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

Optimal. Leaf size=454 \[ \frac{\tan (c+d x) \left (84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)+224 a^3 b B+280 a b^3 B+35 b^4 (2 A+3 C)\right )}{105 d}+\frac{\left (4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \tan (c+d x) \sec ^3(c+d x) \left (a^2 (412 A b+504 b C)+175 a^3 B+336 a b^2 B+24 A b^3\right )}{840 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left (3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+112 a^3 b B+91 a b^3 B+4 A b^4\right )}{105 d}+\frac{\tan (c+d x) \sec (c+d x) \left (4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right )}{16 d}+\frac{\tan (c+d x) \sec ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{70 d}+\frac{(7 a B+4 A b) \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{42 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d} \]

[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))*ArcTanh[Sin[c + d*x]])/(16*d)
+ ((224*a^3*b*B + 280*a*b^3*B + 35*b^4*(2*A + 3*C) + 84*a^2*b^2*(4*A + 5*C) + 8*a^4*(6*A + 7*C))*Tan[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))*Sec[c + d*x]*Tan[c
+ d*x])/(16*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))*Sec[c + d
*x]^2*Tan[c + d*x])/(105*d) + (a*(24*A*b^3 + 175*a^3*B + 336*a*b^2*B + a^2*(412*A*b + 504*b*C))*Sec[c + d*x]^3
*Tan[c + d*x])/(840*d) + ((4*A*b^2 + 21*a*b*B + 2*a^2*(6*A + 7*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c
+ d*x])/(70*d) + ((4*A*b + 7*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(42*d) + (A*(a + b*Cos[
c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)

________________________________________________________________________________________

Rubi [A]  time = 1.39733, antiderivative size = 454, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195, Rules used = {3047, 3031, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{\tan (c+d x) \left (84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)+224 a^3 b B+280 a b^3 B+35 b^4 (2 A+3 C)\right )}{105 d}+\frac{\left (4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \tan (c+d x) \sec ^3(c+d x) \left (a^2 (412 A b+504 b C)+175 a^3 B+336 a b^2 B+24 A b^3\right )}{840 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left (3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+112 a^3 b B+91 a b^3 B+4 A b^4\right )}{105 d}+\frac{\tan (c+d x) \sec (c+d x) \left (4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right )}{16 d}+\frac{\tan (c+d x) \sec ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{70 d}+\frac{(7 a B+4 A b) \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{42 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,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))*ArcTanh[Sin[c + d*x]])/(16*d)
+ ((224*a^3*b*B + 280*a*b^3*B + 35*b^4*(2*A + 3*C) + 84*a^2*b^2*(4*A + 5*C) + 8*a^4*(6*A + 7*C))*Tan[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))*Sec[c + d*x]*Tan[c
+ d*x])/(16*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))*Sec[c + d
*x]^2*Tan[c + d*x])/(105*d) + (a*(24*A*b^3 + 175*a^3*B + 336*a*b^2*B + a^2*(412*A*b + 504*b*C))*Sec[c + d*x]^3
*Tan[c + d*x])/(840*d) + ((4*A*b^2 + 21*a*b*B + 2*a^2*(6*A + 7*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c
+ d*x])/(70*d) + ((4*A*b + 7*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(42*d) + (A*(a + b*Cos[
c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3031

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3021

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rubi steps

\begin{align*} \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx &=\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{7} \int (a+b \cos (c+d x))^3 \left (4 A b+7 a B+(6 a A+7 b B+7 a C) \cos (c+d x)+b (2 A+7 C) \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx\\ &=\frac{(4 A b+7 a B) (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{42} \int (a+b \cos (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 ) \cos (c+d x)+2 b (10 A b+7 a B+21 b C) \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx\\ &=\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{(4 A b+7 a B) (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{210} \int (a+b \cos (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 ) \cos (c+d x)+2 b \left (98 a b B+6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, 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 ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{(4 A b+7 a B) (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}-\frac{1}{840} \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 )-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 ) \cos (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 ) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\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 ) \sec ^2(c+d x) \tan (c+d x)}{105 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 ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{(4 A b+7 a B) (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}-\frac{\int \left (-315 \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 )-24 \left (224 a^3 b B+280 a b^3 B+35 b^4 (2 A+3 C)+84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)\right ) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{2520}\\ &=\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 ) \sec ^2(c+d x) \tan (c+d x)}{105 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 ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{(4 A b+7 a B) (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}-\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 \sec ^3(c+d x) \, dx-\frac{1}{105} \left (-224 a^3 b B-280 a b^3 B-35 b^4 (2 A+3 C)-84 a^2 b^2 (4 A+5 C)-8 a^4 (6 A+7 C)\right ) \int \sec ^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 ) \sec (c+d x) \tan (c+d x)}{16 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 ) \sec ^2(c+d x) \tan (c+d x)}{105 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 ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{(4 A b+7 a B) (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 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 ) \int \sec (c+d x) \, dx-\frac{\left (224 a^3 b B+280 a b^3 B+35 b^4 (2 A+3 C)+84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (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 ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left (224 a^3 b B+280 a b^3 B+35 b^4 (2 A+3 C)+84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)\right ) \tan (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 ) \sec (c+d x) \tan (c+d x)}{16 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 ) \sec ^2(c+d x) \tan (c+d x)}{105 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 ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac{\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac{(4 A b+7 a B) (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 3.50877, size = 341, normalized size = 0.75 \[ \frac{105 \left (4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (16 \left (21 a^2 \tan ^4(c+d x) \left (a^2 (3 A+C)+4 a b B+6 A b^2\right )+35 \tan ^2(c+d x) \left (6 a^2 b^2 (2 A+C)+a^4 (3 A+2 C)+8 a^3 b B+4 a b^3 B+A b^4\right )+105 \left (6 a^2 b^2 (A+C)+a^4 (A+C)+4 a^3 b B+4 a b^3 B+b^4 (A+C)\right )+15 a^4 A \tan ^6(c+d x)\right )+70 a \sec ^3(c+d x) \left (4 a^2 b (5 A+6 C)+5 a^3 B+36 a b^2 B+24 A b^3\right )+105 \sec (c+d x) \left (4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right )+280 a^3 (a B+4 A b) \sec ^5(c+d x)\right )}{1680 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(105*(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))*ArcTanh[Sin[c + d*x]] + Ta
n[c + d*x]*(105*(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))*Sec[c + d*x] +
70*a*(24*A*b^3 + 5*a^3*B + 36*a*b^2*B + 4*a^2*b*(5*A + 6*C))*Sec[c + d*x]^3 + 280*a^3*(4*A*b + a*B)*Sec[c + d*
x]^5 + 16*(105*(4*a^3*b*B + 4*a*b^3*B + a^4*(A + C) + 6*a^2*b^2*(A + C) + b^4*(A + C)) + 35*(A*b^4 + 8*a^3*b*B
+ 4*a*b^3*B + 6*a^2*b^2*(2*A + C) + a^4*(3*A + 2*C))*Tan[c + d*x]^2 + 21*a^2*(6*A*b^2 + 4*a*b*B + a^2*(3*A +
C))*Tan[c + d*x]^4 + 15*a^4*A*Tan[c + d*x]^6)))/(1680*d)

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Maple [B]  time = 0.1, size = 905, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d*b^4*B*ln(sec(d*x+c)+tan(d*x+c))+2/d*a^2*b^2*C*tan(d*x+c)*sec(d*x+c)^2+2/3/d*A*a^3*b*tan(d*x+c)*sec(d*x+c
)^5+1/d*a^3*b*C*tan(d*x+c)*sec(d*x+c)^3+1/d*a*A*b^3*tan(d*x+c)*sec(d*x+c)^3+2/d*C*a*b^3*tan(d*x+c)*sec(d*x+c)+
6/5/d*a^2*A*b^2*tan(d*x+c)*sec(d*x+c)^4+3/2/d*a*A*b^3*sec(d*x+c)*tan(d*x+c)+8/5/d*a^2*A*b^2*tan(d*x+c)*sec(d*x
+c)^2+5/6/d*A*a^3*b*tan(d*x+c)*sec(d*x+c)^3+3/2/d*a^3*b*C*sec(d*x+c)*tan(d*x+c)+5/4/d*A*a^3*b*sec(d*x+c)*tan(d
*x+c)+1/d*C*b^4*tan(d*x+c)+2/3/d*A*b^4*tan(d*x+c)+8/15/d*a^4*C*tan(d*x+c)+5/16/d*a^4*B*ln(sec(d*x+c)+tan(d*x+c
))+16/35/d*A*a^4*tan(d*x+c)+9/4/d*a^2*b^2*B*ln(sec(d*x+c)+tan(d*x+c))+32/15/d*a^3*b*B*tan(d*x+c)+1/3/d*A*b^4*t
an(d*x+c)*sec(d*x+c)^2+1/7/d*A*a^4*tan(d*x+c)*sec(d*x+c)^6+1/5/d*a^4*C*tan(d*x+c)*sec(d*x+c)^4+2/d*C*a*b^3*ln(
sec(d*x+c)+tan(d*x+c))+4/d*a^2*b^2*C*tan(d*x+c)+3/2/d*a*A*b^3*ln(sec(d*x+c)+tan(d*x+c))+16/5/d*a^2*A*b^2*tan(d
*x+c)+3/2/d*a^3*b*C*ln(sec(d*x+c)+tan(d*x+c))+5/4/d*A*a^3*b*ln(sec(d*x+c)+tan(d*x+c))+5/24/d*a^4*B*tan(d*x+c)*
sec(d*x+c)^3+5/16/d*a^4*B*sec(d*x+c)*tan(d*x+c)+6/35/d*A*a^4*tan(d*x+c)*sec(d*x+c)^4+4/15/d*a^4*C*tan(d*x+c)*s
ec(d*x+c)^2+8/35/d*A*a^4*tan(d*x+c)*sec(d*x+c)^2+3/2/d*a^2*b^2*B*tan(d*x+c)*sec(d*x+c)^3+4/3/d*a*b^3*B*tan(d*x
+c)*sec(d*x+c)^2+4/5/d*a^3*b*B*tan(d*x+c)*sec(d*x+c)^4+9/4/d*a^2*b^2*B*sec(d*x+c)*tan(d*x+c)+16/15/d*a^3*b*B*t
an(d*x+c)*sec(d*x+c)^2+8/3/d*a*b^3*B*tan(d*x+c)+1/6/d*a^4*B*tan(d*x+c)*sec(d*x+c)^5+1/2/d*b^4*B*sec(d*x+c)*tan
(d*x+c)

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Maxima [A]  time = 1.04805, size = 1007, normalized size = 2.22 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3360*(96*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*tan(d*x + c))*A*a^4 + 224*(3*tan(d*x
+ c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^4 + 896*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x
+ c))*B*a^3*b + 1344*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^2*b^2 + 6720*(tan(d*x + c)^3
+ 3*tan(d*x + c))*C*a^2*b^2 + 4480*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a*b^3 + 1120*(tan(d*x + c)^3 + 3*tan(d
*x + c))*A*b^4 - 35*B*a^4*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin
(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 140*A*a^3*b*(2*(1
5*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2
- 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 840*C*a^3*b*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c
))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 1260*B*a^2*b
^2*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) +
3*log(sin(d*x + c) - 1)) - 840*A*a*b^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)
^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 3360*C*a*b^3*(2*sin(d*x + c)/(sin(d*x + c)^2 -
1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 840*B*b^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin
(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 3360*C*b^4*tan(d*x + c))/d

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Fricas [A]  time = 2.23407, size = 1085, normalized size = 2.39 \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 )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 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 )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (8 \,{\left (6 \, A + 7 \, C\right )} a^{4} + 224 \, B a^{3} b + 84 \,{\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 280 \, B a b^{3} + 35 \,{\left (2 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} + 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 )^{5} + 240 \, A a^{4} + 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 )^{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} + 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 )^{2} + 280 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3360*(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)^7*log(
sin(d*x + c) + 1) - 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)^7*log(-sin(d*x + c) + 1) + 2*(16*(8*(6*A + 7*C)*a^4 + 224*B*a^3*b + 84*(4*A + 5*C)*a^2*b^2 + 280*B*a*b^3
+ 35*(2*A + 3*C)*b^4)*cos(d*x + c)^6 + 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)^5 + 240*A*a^4 + 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)^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 + 48*((6*A + 7*C)*a^4 + 28*B*a^3*b + 42*A*a^2*b^2)*cos(d*x + c)^2 + 280*(B*a^4 + 4*A*a^3*b)*cos(d*x + c))*
sin(d*x + c))/(d*cos(d*x + c)^7)

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

[Out]

Timed out

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Giac [B]  time = 1.33971, size = 2549, normalized size = 5.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^8,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)*log(abs(tan
(1/2*d*x + 1/2*c) + 1)) - 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)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 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 + 6
720*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)^1
3 - 840*B*b^4*tan(1/2*d*x + 1/2*c)^13 + 1680*C*b^4*tan(1/2*d*x + 1/2*c)^13 - 3360*A*a^4*tan(1/2*d*x + 1/2*c)^1
1 + 980*B*a^4*tan(1/2*d*x + 1/2*c)^11 - 5600*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)^1
1 + 14448*A*a^4*tan(1/2*d*x + 1/2*c)^9 - 2975*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*ta
n(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*tan(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*tan(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*tan(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 + 252
00*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 - 10
080*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))/(tan(1/2*d*x + 1/2*c)^2 - 1)^7)/d