*x + 480*a*b*(2*b^2*(3*A
+ 4*C) + a^2*(5*A + 6*C))*Sin[c + d*x] + 15*(16*A*b^4 + 96*a^2*b^2*(A + C) + a^4*(15*A + 16*C))*Sin[2*(c + d*
x)] + 400*a^3*A*b*Sin[3*(c + d*x)] + 320*a*A*b^3*Sin[3*(c + d*x)] + 320*a^3*b*C*Sin[3*(c + d*x)] + 45*a^4*A*Si
n[4*(c + d*x)] + 180*a^2*A*b^2*Sin[4*(c + d*x)] + 30*a^4*C*Sin[4*(c + d*x)] + 48*a^3*A*b*Sin[5*(c + d*x)] + 5*
a^4*A*Sin[6*(c + d*x)])/(960*d)

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.68

method	result	size
parallelrisch	\(\frac {\left (\left (225 A +240 C \right ) a^{4}+1440 a^{2} b^{2} \left (A +C \right )+240 A \,b^{4}\right ) \sin \left (2 d x +2 c \right )+400 a b \left (\left (A +\frac {4 C}{5}\right ) a^{2}+\frac {4 A \,b^{2}}{5}\right ) \sin \left (3 d x +3 c \right )+45 a^{2} \left (a^{2} \left (A +\frac {2 C}{3}\right )+4 A \,b^{2}\right ) \sin \left (4 d x +4 c \right )+48 A \,a^{3} b \sin \left (5 d x +5 c \right )+5 a^{4} A \sin \left (6 d x +6 c \right )+2400 a b \left (a^{2} \left (A +\frac {6 C}{5}\right )+\frac {6 b^{2} \left (A +\frac {4 C}{3}\right )}{5}\right ) \sin \left (d x +c \right )+300 x d \left (\left (A +\frac {6 C}{5}\right ) a^{4}+\frac {36 b^{2} \left (A +\frac {4 C}{3}\right ) a^{2}}{5}+\frac {8 b^{4} \left (A +2 C \right )}{5}\right )}{960 d}\)	\(203\)
derivativedivides	\(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 A \,a^{3} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 A \,a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {4 a^{3} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+A \,b^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{2} b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 C a \,b^{3} \sin \left (d x +c \right )+C \,b^{4} \left (d x +c \right )}{d}\)	\(294\)
default	\(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 A \,a^{3} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 A \,a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {4 a^{3} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+A \,b^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{2} b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 C a \,b^{3} \sin \left (d x +c \right )+C \,b^{4} \left (d x +c \right )}{d}\)	\(294\)
risch	\(\frac {5 a^{4} A x}{16}+\frac {9 A \,a^{2} b^{2} x}{4}+\frac {x A \,b^{4}}{2}+\frac {3 a^{4} x C}{8}+3 x C \,a^{2} b^{2}+x C \,b^{4}+\frac {5 \sin \left (d x +c \right ) A \,a^{3} b}{2 d}+\frac {3 \sin \left (d x +c \right ) a A \,b^{3}}{d}+\frac {3 \sin \left (d x +c \right ) a^{3} b C}{d}+\frac {4 \sin \left (d x +c \right ) C a \,b^{3}}{d}+\frac {a^{4} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {A \,a^{3} b \sin \left (5 d x +5 c \right )}{20 d}+\frac {3 a^{4} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{2} b^{2}}{16 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} C}{32 d}+\frac {5 A \,a^{3} b \sin \left (3 d x +3 c \right )}{12 d}+\frac {\sin \left (3 d x +3 c \right ) a A \,b^{3}}{3 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3} b C}{3 d}+\frac {15 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+\frac {3 \sin \left (2 d x +2 c \right ) A \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{4}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{4} C}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) C \,a^{2} b^{2}}{2 d}\)	\(360\)

[In]

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

[Out]

1/960*(((225*A+240*C)*a^4+1440*a^2*b^2*(A+C)+240*A*b^4)*sin(2*d*x+2*c)+400*a*b*((A+4/5*C)*a^2+4/5*A*b^2)*sin(3
*d*x+3*c)+45*a^2*(a^2*(A+2/3*C)+4*A*b^2)*sin(4*d*x+4*c)+48*A*a^3*b*sin(5*d*x+5*c)+5*a^4*A*sin(6*d*x+6*c)+2400*
a*b*(a^2*(A+6/5*C)+6/5*b^2*(A+4/3*C))*sin(d*x+c)+300*x*d*((A+6/5*C)*a^4+36/5*b^2*(A+4/3*C)*a^2+8/5*b^4*(A+2*C)
))/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.71 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 12 \, {\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \, {\left (A + 2 \, C\right )} b^{4}\right )} d x + {\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 192 \, A a^{3} b \cos \left (d x + c\right )^{4} + 128 \, {\left (4 \, A + 5 \, C\right )} a^{3} b + 320 \, {\left (2 \, A + 3 \, C\right )} a b^{3} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 36 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 64 \, {\left ({\left (4 \, A + 5 \, C\right )} a^{3} b + 5 \, A a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 12 \, {\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]

[In]

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

[Out]

1/240*(15*((5*A + 6*C)*a^4 + 12*(3*A + 4*C)*a^2*b^2 + 8*(A + 2*C)*b^4)*d*x + (40*A*a^4*cos(d*x + c)^5 + 192*A*
a^3*b*cos(d*x + c)^4 + 128*(4*A + 5*C)*a^3*b + 320*(2*A + 3*C)*a*b^3 + 10*((5*A + 6*C)*a^4 + 36*A*a^2*b^2)*cos
(d*x + c)^3 + 64*((4*A + 5*C)*a^3*b + 5*A*a*b^3)*cos(d*x + c)^2 + 15*((5*A + 6*C)*a^4 + 12*(3*A + 4*C)*a^2*b^2
+ 8*A*b^4)*cos(d*x + c))*sin(d*x + c))/d

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**6*(a+b*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 = 283, normalized size of antiderivative = 0.95 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} b + 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} b - 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} - 1440 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b^{2} + 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{3} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{4} - 960 \, {\left (d x + c\right )} C b^{4} - 3840 \, C a b^{3} \sin \left (d x + c\right )}{960 \, d} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1034 vs. \(2 (284) = 568\).

Time = 0.36 (sec) , antiderivative size = 1034, normalized size of antiderivative = 3.47 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/240*(15*(5*A*a^4 + 6*C*a^4 + 36*A*a^2*b^2 + 48*C*a^2*b^2 + 8*A*b^4 + 16*C*b^4)*(d*x + c) - 2*(165*A*a^4*tan(
1/2*d*x + 1/2*c)^11 + 150*C*a^4*tan(1/2*d*x + 1/2*c)^11 - 960*A*a^3*b*tan(1/2*d*x + 1/2*c)^11 - 960*C*a^3*b*ta
n(1/2*d*x + 1/2*c)^11 + 900*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 720*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 960*A*
a*b^3*tan(1/2*d*x + 1/2*c)^11 - 960*C*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^11 - 25*A
*a^4*tan(1/2*d*x + 1/2*c)^9 + 210*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 2240*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 3520*C*
a^3*b*tan(1/2*d*x + 1/2*c)^9 + 1260*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 2160*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 -
3520*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 4800*C*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 360*A*b^4*tan(1/2*d*x + 1/2*c)^9
+ 450*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 60*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 4992*A*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 5
760*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 360*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 1440*C*a^2*b^2*tan(1/2*d*x + 1/2*c
)^7 - 5760*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 9600*C*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 240*A*b^4*tan(1/2*d*x + 1/2*
c)^7 - 450*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 60*C*a^4*tan(1/2*d*x + 1/2*c)^5 - 4992*A*a^3*b*tan(1/2*d*x + 1/2*c)^
5 - 5760*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 360*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 - 1440*C*a^2*b^2*tan(1/2*d*x +
1/2*c)^5 - 5760*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 9600*C*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 240*A*b^4*tan(1/2*d*x +
1/2*c)^5 + 25*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 210*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 2240*A*a^3*b*tan(1/2*d*x + 1/
2*c)^3 - 3520*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 1260*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 2160*C*a^2*b^2*tan(1/2*
d*x + 1/2*c)^3 - 3520*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 4800*C*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 360*A*b^4*tan(1/2
*d*x + 1/2*c)^3 - 165*A*a^4*tan(1/2*d*x + 1/2*c) - 150*C*a^4*tan(1/2*d*x + 1/2*c) - 960*A*a^3*b*tan(1/2*d*x +
1/2*c) - 960*C*a^3*b*tan(1/2*d*x + 1/2*c) - 900*A*a^2*b^2*tan(1/2*d*x + 1/2*c) - 720*C*a^2*b^2*tan(1/2*d*x + 1
/2*c) - 960*A*a*b^3*tan(1/2*d*x + 1/2*c) - 960*C*a*b^3*tan(1/2*d*x + 1/2*c) - 120*A*b^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 = 16.55 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.20 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {5\,A\,a^4\,x}{16}+\frac {A\,b^4\,x}{2}+\frac {3\,C\,a^4\,x}{8}+C\,b^4\,x+\frac {9\,A\,a^2\,b^2\,x}{4}+3\,C\,a^2\,b^2\,x+\frac {15\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,A\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {A\,a^4\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {A\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {A\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {5\,A\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {A\,a^3\,b\,\sin \left (5\,c+5\,d\,x\right )}{20\,d}+\frac {C\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {3\,C\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {3\,A\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {5\,A\,a^3\,b\,\sin \left (c+d\,x\right )}{2\,d}+\frac {4\,C\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a^3\,b\,\sin \left (c+d\,x\right )}{d} \]

[In]

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

[Out]

(5*A*a^4*x)/16 + (A*b^4*x)/2 + (3*C*a^4*x)/8 + C*b^4*x + (9*A*a^2*b^2*x)/4 + 3*C*a^2*b^2*x + (15*A*a^4*sin(2*c
+ 2*d*x))/(64*d) + (3*A*a^4*sin(4*c + 4*d*x))/(64*d) + (A*a^4*sin(6*c + 6*d*x))/(192*d) + (A*b^4*sin(2*c + 2*
d*x))/(4*d) + (C*a^4*sin(2*c + 2*d*x))/(4*d) + (C*a^4*sin(4*c + 4*d*x))/(32*d) + (A*a*b^3*sin(3*c + 3*d*x))/(3
*d) + (5*A*a^3*b*sin(3*c + 3*d*x))/(12*d) + (A*a^3*b*sin(5*c + 5*d*x))/(20*d) + (C*a^3*b*sin(3*c + 3*d*x))/(3*
d) + (3*A*a^2*b^2*sin(2*c + 2*d*x))/(2*d) + (3*A*a^2*b^2*sin(4*c + 4*d*x))/(16*d) + (3*C*a^2*b^2*sin(2*c + 2*d
*x))/(2*d) + (3*A*a*b^3*sin(c + d*x))/d + (5*A*a^3*b*sin(c + d*x))/(2*d) + (4*C*a*b^3*sin(c + d*x))/d + (3*C*a
^3*b*sin(c + d*x))/d