3.9.86 \(\int \cos ^6(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [886]

Optimal. Leaf size=320 \[ \frac {1}{16} \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) x+\frac {\left (12 a^3 B+42 a b^2 B+9 a^2 b (4 A+5 C)+b^3 (11 A+15 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {\left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) \sin ^3(c+d x)}{15 d} \]

[Out]

1/16*(18*a^2*b*B+8*b^3*B+6*a*b^2*(3*A+4*C)+a^3*(5*A+6*C))*x+1/15*(12*a^3*B+42*a*b^2*B+9*a^2*b*(4*A+5*C)+b^3*(1
1*A+15*C))*sin(d*x+c)/d+1/16*(18*a^2*b*B+8*b^3*B+6*a*b^2*(3*A+4*C)+a^3*(5*A+6*C))*cos(d*x+c)*sin(d*x+c)/d+1/12
0*a*(6*A*b^2+42*a*b*B+5*a^2*(5*A+6*C))*cos(d*x+c)^3*sin(d*x+c)/d+1/10*(A*b+2*B*a)*cos(d*x+c)^4*(a+b*sec(d*x+c)
)^2*sin(d*x+c)/d+1/6*A*cos(d*x+c)^5*(a+b*sec(d*x+c))^3*sin(d*x+c)/d-1/15*(A*b^3+4*a^3*B+12*a*b^2*B+3*a^2*b*(4*
A+5*C))*sin(d*x+c)^3/d

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Rubi [A]
time = 0.61, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4179, 4159, 4132, 2715, 8, 4129, 3092} \begin {gather*} \frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{120 d}-\frac {\sin ^3(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{15 d}+\frac {\sin (c+d x) \left (12 a^3 B+9 a^2 b (4 A+5 C)+42 a b^2 B+b^3 (11 A+15 C)\right )}{15 d}+\frac {\sin (c+d x) \cos (c+d x) \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right )}{16 d}+\frac {1}{16} x \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right )+\frac {(2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{10 d}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

((18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*x)/16 + ((12*a^3*B + 42*a*b^2*B + 9*a^2*b*(4*A
+ 5*C) + b^3*(11*A + 15*C))*Sin[c + d*x])/(15*d) + ((18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A +
6*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(6*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 6*C))*Cos[c + d*x]^3*Sin[c + d
*x])/(120*d) + ((A*b + 2*a*B)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(10*d) + (A*Cos[c + d*x]^5*(
a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) - ((A*b^3 + 4*a^3*B + 12*a*b^2*B + 3*a^2*b*(4*A + 5*C))*Sin[c + d*x]
^3)/(15*d)

Rule 8

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

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 3092

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[-f^(-1), Subst[I
nt[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2
, 0]

Rule 4129

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Int[(C + A*Sin[e + f*
x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[{e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]

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]

Rubi steps

\begin {align*} \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (3 (A b+2 a B)+(5 a A+6 b B+6 a C) \sec (c+d x)+2 b (A+3 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{30} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)+\left (24 a^2 B+30 b^2 B+a b (47 A+60 C)\right ) \sec (c+d x)+2 b (8 A b+6 a B+15 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {1}{120} \int \cos ^3(c+d x) \left (-24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right )-15 \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \sec (c+d x)-8 b^2 (8 A b+6 a B+15 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {1}{120} \int \cos ^3(c+d x) \left (-24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right )-8 b^2 (8 A b+6 a B+15 b C) \sec ^2(c+d x)\right ) \, dx-\frac {1}{8} \left (-18 a^2 b B-8 b^3 B-6 a b^2 (3 A+4 C)-a^3 (5 A+6 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {1}{120} \int \cos (c+d x) \left (-8 b^2 (8 A b+6 a B+15 b C)-24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) \cos ^2(c+d x)\right ) \, dx-\frac {1}{16} \left (-18 a^2 b B-8 b^3 B-6 a b^2 (3 A+4 C)-a^3 (5 A+6 C)\right ) \int 1 \, dx\\ &=\frac {1}{16} \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) x+\frac {\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {\text {Subst}\left (\int \left (-8 b^2 (8 A b+6 a B+15 b C)-24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right )+24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{120 d}\\ &=\frac {1}{16} \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) x+\frac {\left (12 a^3 B+42 a b^2 B+9 a^2 b (4 A+5 C)+b^3 (11 A+15 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {\left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) \sin ^3(c+d x)}{15 d}\\ \end {align*}

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Mathematica [A]
time = 1.20, size = 369, normalized size = 1.15 \begin {gather*} \frac {300 a^3 A c+1080 a A b^2 c+1080 a^2 b B c+480 b^3 B c+360 a^3 c C+1440 a b^2 c C+300 a^3 A d x+1080 a A b^2 d x+1080 a^2 b B d x+480 b^3 B d x+360 a^3 C d x+1440 a b^2 C d x+120 \left (5 a^3 B+18 a b^2 B+2 b^3 (3 A+4 C)+3 a^2 b (5 A+6 C)\right ) \sin (c+d x)+15 \left (48 a^2 b B+16 b^3 B+48 a b^2 (A+C)+a^3 (15 A+16 C)\right ) \sin (2 (c+d x))+300 a^2 A b \sin (3 (c+d x))+80 A b^3 \sin (3 (c+d x))+100 a^3 B \sin (3 (c+d x))+240 a b^2 B \sin (3 (c+d x))+240 a^2 b C \sin (3 (c+d x))+45 a^3 A \sin (4 (c+d x))+90 a A b^2 \sin (4 (c+d x))+90 a^2 b B \sin (4 (c+d x))+30 a^3 C \sin (4 (c+d x))+36 a^2 A b \sin (5 (c+d x))+12 a^3 B \sin (5 (c+d x))+5 a^3 A \sin (6 (c+d x))}{960 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(300*a^3*A*c + 1080*a*A*b^2*c + 1080*a^2*b*B*c + 480*b^3*B*c + 360*a^3*c*C + 1440*a*b^2*c*C + 300*a^3*A*d*x +
1080*a*A*b^2*d*x + 1080*a^2*b*B*d*x + 480*b^3*B*d*x + 360*a^3*C*d*x + 1440*a*b^2*C*d*x + 120*(5*a^3*B + 18*a*b
^2*B + 2*b^3*(3*A + 4*C) + 3*a^2*b*(5*A + 6*C))*Sin[c + d*x] + 15*(48*a^2*b*B + 16*b^3*B + 48*a*b^2*(A + C) +
a^3*(15*A + 16*C))*Sin[2*(c + d*x)] + 300*a^2*A*b*Sin[3*(c + d*x)] + 80*A*b^3*Sin[3*(c + d*x)] + 100*a^3*B*Sin
[3*(c + d*x)] + 240*a*b^2*B*Sin[3*(c + d*x)] + 240*a^2*b*C*Sin[3*(c + d*x)] + 45*a^3*A*Sin[4*(c + d*x)] + 90*a
*A*b^2*Sin[4*(c + d*x)] + 90*a^2*b*B*Sin[4*(c + d*x)] + 30*a^3*C*Sin[4*(c + d*x)] + 36*a^2*A*b*Sin[5*(c + d*x)
] + 12*a^3*B*Sin[5*(c + d*x)] + 5*a^3*A*Sin[6*(c + d*x)])/(960*d)

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Maple [A]
time = 0.10, size = 370, normalized size = 1.16

method	result	size
derivativedivides	\(\frac {\frac {A \,b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+b^{3} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,b^{3} \sin \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a \,b^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 C \,b^{2} a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {3 A \,a^{2} b \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{2} b B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{2} b C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+A \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{3} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(370\)
default	\(\frac {\frac {A \,b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+b^{3} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,b^{3} \sin \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a \,b^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 C \,b^{2} a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {3 A \,a^{2} b \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{2} b B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{2} b C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+A \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{3} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(370\)
risch	\(\frac {5 a^{3} A x}{16}+\frac {9 \sin \left (d x +c \right ) a \,b^{2} B}{4 d}+\frac {9 \sin \left (d x +c \right ) a^{2} b C}{4 d}+\frac {3 \sin \left (4 d x +4 c \right ) a^{2} b B}{32 d}+\frac {\sin \left (3 d x +3 c \right ) a \,b^{2} B}{4 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} b C}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) C \,b^{2} a}{4 d}+\frac {5 A \,a^{2} b \sin \left (3 d x +3 c \right )}{16 d}+\frac {3 \sin \left (2 d x +2 c \right ) a A \,b^{2}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{2} b B}{4 d}+\frac {\sin \left (3 d x +3 c \right ) A \,b^{3}}{12 d}+\frac {5 a^{3} B \sin \left (3 d x +3 c \right )}{48 d}+\frac {3 \sin \left (d x +c \right ) A \,b^{3}}{4 d}+\frac {A \,a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {15 \sin \left (2 d x +2 c \right ) A \,a^{3}}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{4 d}+\frac {3 C a \,b^{2} x}{2}+\frac {3 A \,a^{3} \sin \left (4 d x +4 c \right )}{64 d}+\frac {9 B \,a^{2} b x}{8}+\frac {5 a^{3} B \sin \left (d x +c \right )}{8 d}+\frac {9 A a \,b^{2} x}{8}+\frac {\sin \left (4 d x +4 c \right ) a^{3} C}{32 d}+\frac {\sin \left (2 d x +2 c \right ) b^{3} B}{4 d}+\frac {\sin \left (d x +c \right ) C \,b^{3}}{d}+\frac {a^{3} B \sin \left (5 d x +5 c \right )}{80 d}+\frac {x \,b^{3} B}{2}+\frac {3 C \,a^{3} x}{8}+\frac {15 \sin \left (d x +c \right ) A \,a^{2} b}{8 d}+\frac {3 \sin \left (5 d x +5 c \right ) A \,a^{2} b}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) a A \,b^{2}}{32 d}\)	\(472\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/3*A*b^3*(2+cos(d*x+c)^2)*sin(d*x+c)+b^3*B*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+C*b^3*sin(d*x+c)+3*
a*A*b^2*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+a*b^2*B*(2+cos(d*x+c)^2)*sin(d*x+c)+3*C*b
^2*a*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+3/5*A*a^2*b*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+3*a^
2*b*B*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+a^2*b*C*(2+cos(d*x+c)^2)*sin(d*x+c)+A*a^3*(
1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+1/5*a^3*B*(8/3+cos(d*x+c)^4+4/
3*cos(d*x+c)^2)*sin(d*x+c)+a^3*C*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c))

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Maxima [A]
time = 0.28, size = 360, normalized size = 1.12 \begin {gather*} -\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^{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} - 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^{3} - 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^{2} b - 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^{2} b + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b - 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 b^{2} + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{2} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{2} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{3} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{3} - 960 \, C b^{3} \sin \left (d x + c\right )}{960 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+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^3 - 64*(3*sin(
d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^3 - 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x +
2*c))*C*a^3 - 192*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^2*b - 90*(12*d*x + 12*c + sin(
4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^2*b + 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^2*b - 90*(12*d*x + 12*c
+ sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a*b^2 + 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a*b^2 - 720*(2*d*x
+ 2*c + sin(2*d*x + 2*c))*C*a*b^2 + 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*b^3 - 240*(2*d*x + 2*c + sin(2*d*
x + 2*c))*B*b^3 - 960*C*b^3*sin(d*x + c))/d

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Fricas [A]
time = 2.98, size = 256, normalized size = 0.80 \begin {gather*} \frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} d x + {\left (40 \, A a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{4} + 128 \, B a^{3} + 96 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 480 \, B a b^{2} + 80 \, {\left (2 \, A + 3 \, C\right )} b^{3} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (4 \, B a^{3} + 3 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/240*(15*((5*A + 6*C)*a^3 + 18*B*a^2*b + 6*(3*A + 4*C)*a*b^2 + 8*B*b^3)*d*x + (40*A*a^3*cos(d*x + c)^5 + 48*(
B*a^3 + 3*A*a^2*b)*cos(d*x + c)^4 + 128*B*a^3 + 96*(4*A + 5*C)*a^2*b + 480*B*a*b^2 + 80*(2*A + 3*C)*b^3 + 10*(
(5*A + 6*C)*a^3 + 18*B*a^2*b + 18*A*a*b^2)*cos(d*x + c)^3 + 16*(4*B*a^3 + 3*(4*A + 5*C)*a^2*b + 15*B*a*b^2 + 5
*A*b^3)*cos(d*x + c)^2 + 15*((5*A + 6*C)*a^3 + 18*B*a^2*b + 6*(3*A + 4*C)*a*b^2 + 8*B*b^3)*cos(d*x + c))*sin(d
*x + c))/d

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*(a+b*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1307 vs. \(2 (306) = 612\).
time = 0.56, size = 1307, normalized size = 4.08 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/240*(15*(5*A*a^3 + 6*C*a^3 + 18*B*a^2*b + 18*A*a*b^2 + 24*C*a*b^2 + 8*B*b^3)*(d*x + c) - 2*(165*A*a^3*tan(1/
2*d*x + 1/2*c)^11 - 240*B*a^3*tan(1/2*d*x + 1/2*c)^11 + 150*C*a^3*tan(1/2*d*x + 1/2*c)^11 - 720*A*a^2*b*tan(1/
2*d*x + 1/2*c)^11 + 450*B*a^2*b*tan(1/2*d*x + 1/2*c)^11 - 720*C*a^2*b*tan(1/2*d*x + 1/2*c)^11 + 450*A*a*b^2*ta
n(1/2*d*x + 1/2*c)^11 - 720*B*a*b^2*tan(1/2*d*x + 1/2*c)^11 + 360*C*a*b^2*tan(1/2*d*x + 1/2*c)^11 - 240*A*b^3*
tan(1/2*d*x + 1/2*c)^11 + 120*B*b^3*tan(1/2*d*x + 1/2*c)^11 - 240*C*b^3*tan(1/2*d*x + 1/2*c)^11 - 25*A*a^3*tan
(1/2*d*x + 1/2*c)^9 - 560*B*a^3*tan(1/2*d*x + 1/2*c)^9 + 210*C*a^3*tan(1/2*d*x + 1/2*c)^9 - 1680*A*a^2*b*tan(1
/2*d*x + 1/2*c)^9 + 630*B*a^2*b*tan(1/2*d*x + 1/2*c)^9 - 2640*C*a^2*b*tan(1/2*d*x + 1/2*c)^9 + 630*A*a*b^2*tan
(1/2*d*x + 1/2*c)^9 - 2640*B*a*b^2*tan(1/2*d*x + 1/2*c)^9 + 1080*C*a*b^2*tan(1/2*d*x + 1/2*c)^9 - 880*A*b^3*ta
n(1/2*d*x + 1/2*c)^9 + 360*B*b^3*tan(1/2*d*x + 1/2*c)^9 - 1200*C*b^3*tan(1/2*d*x + 1/2*c)^9 + 450*A*a^3*tan(1/
2*d*x + 1/2*c)^7 - 1248*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 60*C*a^3*tan(1/2*d*x + 1/2*c)^7 - 3744*A*a^2*b*tan(1/2*
d*x + 1/2*c)^7 + 180*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 4320*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 180*A*a*b^2*tan(1/
2*d*x + 1/2*c)^7 - 4320*B*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 720*C*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 1440*A*b^3*tan(1
/2*d*x + 1/2*c)^7 + 240*B*b^3*tan(1/2*d*x + 1/2*c)^7 - 2400*C*b^3*tan(1/2*d*x + 1/2*c)^7 - 450*A*a^3*tan(1/2*d
*x + 1/2*c)^5 - 1248*B*a^3*tan(1/2*d*x + 1/2*c)^5 - 60*C*a^3*tan(1/2*d*x + 1/2*c)^5 - 3744*A*a^2*b*tan(1/2*d*x
+ 1/2*c)^5 - 180*B*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 4320*C*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 180*A*a*b^2*tan(1/2*d
*x + 1/2*c)^5 - 4320*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 720*C*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 1440*A*b^3*tan(1/2*
d*x + 1/2*c)^5 - 240*B*b^3*tan(1/2*d*x + 1/2*c)^5 - 2400*C*b^3*tan(1/2*d*x + 1/2*c)^5 + 25*A*a^3*tan(1/2*d*x +
1/2*c)^3 - 560*B*a^3*tan(1/2*d*x + 1/2*c)^3 - 210*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 1680*A*a^2*b*tan(1/2*d*x + 1
/2*c)^3 - 630*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 2640*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 630*A*a*b^2*tan(1/2*d*x +
1/2*c)^3 - 2640*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 1080*C*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 880*A*b^3*tan(1/2*d*x
+ 1/2*c)^3 - 360*B*b^3*tan(1/2*d*x + 1/2*c)^3 - 1200*C*b^3*tan(1/2*d*x + 1/2*c)^3 - 165*A*a^3*tan(1/2*d*x + 1/
2*c) - 240*B*a^3*tan(1/2*d*x + 1/2*c) - 150*C*a^3*tan(1/2*d*x + 1/2*c) - 720*A*a^2*b*tan(1/2*d*x + 1/2*c) - 45
0*B*a^2*b*tan(1/2*d*x + 1/2*c) - 720*C*a^2*b*tan(1/2*d*x + 1/2*c) - 450*A*a*b^2*tan(1/2*d*x + 1/2*c) - 720*B*a
*b^2*tan(1/2*d*x + 1/2*c) - 360*C*a*b^2*tan(1/2*d*x + 1/2*c) - 240*A*b^3*tan(1/2*d*x + 1/2*c) - 120*B*b^3*tan(
1/2*d*x + 1/2*c) - 240*C*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d

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Mupad [B]
time = 5.82, size = 471, normalized size = 1.47 \begin {gather*} \frac {5\,A\,a^3\,x}{16}+\frac {B\,b^3\,x}{2}+\frac {3\,C\,a^3\,x}{8}+\frac {9\,A\,a\,b^2\,x}{8}+\frac {9\,B\,a^2\,b\,x}{8}+\frac {3\,C\,a\,b^2\,x}{2}+\frac {3\,A\,b^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {5\,B\,a^3\,\sin \left (c+d\,x\right )}{8\,d}+\frac {C\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {15\,A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,A\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {A\,a^3\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {A\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {5\,B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,a^3\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {B\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,A\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,A\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{16\,d}+\frac {3\,A\,a\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,A\,a^2\,b\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {3\,B\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {3\,B\,a^2\,b\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,C\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {15\,A\,a^2\,b\,\sin \left (c+d\,x\right )}{8\,d}+\frac {9\,B\,a\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {9\,C\,a^2\,b\,\sin \left (c+d\,x\right )}{4\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6*(a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

(5*A*a^3*x)/16 + (B*b^3*x)/2 + (3*C*a^3*x)/8 + (9*A*a*b^2*x)/8 + (9*B*a^2*b*x)/8 + (3*C*a*b^2*x)/2 + (3*A*b^3*
sin(c + d*x))/(4*d) + (5*B*a^3*sin(c + d*x))/(8*d) + (C*b^3*sin(c + d*x))/d + (15*A*a^3*sin(2*c + 2*d*x))/(64*
d) + (3*A*a^3*sin(4*c + 4*d*x))/(64*d) + (A*a^3*sin(6*c + 6*d*x))/(192*d) + (A*b^3*sin(3*c + 3*d*x))/(12*d) +
(5*B*a^3*sin(3*c + 3*d*x))/(48*d) + (B*a^3*sin(5*c + 5*d*x))/(80*d) + (B*b^3*sin(2*c + 2*d*x))/(4*d) + (C*a^3*
sin(2*c + 2*d*x))/(4*d) + (C*a^3*sin(4*c + 4*d*x))/(32*d) + (3*A*a*b^2*sin(2*c + 2*d*x))/(4*d) + (5*A*a^2*b*si
n(3*c + 3*d*x))/(16*d) + (3*A*a*b^2*sin(4*c + 4*d*x))/(32*d) + (3*A*a^2*b*sin(5*c + 5*d*x))/(80*d) + (3*B*a^2*
b*sin(2*c + 2*d*x))/(4*d) + (B*a*b^2*sin(3*c + 3*d*x))/(4*d) + (3*B*a^2*b*sin(4*c + 4*d*x))/(32*d) + (3*C*a*b^
2*sin(2*c + 2*d*x))/(4*d) + (C*a^2*b*sin(3*c + 3*d*x))/(4*d) + (15*A*a^2*b*sin(c + d*x))/(8*d) + (9*B*a*b^2*si
n(c + d*x))/(4*d) + (9*C*a^2*b*sin(c + d*x))/(4*d)

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