3.446 \(\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=213 \[ -\frac{2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac{a^4 (7 A+8 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (7 A+8 B+10 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (7 A+8 B+10 C)+\frac{(2 A+3 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{15 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d} \]

[Out]

(7*a^4*(7*A + 8*B + 10*C)*x)/16 + (4*a^4*(7*A + 8*B + 10*C)*Sin[c + d*x])/(5*d) + (27*a^4*(7*A + 8*B + 10*C)*C
os[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(7*A + 8*B + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((2*A + 3*B)*
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 + 8*B + 10*C)*Sin[c + d*x]^3)/(15*d)

________________________________________________________________________________________

Rubi [A]  time = 0.40923, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4086, 4013, 3791, 2637, 2635, 8, 2633} \[ -\frac{2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac{a^4 (7 A+8 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (7 A+8 B+10 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (7 A+8 B+10 C)+\frac{(2 A+3 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{15 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*a^4*(7*A + 8*B + 10*C)*x)/16 + (4*a^4*(7*A + 8*B + 10*C)*Sin[c + d*x])/(5*d) + (27*a^4*(7*A + 8*B + 10*C)*C
os[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(7*A + 8*B + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((2*A + 3*B)*
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 + 8*B + 10*C)*Sin[c + d*x]^3)/(15*d)

Rule 4086

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/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m -
b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 -
 b^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])

Rule 4013

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 3791

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 2637

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

Rule 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 2633

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]

Rubi steps

\begin{align*} \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\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 (2 a (2 A+3 B)+a (A+6 C) \sec (c+d x)) \, dx}{6 a}\\ &=\frac{(2 A+3 B) \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+8 B+10 C) \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \, dx\\ &=\frac{(2 A+3 B) \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+8 B+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+8 B+10 C) x+\frac{(2 A+3 B) \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+8 B+10 C)\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{5} \left (2 a^4 (7 A+8 B+10 C)\right ) \int \cos (c+d x) \, dx+\frac{1}{5} \left (2 a^4 (7 A+8 B+10 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{5} \left (3 a^4 (7 A+8 B+10 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{1}{10} a^4 (7 A+8 B+10 C) x+\frac{2 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac{3 a^4 (7 A+8 B+10 C) \cos (c+d x) \sin (c+d x)}{10 d}+\frac{a^4 (7 A+8 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{(2 A+3 B) \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+8 B+10 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{10} \left (3 a^4 (7 A+8 B+10 C)\right ) \int 1 \, dx-\frac{\left (2 a^4 (7 A+8 B+10 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac{2}{5} a^4 (7 A+8 B+10 C) x+\frac{4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac{27 a^4 (7 A+8 B+10 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac{a^4 (7 A+8 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{(2 A+3 B) \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+8 B+10 C) \sin ^3(c+d x)}{15 d}+\frac{1}{80} \left (3 a^4 (7 A+8 B+10 C)\right ) \int 1 \, dx\\ &=\frac{7}{16} a^4 (7 A+8 B+10 C) x+\frac{4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac{27 a^4 (7 A+8 B+10 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac{a^4 (7 A+8 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{(2 A+3 B) \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+8 B+10 C) \sin ^3(c+d x)}{15 d}\\ \end{align*}

Mathematica [A]  time = 0.496599, size = 163, normalized size = 0.77 \[ \frac{a^4 (120 (44 A+49 B+56 C) \sin (c+d x)+15 (127 A+128 B+112 C) \sin (2 (c+d x))+720 A \sin (3 (c+d x))+225 A \sin (4 (c+d x))+48 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+2940 A d x+580 B \sin (3 (c+d x))+120 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+3360 B d x+320 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+4200 C d x)}{960 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(2940*A*d*x + 3360*B*d*x + 4200*C*d*x + 120*(44*A + 49*B + 56*C)*Sin[c + d*x] + 15*(127*A + 128*B + 112*C
)*Sin[2*(c + d*x)] + 720*A*Sin[3*(c + d*x)] + 580*B*Sin[3*(c + d*x)] + 320*C*Sin[3*(c + d*x)] + 225*A*Sin[4*(c
 + d*x)] + 120*B*Sin[4*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 48*A*Sin[5*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 5*
A*Sin[6*(c + d*x)]))/(960*d)

________________________________________________________________________________________

Maple [B]  time = 0.13, size = 416, normalized size = 2. \begin{align*}{\frac{1}{d} \left ( A{a}^{4} \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +{\frac{4\,A{a}^{4}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+{\frac{B{a}^{4}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+6\,A{a}^{4} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +4\,B{a}^{4} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{a}^{4}C \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{4\,A{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,B{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +{\frac{4\,{a}^{4}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+A{a}^{4} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +4\,B{a}^{4} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +6\,{a}^{4}C \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +B{a}^{4}\sin \left ( dx+c \right ) +4\,{a}^{4}C\sin \left ( dx+c \right ) +{a}^{4}C \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 0.972799, size = 540, normalized size = 2.54 \begin{align*} \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} + 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^{4} - 1920 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 120 \,{\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} + 960 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B 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} + 960 \, B a^{4} \sin \left (d x + c\right ) + 3840 \, C a^{4} \sin \left (d x + c\right )}{960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+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 + 64
*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^4 - 1920*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^4
 + 120*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 + 960*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*
a^4 - 1280*(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 + 960*B*a^4*sin(d*x + c) + 3840*C*
a^4*sin(d*x + c))/d

________________________________________________________________________________________

Fricas [A]  time = 0.528742, size = 378, normalized size = 1.77 \begin{align*} \frac{105 \,{\left (7 \, A + 8 \, B + 10 \, C\right )} a^{4} d x +{\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 48 \,{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{4} + 10 \,{\left (41 \, A + 24 \, B + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 32 \,{\left (18 \, A + 17 \, B + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \,{\left (49 \, A + 56 \, B + 54 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \,{\left (72 \, A + 83 \, B + 100 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(105*(7*A + 8*B + 10*C)*a^4*d*x + (40*A*a^4*cos(d*x + c)^5 + 48*(4*A + B)*a^4*cos(d*x + c)^4 + 10*(41*A
+ 24*B + 6*C)*a^4*cos(d*x + c)^3 + 32*(18*A + 17*B + 10*C)*a^4*cos(d*x + c)^2 + 15*(49*A + 56*B + 54*C)*a^4*co
s(d*x + c) + 16*(72*A + 83*B + 100*C)*a^4)*sin(d*x + c))/d

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.3357, size = 473, normalized size = 2.22 \begin{align*} \frac{105 \,{\left (7 \, A a^{4} + 8 \, B 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} + 840 \, B 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} + 4760 \, B 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} + 11088 \, B 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} + 13488 \, B 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} + 9320 \, B 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 ) + 3000 \, B 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(105*(7*A*a^4 + 8*B*a^4 + 10*C*a^4)*(d*x + c) + 2*(735*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 840*B*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 + 4760*B*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 + 11088*B*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 + 13488*B*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 + 9320*B*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) + 3000*B*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