3.5.48 \(\int \cos ^8(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [448]

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

[Out]

1/128*a^4*(323*A+352*B+392*C)*x+1/35*a^4*(208*A+227*B+252*C)*sin(d*x+c)/d+1/128*a^4*(323*A+352*B+392*C)*cos(d*
x+c)*sin(d*x+c)/d+1/2240*a^4*(2007*A+2208*B+2408*C)*cos(d*x+c)^3*sin(d*x+c)/d+1/14*a*(A+2*B)*cos(d*x+c)^6*(a+a
*sec(d*x+c))^3*sin(d*x+c)/d+1/8*A*cos(d*x+c)^7*(a+a*sec(d*x+c))^4*sin(d*x+c)/d+1/336*(61*A+80*B+56*C)*cos(d*x+
c)^5*(a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/d+7/120*(7*A+8*B+8*C)*cos(d*x+c)^4*(a^4+a^4*sec(d*x+c))*sin(d*x+c)/d-1/
105*a^4*(208*A+227*B+252*C)*sin(d*x+c)^3/d

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Rubi [A]
time = 0.56, antiderivative size = 303, normalized size of antiderivative = 1.00, 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 = {4171, 4102, 4081, 3872, 2713, 2715, 8} \begin {gather*} -\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 {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 {(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 {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} \end {gather*}

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 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 3872

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 4081

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 4102

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 4171

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])

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 ) \text {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*}

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Mathematica [A]
time = 2.14, size = 237, normalized size = 0.78 \begin {gather*} \frac {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))+2240 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} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(576\) vs. \(2(286)=572\).
time = 1.49, size = 577, normalized size = 1.90 Too large to display

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,method=_RETURNVERBOSE)

[Out]

1/d*(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*a^4*B*(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)+4/5*A*a^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4*a^4
*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)+4/3*a^4*C*(2+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*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+6/5*a^4*B*(8/3+cos(d*x+c)^4+4
/3*cos(d*x+c)^2)*sin(d*x+c)+6*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)+4/7*A*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)+4*a^4*B*(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)+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+35/128*c)+1/7*a^4*B*(
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))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (286) = 572\).
time = 0.30, size = 579, normalized size = 1.91 \begin {gather*} -\frac {12288 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} A a^{4} - 28672 \, {\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} + 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 3360 \, {\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} - 3360 \, {\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} + 3072 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} B a^{4} - 43008 \, {\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} + 2240 \, {\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 )} B a^{4} + 35840 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 13440 \, {\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} - 28672 \, {\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 )} C a^{4} + 560 \, {\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 )} C a^{4} + 143360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 20160 \, {\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} - 26880 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4}}{107520 \, d} \end {gather*}

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 = 2.52, size = 191, normalized size = 0.63 \begin {gather*} \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 {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.56, size = 452, normalized size = 1.49 \begin {gather*} \frac {105 \, {\left (323 \, A a^{4} + 352 \, B a^{4} + 392 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (33915 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 36960 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 41160 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 260015 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 283360 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 315560 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 865963 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 943712 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1050952 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1632119 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1778656 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1980776 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1872009 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2090016 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2277016 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1442133 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1479072 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1658552 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 528465 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 648480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 759640 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 181125 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 178080 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 173880 \, 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 )}^{8}}}{13440 \, d} \end {gather*}

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

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Mupad [B]
time = 5.98, size = 421, normalized size = 1.39 \begin {gather*} \frac {\left (\frac {323\,A\,a^4}{64}+\frac {11\,B\,a^4}{2}+\frac {49\,C\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+\left (\frac {7429\,A\,a^4}{192}+\frac {253\,B\,a^4}{6}+\frac {1127\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {123709\,A\,a^4}{960}+\frac {4213\,B\,a^4}{30}+\frac {18767\,C\,a^4}{120}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1632119\,A\,a^4}{6720}+\frac {55583\,B\,a^4}{210}+\frac {35371\,C\,a^4}{120}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {624003\,A\,a^4}{2240}+\frac {21771\,B\,a^4}{70}+\frac {40661\,C\,a^4}{120}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {68673\,A\,a^4}{320}+\frac {2201\,B\,a^4}{10}+\frac {29617\,C\,a^4}{120}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {5033\,A\,a^4}{64}+\frac {193\,B\,a^4}{2}+\frac {2713\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {1725\,A\,a^4}{64}+\frac {53\,B\,a^4}{2}+\frac {207\,C\,a^4}{8}\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 )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (323\,A+352\,B+392\,C\right )}{64\,\left (\frac {323\,A\,a^4}{64}+\frac {11\,B\,a^4}{2}+\frac {49\,C\,a^4}{8}\right )}\right )\,\left (323\,A+352\,B+392\,C\right )}{64\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^8*(a + a/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

(tan(c/2 + (d*x)/2)^15*((323*A*a^4)/64 + (11*B*a^4)/2 + (49*C*a^4)/8) + tan(c/2 + (d*x)/2)^3*((5033*A*a^4)/64
+ (193*B*a^4)/2 + (2713*C*a^4)/24) + tan(c/2 + (d*x)/2)^13*((7429*A*a^4)/192 + (253*B*a^4)/6 + (1127*C*a^4)/24
) + tan(c/2 + (d*x)/2)^5*((68673*A*a^4)/320 + (2201*B*a^4)/10 + (29617*C*a^4)/120) + tan(c/2 + (d*x)/2)^11*((1
23709*A*a^4)/960 + (4213*B*a^4)/30 + (18767*C*a^4)/120) + tan(c/2 + (d*x)/2)^7*((624003*A*a^4)/2240 + (21771*B
*a^4)/70 + (40661*C*a^4)/120) + tan(c/2 + (d*x)/2)^9*((1632119*A*a^4)/6720 + (55583*B*a^4)/210 + (35371*C*a^4)
/120) + tan(c/2 + (d*x)/2)*((1725*A*a^4)/64 + (53*B*a^4)/2 + (207*C*a^4)/8))/(d*(8*tan(c/2 + (d*x)/2)^2 + 28*t
an(c/2 + (d*x)/2)^4 + 56*tan(c/2 + (d*x)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 + 56*tan(c/2 + (d*x)/2)^10 + 28*tan(c/
2 + (d*x)/2)^12 + 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1)) + (a^4*atan((a^4*tan(c/2 + (d*x)/2)*(3
23*A + 352*B + 392*C))/(64*((323*A*a^4)/64 + (11*B*a^4)/2 + (49*C*a^4)/8)))*(323*A + 352*B + 392*C))/(64*d)

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