3.448 \(\int \cos ^8(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=303 \[ -\frac{a^4 (208 A+227 B+252 C) \sin ^3(c+d x)}{105 d}+\frac{a^4 (208 A+227 B+252 C) \sin (c+d x)}{35 d}+\frac{a^4 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac{a^4 (323 A+352 B+392 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{(61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{336 d}+\frac{7 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{120 d}+\frac{1}{128} a^4 x (323 A+352 B+392 C)+\frac{a (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{14 d}+\frac{A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d} \]

[Out]

(a^4*(323*A + 352*B + 392*C)*x)/128 + (a^4*(208*A + 227*B + 252*C)*Sin[c + d*x])/(35*d) + (a^4*(323*A + 352*B
+ 392*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^4*(2007*A + 2208*B + 2408*C)*Cos[c + d*x]^3*Sin[c + d*x])/(22
40*d) + (a*(A + 2*B)*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(14*d) + (A*Cos[c + d*x]^7*(a + a*Sec
[c + d*x])^4*Sin[c + d*x])/(8*d) + ((61*A + 80*B + 56*C)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x
])/(336*d) + (7*(7*A + 8*(B + C))*Cos[c + d*x]^4*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(120*d) - (a^4*(208*A
+ 227*B + 252*C)*Sin[c + d*x]^3)/(105*d)

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Rubi [A]  time = 0.794197, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4086, 4017, 3996, 3787, 2633, 2635, 8} \[ -\frac{a^4 (208 A+227 B+252 C) \sin ^3(c+d x)}{105 d}+\frac{a^4 (208 A+227 B+252 C) \sin (c+d x)}{35 d}+\frac{a^4 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac{a^4 (323 A+352 B+392 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{(61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{336 d}+\frac{7 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{120 d}+\frac{1}{128} a^4 x (323 A+352 B+392 C)+\frac{a (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{14 d}+\frac{A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^4*(323*A + 352*B + 392*C)*x)/128 + (a^4*(208*A + 227*B + 252*C)*Sin[c + d*x])/(35*d) + (a^4*(323*A + 352*B
+ 392*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^4*(2007*A + 2208*B + 2408*C)*Cos[c + d*x]^3*Sin[c + d*x])/(22
40*d) + (a*(A + 2*B)*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(14*d) + (A*Cos[c + d*x]^7*(a + a*Sec
[c + d*x])^4*Sin[c + d*x])/(8*d) + ((61*A + 80*B + 56*C)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x
])/(336*d) + (7*(7*A + 8*(B + C))*Cos[c + d*x]^4*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(120*d) - (a^4*(208*A
+ 227*B + 252*C)*Sin[c + d*x]^3)/(105*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 4017

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]

Rule 3996

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 3787

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 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]

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]

Rubi steps

\begin{align*} \int \cos ^8(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 ^7(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{8 d}+\frac{\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (4 a (A+2 B)+a (3 A+8 C) \sec (c+d x)) \, dx}{8 a}\\ &=\frac{a (A+2 B) \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{14 d}+\frac{A \cos ^7(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{8 d}+\frac{\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (a^2 (61 A+80 B+56 C)+a^2 (33 A+24 B+56 C) \sec (c+d x)\right ) \, dx}{56 a}\\ &=\frac{a (A+2 B) \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{14 d}+\frac{A \cos ^7(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{8 d}+\frac{(61 A+80 B+56 C) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac{\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (98 a^3 (7 A+8 (B+C))+3 a^3 (127 A+128 B+168 C) \sec (c+d x)\right ) \, dx}{336 a}\\ &=\frac{a (A+2 B) \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{14 d}+\frac{A \cos ^7(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{8 d}+\frac{(61 A+80 B+56 C) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac{7 (7 A+8 (B+C)) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac{\int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (3 a^4 (2007 A+2208 B+2408 C)+3 a^4 (1321 A+1424 B+1624 C) \sec (c+d x)\right ) \, dx}{1680 a}\\ &=\frac{a^4 (2007 A+2208 B+2408 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac{a (A+2 B) \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{14 d}+\frac{A \cos ^7(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{8 d}+\frac{(61 A+80 B+56 C) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac{7 (7 A+8 (B+C)) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}-\frac{\int \cos ^3(c+d x) \left (-192 a^5 (208 A+227 B+252 C)-105 a^5 (323 A+352 B+392 C) \sec (c+d x)\right ) \, dx}{6720 a}\\ &=\frac{a^4 (2007 A+2208 B+2408 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac{a (A+2 B) \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{14 d}+\frac{A \cos ^7(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{8 d}+\frac{(61 A+80 B+56 C) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac{7 (7 A+8 (B+C)) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac{1}{35} \left (a^4 (208 A+227 B+252 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{64} \left (a^4 (323 A+352 B+392 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{a^4 (323 A+352 B+392 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{a^4 (2007 A+2208 B+2408 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac{a (A+2 B) \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{14 d}+\frac{A \cos ^7(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{8 d}+\frac{(61 A+80 B+56 C) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac{7 (7 A+8 (B+C)) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac{1}{128} \left (a^4 (323 A+352 B+392 C)\right ) \int 1 \, dx-\frac{\left (a^4 (208 A+227 B+252 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac{1}{128} a^4 (323 A+352 B+392 C) x+\frac{a^4 (208 A+227 B+252 C) \sin (c+d x)}{35 d}+\frac{a^4 (323 A+352 B+392 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{a^4 (2007 A+2208 B+2408 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac{a (A+2 B) \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{14 d}+\frac{A \cos ^7(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{8 d}+\frac{(61 A+80 B+56 C) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac{7 (7 A+8 (B+C)) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}-\frac{a^4 (208 A+227 B+252 C) \sin ^3(c+d x)}{105 d}\\ \end{align*}

Mathematica [A]  time = 1.9343, size = 237, normalized size = 0.78 \[ \frac{a^4 (1680 (300 A+323 B+352 C) \sin (c+d x)+1680 (120 A+124 B+127 C) \sin (2 (c+d x))+91840 A \sin (3 (c+d x))+39480 A \sin (4 (c+d x))+14784 A \sin (5 (c+d x))+4480 A \sin (6 (c+d x))+960 A \sin (7 (c+d x))+105 A \sin (8 (c+d x))+106680 A c+271320 A d x+87920 B \sin (3 (c+d x))+33600 B \sin (4 (c+d x))+10416 B \sin (5 (c+d x))+2240 B \sin (6 (c+d x))+240 B \sin (7 (c+d x))+295680 B c+295680 B d x+80640 C \sin (3 (c+d x))+25200 C \sin (4 (c+d x))+5376 C \sin (5 (c+d x))+560 C \sin (6 (c+d x))+329280 C d x)}{107520 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^4*(106680*A*c + 295680*B*c + 271320*A*d*x + 295680*B*d*x + 329280*C*d*x + 1680*(300*A + 323*B + 352*C)*Sin[
c + d*x] + 1680*(120*A + 124*B + 127*C)*Sin[2*(c + d*x)] + 91840*A*Sin[3*(c + d*x)] + 87920*B*Sin[3*(c + d*x)]
+ 80640*C*Sin[3*(c + d*x)] + 39480*A*Sin[4*(c + d*x)] + 33600*B*Sin[4*(c + d*x)] + 25200*C*Sin[4*(c + d*x)] +
14784*A*Sin[5*(c + d*x)] + 10416*B*Sin[5*(c + d*x)] + 5376*C*Sin[5*(c + d*x)] + 4480*A*Sin[6*(c + d*x)] + 224
0*B*Sin[6*(c + d*x)] + 560*C*Sin[6*(c + d*x)] + 960*A*Sin[7*(c + d*x)] + 240*B*Sin[7*(c + d*x)] + 105*A*Sin[8*
(c + d*x)]))/(107520*d)

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Maple [B]  time = 0.221, size = 577, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*(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*(1/8*(cos(d*x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos(d*x+c))*sin(d*x+c)+35/128*d*x+35/1
28*c)+1/7*B*a^4*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c)+a^4*C*(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)+4/7*A*a^4*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*co
s(d*x+c)^2)*sin(d*x+c)+4*B*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^4*C*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+6*A*a^4*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*co
s(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+6/5*B*a^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+6*a^4*C*(1/4*(c
os(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4/5*A*a^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+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)+4/3*a^4*C*(2+cos(d*x+c)^2)*sin(d*x+c)+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)+1/3*B*a^4*(2+cos(d*x+c)^2)*sin(d*x+c)+a^4*C*(
1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))

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Maxima [B]  time = 0.99332, size = 782, normalized size = 2.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/107520*(12288*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*A*a^4 - 28672*(3
*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 + 35*(128*sin(2*d*x + 2*c)^3 - 840*d*x - 840*c -
3*sin(8*d*x + 8*c) - 168*sin(4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*A*a^4 + 3360*(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 - 3360*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*
x + 2*c))*A*a^4 + 3072*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*B*a^4 - 43
008*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^4 + 2240*(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))*B*a^4 + 35840*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^4 - 13440*(1
2*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 - 28672*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15
*sin(d*x + c))*C*a^4 + 560*(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))*C
*a^4 + 143360*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 - 20160*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x
+ 2*c))*C*a^4 - 26880*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^4)/d

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Fricas [A]  time = 0.547145, size = 536, normalized size = 1.77 \begin{align*} \frac{105 \,{\left (323 \, A + 352 \, B + 392 \, C\right )} a^{4} d x +{\left (1680 \, A a^{4} \cos \left (d x + c\right )^{7} + 1920 \,{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{6} + 280 \,{\left (55 \, A + 32 \, B + 8 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 1536 \,{\left (13 \, A + 12 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \,{\left (323 \, A + 352 \, B + 328 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 128 \,{\left (208 \, A + 227 \, B + 252 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \,{\left (323 \, A + 352 \, B + 392 \, C\right )} a^{4} \cos \left (d x + c\right ) + 256 \,{\left (208 \, A + 227 \, B + 252 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{13440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/13440*(105*(323*A + 352*B + 392*C)*a^4*d*x + (1680*A*a^4*cos(d*x + c)^7 + 1920*(4*A + B)*a^4*cos(d*x + c)^6
+ 280*(55*A + 32*B + 8*C)*a^4*cos(d*x + c)^5 + 1536*(13*A + 12*B + 7*C)*a^4*cos(d*x + c)^4 + 70*(323*A + 352*B
+ 328*C)*a^4*cos(d*x + c)^3 + 128*(208*A + 227*B + 252*C)*a^4*cos(d*x + c)^2 + 105*(323*A + 352*B + 392*C)*a^
4*cos(d*x + c) + 256*(208*A + 227*B + 252*C)*a^4)*sin(d*x + c))/d

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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)**8*(a+a*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Timed out

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Giac [A]  time = 1.31838, size = 610, normalized size = 2.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/13440*(105*(323*A*a^4 + 352*B*a^4 + 392*C*a^4)*(d*x + c) + 2*(33915*A*a^4*tan(1/2*d*x + 1/2*c)^15 + 36960*B*
a^4*tan(1/2*d*x + 1/2*c)^15 + 41160*C*a^4*tan(1/2*d*x + 1/2*c)^15 + 260015*A*a^4*tan(1/2*d*x + 1/2*c)^13 + 283
360*B*a^4*tan(1/2*d*x + 1/2*c)^13 + 315560*C*a^4*tan(1/2*d*x + 1/2*c)^13 + 865963*A*a^4*tan(1/2*d*x + 1/2*c)^1
1 + 943712*B*a^4*tan(1/2*d*x + 1/2*c)^11 + 1050952*C*a^4*tan(1/2*d*x + 1/2*c)^11 + 1632119*A*a^4*tan(1/2*d*x +
1/2*c)^9 + 1778656*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 1980776*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 1872009*A*a^4*tan(1/
2*d*x + 1/2*c)^7 + 2090016*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 2277016*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 1442133*A*a^4
*tan(1/2*d*x + 1/2*c)^5 + 1479072*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 1658552*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 528465
*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 648480*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 759640*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 18
1125*A*a^4*tan(1/2*d*x + 1/2*c) + 178080*B*a^4*tan(1/2*d*x + 1/2*c) + 173880*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(
1/2*d*x + 1/2*c)^2 + 1)^8)/d