3.447 \(\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=278 \[ \frac{a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac{a^4 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{840 d}+\frac{a^4 (44 A+49 B+56 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{70 d}+\frac{(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{840 d}+\frac{1}{16} a^4 x (44 A+49 B+56 C)+\frac{a (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{42 d}+\frac{A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d} \]

[Out]

(a^4*(44*A + 49*B + 56*C)*x)/16 + (a^4*(454*A + 504*B + 581*C)*Sin[c + d*x])/(105*d) + (a^4*(44*A + 49*B + 56*
C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^4*(988*A + 1113*B + 1232*C)*Cos[c + d*x]^2*Sin[c + d*x])/(840*d) + (
a*(4*A + 7*B)*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(42*d) + (A*Cos[c + d*x]^6*(a + a*Sec[c + d*
x])^4*Sin[c + d*x])/(7*d) + ((16*A + 21*B + 14*C)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(70*
d) + ((436*A + 511*B + 504*C)*Cos[c + d*x]^3*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(840*d)

________________________________________________________________________________________

Rubi [A]  time = 0.763355, antiderivative size = 278, normalized size of antiderivative = 1., 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 = {4086, 4017, 3996, 3787, 2635, 8, 2637} \[ \frac{a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac{a^4 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{840 d}+\frac{a^4 (44 A+49 B+56 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{70 d}+\frac{(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{840 d}+\frac{1}{16} a^4 x (44 A+49 B+56 C)+\frac{a (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{42 d}+\frac{A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^4*(44*A + 49*B + 56*C)*x)/16 + (a^4*(454*A + 504*B + 581*C)*Sin[c + d*x])/(105*d) + (a^4*(44*A + 49*B + 56*
C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^4*(988*A + 1113*B + 1232*C)*Cos[c + d*x]^2*Sin[c + d*x])/(840*d) + (
a*(4*A + 7*B)*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(42*d) + (A*Cos[c + d*x]^6*(a + a*Sec[c + d*
x])^4*Sin[c + d*x])/(7*d) + ((16*A + 21*B + 14*C)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(70*
d) + ((436*A + 511*B + 504*C)*Cos[c + d*x]^3*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(840*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 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 2637

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

Rubi steps

\begin{align*} \int \cos ^7(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 ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 (a (4 A+7 B)+a (2 A+7 C) \sec (c+d x)) \, dx}{7 a}\\ &=\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 \left (3 a^2 (16 A+21 B+14 C)+2 a^2 (10 A+7 B+21 C) \sec (c+d x)\right ) \, dx}{42 a}\\ &=\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (a^3 (436 A+511 B+504 C)+98 a^3 (2 A+2 B+3 C) \sec (c+d x)\right ) \, dx}{210 a}\\ &=\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}+\frac{\int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (3 a^4 (988 A+1113 B+1232 C)+6 a^4 (276 A+301 B+364 C) \sec (c+d x)\right ) \, dx}{840 a}\\ &=\frac{a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}-\frac{\int \cos ^2(c+d x) \left (-315 a^5 (44 A+49 B+56 C)-24 a^5 (454 A+504 B+581 C) \sec (c+d x)\right ) \, dx}{2520 a}\\ &=\frac{a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}+\frac{1}{8} \left (a^4 (44 A+49 B+56 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{105} \left (a^4 (454 A+504 B+581 C)\right ) \int \cos (c+d x) \, dx\\ &=\frac{a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac{a^4 (44 A+49 B+56 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}+\frac{1}{16} \left (a^4 (44 A+49 B+56 C)\right ) \int 1 \, dx\\ &=\frac{1}{16} a^4 (44 A+49 B+56 C) x+\frac{a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac{a^4 (44 A+49 B+56 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac{a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac{(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}\\ \end{align*}

Mathematica [A]  time = 1.00692, size = 204, normalized size = 0.73 \[ \frac{a^4 (105 (323 A+352 B+392 C) \sin (c+d x)+105 (124 A+127 B+128 C) \sin (2 (c+d x))+5495 A \sin (3 (c+d x))+2100 A \sin (4 (c+d x))+651 A \sin (5 (c+d x))+140 A \sin (6 (c+d x))+15 A \sin (7 (c+d x))+11760 A c+18480 A d x+5040 B \sin (3 (c+d x))+1575 B \sin (4 (c+d x))+336 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+20580 B c+20580 B d x+4060 C \sin (3 (c+d x))+840 C \sin (4 (c+d x))+84 C \sin (5 (c+d x))+23520 C d x)}{6720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^4*(11760*A*c + 20580*B*c + 18480*A*d*x + 20580*B*d*x + 23520*C*d*x + 105*(323*A + 352*B + 392*C)*Sin[c + d*
x] + 105*(124*A + 127*B + 128*C)*Sin[2*(c + d*x)] + 5495*A*Sin[3*(c + d*x)] + 5040*B*Sin[3*(c + d*x)] + 4060*C
*Sin[3*(c + d*x)] + 2100*A*Sin[4*(c + d*x)] + 1575*B*Sin[4*(c + d*x)] + 840*C*Sin[4*(c + d*x)] + 651*A*Sin[5*(
c + d*x)] + 336*B*Sin[5*(c + d*x)] + 84*C*Sin[5*(c + d*x)] + 140*A*Sin[6*(c + d*x)] + 35*B*Sin[6*(c + d*x)] +
15*A*Sin[7*(c + d*x)]))/(6720*d)

________________________________________________________________________________________

Maple [A]  time = 0.141, size = 490, normalized size = 1.8 \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)^7*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.985393, size = 652, normalized size = 2.35 \begin{align*} -\frac{192 \,{\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} - 2688 \,{\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} + 140 \,{\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} + 2240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 840 \,{\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} - 1792 \,{\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} + 35 \,{\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} + 8960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 1260 \,{\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} - 1680 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 448 \,{\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} + 13440 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 840 \,{\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} - 6720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 6720 \, C a^{4} \sin \left (d x + c\right )}{6720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/6720*(192*(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 - 2688*(3*sin(
d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 + 140*(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 + 2240*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 - 840*(12*d*x + 12*c +
sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 - 1792*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B
*a^4 + 35*(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 + 8960*(sin(
d*x + c)^3 - 3*sin(d*x + c))*B*a^4 - 1260*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 - 1680
*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4 - 448*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^4 +
13440*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 - 840*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C
*a^4 - 6720*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^4 - 6720*C*a^4*sin(d*x + c))/d

________________________________________________________________________________________

Fricas [A]  time = 0.529753, size = 454, normalized size = 1.63 \begin{align*} \frac{105 \,{\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} d x +{\left (240 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \,{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{5} + 48 \,{\left (48 \, A + 28 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \,{\left (44 \, A + 41 \, B + 24 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \,{\left (227 \, A + 252 \, B + 238 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \,{\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \,{\left (454 \, A + 504 \, B + 581 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{1680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/1680*(105*(44*A + 49*B + 56*C)*a^4*d*x + (240*A*a^4*cos(d*x + c)^6 + 280*(4*A + B)*a^4*cos(d*x + c)^5 + 48*(
48*A + 28*B + 7*C)*a^4*cos(d*x + c)^4 + 70*(44*A + 41*B + 24*C)*a^4*cos(d*x + c)^3 + 16*(227*A + 252*B + 238*C
)*a^4*cos(d*x + c)^2 + 105*(44*A + 49*B + 56*C)*a^4*cos(d*x + c) + 16*(454*A + 504*B + 581*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)**7*(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 = 1.32531, size = 541, normalized size = 1.95 \begin{align*} \frac{105 \,{\left (44 \, A a^{4} + 49 \, B a^{4} + 56 \, C a^{4}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (4620 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 5145 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 5880 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 30800 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 34300 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 39200 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 87164 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 97069 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 110936 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 135168 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 150528 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 172032 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 126084 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 134099 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 159656 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 58800 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 73220 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 86240 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 22260 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 21735 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 21000 \, 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 )}^{7}}}{1680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/1680*(105*(44*A*a^4 + 49*B*a^4 + 56*C*a^4)*(d*x + c) + 2*(4620*A*a^4*tan(1/2*d*x + 1/2*c)^13 + 5145*B*a^4*ta
n(1/2*d*x + 1/2*c)^13 + 5880*C*a^4*tan(1/2*d*x + 1/2*c)^13 + 30800*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 34300*B*a^4
*tan(1/2*d*x + 1/2*c)^11 + 39200*C*a^4*tan(1/2*d*x + 1/2*c)^11 + 87164*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 97069*B*
a^4*tan(1/2*d*x + 1/2*c)^9 + 110936*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 135168*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 15052
8*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 172032*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 126084*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 1
34099*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 159656*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 58800*A*a^4*tan(1/2*d*x + 1/2*c)^3
+ 73220*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 86240*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 22260*A*a^4*tan(1/2*d*x + 1/2*c) +
21735*B*a^4*tan(1/2*d*x + 1/2*c) + 21000*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^7)/d