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

Optimal. Leaf size=438 \[ -\frac{\sin ^3(c+d x) \left (3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+112 a^3 b B+91 a b^3 B+4 A b^4\right )}{105 d}+\frac{\sin (c+d x) \left (3 a^2 b^2 (162 A+203 C)+12 a^4 (6 A+7 C)+336 a^3 b B+371 a b^3 B+b^4 (74 A+105 C)\right )}{105 d}+\frac{a \sin (c+d x) \cos ^3(c+d x) \left (a^2 (412 A b+504 b C)+175 a^3 B+336 a b^2 B+24 A b^3\right )}{840 d}+\frac{\sin (c+d x) \cos (c+d x) \left (4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right )}{16 d}+\frac{\sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{70 d}+\frac{1}{16} x \left (4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right )+\frac{(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{42 d}+\frac{A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d} \]

[Out]

((5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*x)/16 + ((336*a^3*b*B + 371*a*
b^3*B + 12*a^4*(6*A + 7*C) + b^4*(74*A + 105*C) + 3*a^2*b^2*(162*A + 203*C))*Sin[c + d*x])/(105*d) + ((5*a^4*B
+ 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*
(24*A*b^3 + 175*a^3*B + 336*a*b^2*B + a^2*(412*A*b + 504*b*C))*Cos[c + d*x]^3*Sin[c + d*x])/(840*d) + ((4*A*b^
2 + 21*a*b*B + 2*a^2*(6*A + 7*C))*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(70*d) + ((4*A*b + 7*a*B
)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(42*d) + (A*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*Sin[c
+ d*x])/(7*d) - ((4*A*b^4 + 112*a^3*b*B + 91*a*b^3*B + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*(50*A + 63*C))*Sin[c + d*
x]^3)/(105*d)

________________________________________________________________________________________

Rubi [A]  time = 1.38427, antiderivative size = 438, 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 = {4094, 4074, 4047, 2635, 8, 4044, 3013} \[ -\frac{\sin ^3(c+d x) \left (3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+112 a^3 b B+91 a b^3 B+4 A b^4\right )}{105 d}+\frac{\sin (c+d x) \left (3 a^2 b^2 (162 A+203 C)+12 a^4 (6 A+7 C)+336 a^3 b B+371 a b^3 B+b^4 (74 A+105 C)\right )}{105 d}+\frac{a \sin (c+d x) \cos ^3(c+d x) \left (a^2 (412 A b+504 b C)+175 a^3 B+336 a b^2 B+24 A b^3\right )}{840 d}+\frac{\sin (c+d x) \cos (c+d x) \left (4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right )}{16 d}+\frac{\sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{70 d}+\frac{1}{16} x \left (4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right )+\frac{(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{42 d}+\frac{A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*x)/16 + ((336*a^3*b*B + 371*a*
b^3*B + 12*a^4*(6*A + 7*C) + b^4*(74*A + 105*C) + 3*a^2*b^2*(162*A + 203*C))*Sin[c + d*x])/(105*d) + ((5*a^4*B
+ 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*
(24*A*b^3 + 175*a^3*B + 336*a*b^2*B + a^2*(412*A*b + 504*b*C))*Cos[c + d*x]^3*Sin[c + d*x])/(840*d) + ((4*A*b^
2 + 21*a*b*B + 2*a^2*(6*A + 7*C))*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(70*d) + ((4*A*b + 7*a*B
)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(42*d) + (A*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*Sin[c
+ d*x])/(7*d) - ((4*A*b^4 + 112*a^3*b*B + 91*a*b^3*B + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*(50*A + 63*C))*Sin[c + d*
x]^3)/(105*d)

Rule 4094

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]

Rule 4074

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 4047

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 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 4044

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 3013

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]

Rubi steps

\begin{align*} \int \cos ^7(c+d x) (a+b \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+b \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{1}{7} \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (4 A b+7 a B+(6 a A+7 b B+7 a C) \sec (c+d x)+b (2 A+7 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{1}{42} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (3 \left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right )+\left (68 a A b+35 a^2 B+42 b^2 B+84 a b C\right ) \sec (c+d x)+2 b (10 A b+7 a B+21 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{70 d}+\frac{(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{1}{210} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)+\left (497 a^2 b B+210 b^3 B+24 a^3 (6 A+7 C)+2 a b^2 (244 A+315 C)\right ) \sec (c+d x)+2 b \left (98 a b B+6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{70 d}+\frac{(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}-\frac{1}{840} \int \cos ^3(c+d x) \left (-24 \left (4 A b^4+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right )-105 \left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \sec (c+d x)-8 b^2 \left (98 a b B+6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{70 d}+\frac{(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}-\frac{1}{840} \int \cos ^3(c+d x) \left (-24 \left (4 A b^4+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right )-8 b^2 \left (98 a b B+6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)\right ) \, dx-\frac{1}{8} \left (-5 a^4 B-36 a^2 b^2 B-8 b^4 B-8 a b^3 (3 A+4 C)-4 a^3 b (5 A+6 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{\left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{70 d}+\frac{(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}-\frac{1}{840} \int \cos (c+d x) \left (-8 b^2 \left (98 a b B+6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right )-24 \left (4 A b^4+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \cos ^2(c+d x)\right ) \, dx-\frac{1}{16} \left (-5 a^4 B-36 a^2 b^2 B-8 b^4 B-8 a b^3 (3 A+4 C)-4 a^3 b (5 A+6 C)\right ) \int 1 \, dx\\ &=\frac{1}{16} \left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) x+\frac{\left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{70 d}+\frac{(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac{\operatorname{Subst}\left (\int \left (-24 \left (4 A b^4+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right )-8 b^2 \left (98 a b B+6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right )+24 \left (4 A b^4+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{840 d}\\ &=\frac{1}{16} \left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) x+\frac{\left (336 a^3 b B+371 a b^3 B+12 a^4 (6 A+7 C)+b^4 (74 A+105 C)+3 a^2 b^2 (162 A+203 C)\right ) \sin (c+d x)}{105 d}+\frac{\left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{70 d}+\frac{(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac{A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}-\frac{\left (4 A b^4+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \sin ^3(c+d x)}{105 d}\\ \end{align*}

Mathematica [A]  time = 1.59563, size = 528, normalized size = 1.21 \[ \frac{105 \sin (c+d x) \left (48 a^2 b^2 (5 A+6 C)+5 a^4 (7 A+8 C)+160 a^3 b B+192 a b^3 B+16 b^4 (3 A+4 C)\right )+105 \sin (2 (c+d x)) \left (a^3 (60 A b+64 b C)+96 a^2 b^2 B+15 a^4 B+64 a b^3 (A+C)+16 b^4 B\right )+4200 a^2 A b^2 \sin (3 (c+d x))+504 a^2 A b^2 \sin (5 (c+d x))+1260 a^3 A b \sin (4 (c+d x))+140 a^3 A b \sin (6 (c+d x))+8400 a^3 A b c+8400 a^3 A b d x+735 a^4 A \sin (3 (c+d x))+147 a^4 A \sin (5 (c+d x))+15 a^4 A \sin (7 (c+d x))+1260 a^2 b^2 B \sin (4 (c+d x))+15120 a^2 b^2 B c+15120 a^2 b^2 B d x+3360 a^2 b^2 C \sin (3 (c+d x))+2800 a^3 b B \sin (3 (c+d x))+336 a^3 b B \sin (5 (c+d x))+840 a^3 b C \sin (4 (c+d x))+10080 a^3 b c C+10080 a^3 b C d x+315 a^4 B \sin (4 (c+d x))+35 a^4 B \sin (6 (c+d x))+2100 a^4 B c+2100 a^4 B d x+700 a^4 C \sin (3 (c+d x))+84 a^4 C \sin (5 (c+d x))+840 a A b^3 \sin (4 (c+d x))+10080 a A b^3 c+10080 a A b^3 d x+2240 a b^3 B \sin (3 (c+d x))+13440 a b^3 c C+13440 a b^3 C d x+560 A b^4 \sin (3 (c+d x))+3360 b^4 B c+3360 b^4 B d x}{6720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8400*a^3*A*b*c + 10080*a*A*b^3*c + 2100*a^4*B*c + 15120*a^2*b^2*B*c + 3360*b^4*B*c + 10080*a^3*b*c*C + 13440*
a*b^3*c*C + 8400*a^3*A*b*d*x + 10080*a*A*b^3*d*x + 2100*a^4*B*d*x + 15120*a^2*b^2*B*d*x + 3360*b^4*B*d*x + 100
80*a^3*b*C*d*x + 13440*a*b^3*C*d*x + 105*(160*a^3*b*B + 192*a*b^3*B + 16*b^4*(3*A + 4*C) + 48*a^2*b^2*(5*A + 6
*C) + 5*a^4*(7*A + 8*C))*Sin[c + d*x] + 105*(15*a^4*B + 96*a^2*b^2*B + 16*b^4*B + 64*a*b^3*(A + C) + a^3*(60*A
*b + 64*b*C))*Sin[2*(c + d*x)] + 735*a^4*A*Sin[3*(c + d*x)] + 4200*a^2*A*b^2*Sin[3*(c + d*x)] + 560*A*b^4*Sin[
3*(c + d*x)] + 2800*a^3*b*B*Sin[3*(c + d*x)] + 2240*a*b^3*B*Sin[3*(c + d*x)] + 700*a^4*C*Sin[3*(c + d*x)] + 33
60*a^2*b^2*C*Sin[3*(c + d*x)] + 1260*a^3*A*b*Sin[4*(c + d*x)] + 840*a*A*b^3*Sin[4*(c + d*x)] + 315*a^4*B*Sin[4
*(c + d*x)] + 1260*a^2*b^2*B*Sin[4*(c + d*x)] + 840*a^3*b*C*Sin[4*(c + d*x)] + 147*a^4*A*Sin[5*(c + d*x)] + 50
4*a^2*A*b^2*Sin[5*(c + d*x)] + 336*a^3*b*B*Sin[5*(c + d*x)] + 84*a^4*C*Sin[5*(c + d*x)] + 140*a^3*A*b*Sin[6*(c
+ d*x)] + 35*a^4*B*Sin[6*(c + d*x)] + 15*a^4*A*Sin[7*(c + d*x)])/(6720*d)

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Maple [A]  time = 0.097, size = 505, normalized size = 1.2 \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+b*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^3*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*B*a^3*b*(8/
3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4*a^3*b*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^2*b^2*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+6*a^2*b^2*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)+2*C*a^2*b^2*(2+cos(d*x+c)^2)*sin(d*x+c)+4*A*a*b^3*(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*b^3*B*(2+cos(d*x+c)^2)*sin(d*x+c)+4*C*a*b^3*(1/2*cos(d*x+c)*sin(d*x+c)+1/2
*d*x+1/2*c)+1/3*A*b^4*(2+cos(d*x+c)^2)*sin(d*x+c)+B*b^4*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+C*b^4*sin(d*
x+c))

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Maxima [A]  time = 1.07809, size = 672, normalized size = 1.53 \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} + 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} - 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} + 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^{3} b - 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^{3} b - 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^{3} b - 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^{2} b^{2} - 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^{2} b^{2} + 13440 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b^{2} - 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 b^{3} + 8960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{3} - 6720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} + 2240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{4} - 1680 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{4} - 6720 \, C b^{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+b*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 + 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 - 448*(3*sin(d*x + c)^5 - 10*si
n(d*x + c)^3 + 15*sin(d*x + c))*C*a^4 + 140*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*si
n(2*d*x + 2*c))*A*a^3*b - 1792*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^3*b - 840*(12*d*x
+ 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^3*b - 2688*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(
d*x + c))*A*a^2*b^2 - 1260*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^2*b^2 + 13440*(sin(d*x
+ c)^3 - 3*sin(d*x + c))*C*a^2*b^2 - 840*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a*b^3 + 896
0*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a*b^3 - 6720*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a*b^3 + 2240*(sin(d*x +
c)^3 - 3*sin(d*x + c))*A*b^4 - 1680*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*b^4 - 6720*C*b^4*sin(d*x + c))/d

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Fricas [A]  time = 0.644114, size = 852, normalized size = 1.95 \begin{align*} \frac{105 \,{\left (5 \, B a^{4} + 4 \,{\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 8 \,{\left (3 \, A + 4 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} d x +{\left (240 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{5} + 128 \,{\left (6 \, A + 7 \, C\right )} a^{4} + 3584 \, B a^{3} b + 1344 \,{\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 4480 \, B a b^{3} + 560 \,{\left (2 \, A + 3 \, C\right )} b^{4} + 48 \,{\left ({\left (6 \, A + 7 \, C\right )} a^{4} + 28 \, B a^{3} b + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + 70 \,{\left (5 \, B a^{4} + 4 \,{\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 16 \,{\left (4 \,{\left (6 \, A + 7 \, C\right )} a^{4} + 112 \, B a^{3} b + 42 \,{\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 140 \, B a b^{3} + 35 \, A b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \,{\left (5 \, B a^{4} + 4 \,{\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 8 \,{\left (3 \, A + 4 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )\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+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/1680*(105*(5*B*a^4 + 4*(5*A + 6*C)*a^3*b + 36*B*a^2*b^2 + 8*(3*A + 4*C)*a*b^3 + 8*B*b^4)*d*x + (240*A*a^4*co
s(d*x + c)^6 + 280*(B*a^4 + 4*A*a^3*b)*cos(d*x + c)^5 + 128*(6*A + 7*C)*a^4 + 3584*B*a^3*b + 1344*(4*A + 5*C)*
a^2*b^2 + 4480*B*a*b^3 + 560*(2*A + 3*C)*b^4 + 48*((6*A + 7*C)*a^4 + 28*B*a^3*b + 42*A*a^2*b^2)*cos(d*x + c)^4
+ 70*(5*B*a^4 + 4*(5*A + 6*C)*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3)*cos(d*x + c)^3 + 16*(4*(6*A + 7*C)*a^4 + 112
*B*a^3*b + 42*(4*A + 5*C)*a^2*b^2 + 140*B*a*b^3 + 35*A*b^4)*cos(d*x + c)^2 + 105*(5*B*a^4 + 4*(5*A + 6*C)*a^3*
b + 36*B*a^2*b^2 + 8*(3*A + 4*C)*a*b^3 + 8*B*b^4)*cos(d*x + c))*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)**7*(a+b*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 [B]  time = 1.50933, size = 2450, normalized size = 5.59 \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)^7*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/1680*(105*(5*B*a^4 + 20*A*a^3*b + 24*C*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 32*C*a*b^3 + 8*B*b^4)*(d*x + c) +
2*(1680*A*a^4*tan(1/2*d*x + 1/2*c)^13 - 1155*B*a^4*tan(1/2*d*x + 1/2*c)^13 + 1680*C*a^4*tan(1/2*d*x + 1/2*c)^
13 - 4620*A*a^3*b*tan(1/2*d*x + 1/2*c)^13 + 6720*B*a^3*b*tan(1/2*d*x + 1/2*c)^13 - 4200*C*a^3*b*tan(1/2*d*x +
1/2*c)^13 + 10080*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 - 6300*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 + 10080*C*a^2*b^2
*tan(1/2*d*x + 1/2*c)^13 - 4200*A*a*b^3*tan(1/2*d*x + 1/2*c)^13 + 6720*B*a*b^3*tan(1/2*d*x + 1/2*c)^13 - 3360*
C*a*b^3*tan(1/2*d*x + 1/2*c)^13 + 1680*A*b^4*tan(1/2*d*x + 1/2*c)^13 - 840*B*b^4*tan(1/2*d*x + 1/2*c)^13 + 168
0*C*b^4*tan(1/2*d*x + 1/2*c)^13 + 3360*A*a^4*tan(1/2*d*x + 1/2*c)^11 - 980*B*a^4*tan(1/2*d*x + 1/2*c)^11 + 560
0*C*a^4*tan(1/2*d*x + 1/2*c)^11 - 3920*A*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 22400*B*a^3*b*tan(1/2*d*x + 1/2*c)^11
- 10080*C*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 33600*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 15120*B*a^2*b^2*tan(1/2*d
*x + 1/2*c)^11 + 47040*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 10080*A*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 31360*B*a*b
^3*tan(1/2*d*x + 1/2*c)^11 - 13440*C*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 7840*A*b^4*tan(1/2*d*x + 1/2*c)^11 - 3360
*B*b^4*tan(1/2*d*x + 1/2*c)^11 + 10080*C*b^4*tan(1/2*d*x + 1/2*c)^11 + 14448*A*a^4*tan(1/2*d*x + 1/2*c)^9 - 29
75*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 12656*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 11900*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 +
50624*B*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 7560*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 75936*A*a^2*b^2*tan(1/2*d*x + 1/2
*c)^9 - 11340*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 97440*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 7560*A*a*b^3*tan(1/2
*d*x + 1/2*c)^9 + 64960*B*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 16800*C*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 16240*A*b^4*ta
n(1/2*d*x + 1/2*c)^9 - 4200*B*b^4*tan(1/2*d*x + 1/2*c)^9 + 25200*C*b^4*tan(1/2*d*x + 1/2*c)^9 + 10176*A*a^4*ta
n(1/2*d*x + 1/2*c)^7 + 17472*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 69888*B*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 104832*A*a^
2*b^2*tan(1/2*d*x + 1/2*c)^7 + 120960*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 80640*B*a*b^3*tan(1/2*d*x + 1/2*c)^7
+ 20160*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 33600*C*b^4*tan(1/2*d*x + 1/2*c)^7 + 14448*A*a^4*tan(1/2*d*x + 1/2*c)^5
+ 2975*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 12656*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 11900*A*a^3*b*tan(1/2*d*x + 1/2*c)
^5 + 50624*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 7560*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 75936*A*a^2*b^2*tan(1/2*d*x
+ 1/2*c)^5 + 11340*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 97440*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 7560*A*a*b^3*ta
n(1/2*d*x + 1/2*c)^5 + 64960*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 16800*C*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 16240*A*b
^4*tan(1/2*d*x + 1/2*c)^5 + 4200*B*b^4*tan(1/2*d*x + 1/2*c)^5 + 25200*C*b^4*tan(1/2*d*x + 1/2*c)^5 + 3360*A*a^
4*tan(1/2*d*x + 1/2*c)^3 + 980*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 5600*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 3920*A*a^3*b
*tan(1/2*d*x + 1/2*c)^3 + 22400*B*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 10080*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 33600*
A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 15120*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 47040*C*a^2*b^2*tan(1/2*d*x + 1/2*
c)^3 + 10080*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 31360*B*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 13440*C*a*b^3*tan(1/2*d*x
+ 1/2*c)^3 + 7840*A*b^4*tan(1/2*d*x + 1/2*c)^3 + 3360*B*b^4*tan(1/2*d*x + 1/2*c)^3 + 10080*C*b^4*tan(1/2*d*x
+ 1/2*c)^3 + 1680*A*a^4*tan(1/2*d*x + 1/2*c) + 1155*B*a^4*tan(1/2*d*x + 1/2*c) + 1680*C*a^4*tan(1/2*d*x + 1/2*
c) + 4620*A*a^3*b*tan(1/2*d*x + 1/2*c) + 6720*B*a^3*b*tan(1/2*d*x + 1/2*c) + 4200*C*a^3*b*tan(1/2*d*x + 1/2*c)
+ 10080*A*a^2*b^2*tan(1/2*d*x + 1/2*c) + 6300*B*a^2*b^2*tan(1/2*d*x + 1/2*c) + 10080*C*a^2*b^2*tan(1/2*d*x +
1/2*c) + 4200*A*a*b^3*tan(1/2*d*x + 1/2*c) + 6720*B*a*b^3*tan(1/2*d*x + 1/2*c) + 3360*C*a*b^3*tan(1/2*d*x + 1/
2*c) + 1680*A*b^4*tan(1/2*d*x + 1/2*c) + 840*B*b^4*tan(1/2*d*x + 1/2*c) + 1680*C*b^4*tan(1/2*d*x + 1/2*c))/(ta
n(1/2*d*x + 1/2*c)^2 + 1)^7)/d