\(\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^8(c+d x) \, dx\) [973]

Optimal result

Integrand size = 41, antiderivative size = 454 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {\left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (224 a^3 b B+280 a b^3 B+35 b^4 (2 A+3 C)+84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)\right ) \tan (c+d x)}{105 d}+\frac {\left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {\left (4 A b^4+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{105 d}+\frac {a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac {\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac {(4 A b+7 a B) (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d} \]

[Out]

1/16*(5*B*a^4+36*B*a^2*b^2+8*B*b^4+8*a*b^3*(3*A+4*C)+4*a^3*b*(5*A+6*C))*arctanh(sin(d*x+c))/d+1/105*(224*B*a^3
*b+280*B*a*b^3+35*b^4*(2*A+3*C)+84*a^2*b^2*(4*A+5*C)+8*a^4*(6*A+7*C))*tan(d*x+c)/d+1/16*(5*B*a^4+36*B*a^2*b^2+
8*B*b^4+8*a*b^3*(3*A+4*C)+4*a^3*b*(5*A+6*C))*sec(d*x+c)*tan(d*x+c)/d+1/105*(4*A*b^4+112*B*a^3*b+91*B*a*b^3+4*a
^4*(6*A+7*C)+3*a^2*b^2*(50*A+63*C))*sec(d*x+c)^2*tan(d*x+c)/d+1/840*a*(24*A*b^3+175*B*a^3+336*B*a*b^2+a^2*(412
*A*b+504*C*b))*sec(d*x+c)^3*tan(d*x+c)/d+1/70*(4*A*b^2+21*B*a*b+2*a^2*(6*A+7*C))*(a+b*cos(d*x+c))^2*sec(d*x+c)
^4*tan(d*x+c)/d+1/42*(4*A*b+7*B*a)*(a+b*cos(d*x+c))^3*sec(d*x+c)^5*tan(d*x+c)/d+1/7*A*(a+b*cos(d*x+c))^4*sec(d
*x+c)^6*tan(d*x+c)/d

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {3126, 3110, 3100, 2827, 3853, 3855, 3852, 8} \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {\tan (c+d x) \sec ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{70 d}+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{840 d}+\frac {\left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\tan (c+d x) \left (8 a^4 (6 A+7 C)+224 a^3 b B+84 a^2 b^2 (4 A+5 C)+280 a b^3 B+35 b^4 (2 A+3 C)\right )}{105 d}+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^4 (6 A+7 C)+112 a^3 b B+3 a^2 b^2 (50 A+63 C)+91 a b^3 B+4 A b^4\right )}{105 d}+\frac {\tan (c+d x) \sec (c+d x) \left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right )}{16 d}+\frac {(7 a B+4 A b) \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{42 d}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d} \]

[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 8

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

Rule 2827

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 3100

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 3110

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] &&
NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3126

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 3852

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 3853

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 3855

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

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {arctanh}(\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] (verified)

Time = 3.44 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.75 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {105 \left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (105 \left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \sec (c+d x)+70 a \left (24 A b^3+5 a^3 B+36 a b^2 B+4 a^2 b (5 A+6 C)\right ) \sec ^3(c+d x)+280 a^3 (4 A b+a B) \sec ^5(c+d x)+16 \left (105 \left (4 a^3 b B+4 a b^3 B+a^4 (A+C)+6 a^2 b^2 (A+C)+b^4 (A+C)\right )+35 \left (A b^4+8 a^3 b B+4 a b^3 B+6 a^2 b^2 (2 A+C)+a^4 (3 A+2 C)\right ) \tan ^2(c+d x)+21 a^2 \left (6 A b^2+4 a b B+a^2 (3 A+C)\right ) \tan ^4(c+d x)+15 a^4 A \tan ^6(c+d x)\right )\right )}{1680 d} \]

[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)

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.78

method	result	size
parts	\(-\frac {A \,a^{4} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {C \,b^{4} \tan \left (d x +c \right )}{d}\)	\(354\)
derivativedivides	\(\frac {-A \,a^{4} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+B \,a^{4} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-a^{4} C \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 A \,a^{3} b \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-4 B \,a^{3} b \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 a^{3} b C \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 A \,a^{2} b^{2} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+6 B \,a^{2} b^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 C \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 a A \,b^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 B a \,b^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 C a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-A \,b^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \tan \left (d x +c \right ) b^{4}}{d}\)	\(592\)
default	\(\frac {-A \,a^{4} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+B \,a^{4} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-a^{4} C \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 A \,a^{3} b \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-4 B \,a^{3} b \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 a^{3} b C \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 A \,a^{2} b^{2} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+6 B \,a^{2} b^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 C \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 a A \,b^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 B a \,b^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 C a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-A \,b^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \tan \left (d x +c \right ) b^{4}}{d}\)	\(592\)
parallelrisch	\(\frac {-44100 \left (\frac {\cos \left (7 d x +7 c \right )}{21}+\frac {\cos \left (5 d x +5 c \right )}{3}+\cos \left (3 d x +3 c \right )+\frac {5 \cos \left (d x +c \right )}{3}\right ) \left (\frac {B \,a^{4}}{4}+b \left (A +\frac {6 C}{5}\right ) a^{3}+\frac {9 B \,a^{2} b^{2}}{5}+\frac {6 \left (A +\frac {4 C}{3}\right ) b^{3} a}{5}+\frac {2 B \,b^{4}}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+44100 \left (\frac {\cos \left (7 d x +7 c \right )}{21}+\frac {\cos \left (5 d x +5 c \right )}{3}+\cos \left (3 d x +3 c \right )+\frac {5 \cos \left (d x +c \right )}{3}\right ) \left (\frac {B \,a^{4}}{4}+b \left (A +\frac {6 C}{5}\right ) a^{3}+\frac {9 B \,a^{2} b^{2}}{5}+\frac {6 \left (A +\frac {4 C}{3}\right ) b^{3} a}{5}+\frac {2 B \,b^{4}}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (16128 A +18816 C \right ) a^{4}+75264 B \,a^{3} b +112896 \left (A +\frac {25 C}{28}\right ) b^{2} a^{2}+67200 B a \,b^{3}+16800 \left (A +\frac {9 C}{10}\right ) b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (\left (5376 A +6272 C \right ) a^{4}+25088 B \,a^{3} b +37632 a^{2} \left (A +\frac {5 C}{4}\right ) b^{2}+31360 B a \,b^{3}+7840 \left (A +\frac {15 C}{14}\right ) b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (\left (768 A +896 C \right ) a^{4}+3584 B \,a^{3} b +5376 a^{2} \left (A +\frac {5 C}{4}\right ) b^{2}+4480 B a \,b^{3}+1120 \left (A +\frac {3 C}{2}\right ) b^{4}\right ) \sin \left (7 d x +7 c \right )+\left (19810 B \,a^{4}+79240 \left (A +\frac {186 C}{283}\right ) b \,a^{3}+78120 B \,a^{2} b^{2}+52080 \left (A +\frac {20 C}{31}\right ) b^{3} a +8400 B \,b^{4}\right ) \sin \left (2 d x +2 c \right )+\left (7000 B \,a^{4}+28000 b \left (A +\frac {6 C}{5}\right ) a^{3}+50400 B \,a^{2} b^{2}+33600 \left (A +\frac {4 C}{5}\right ) b^{3} a +6720 B \,b^{4}\right ) \sin \left (4 d x +4 c \right )+\left (1050 B \,a^{4}+4200 b \left (A +\frac {6 C}{5}\right ) a^{3}+7560 B \,a^{2} b^{2}+5040 \left (A +\frac {4 C}{3}\right ) b^{3} a +1680 B \,b^{4}\right ) \sin \left (6 d x +6 c \right )+26880 \sin \left (d x +c \right ) \left (a^{4} \left (A +\frac {C}{2}\right )+2 B \,a^{3} b +3 \left (A +\frac {3 C}{4}\right ) a^{2} b^{2}+\frac {3 B a \,b^{3}}{2}+\frac {3 \left (A +\frac {5 C}{6}\right ) b^{4}}{8}\right )}{1680 d \left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right )}\)	\(637\)
risch	\(\text {Expression too large to display}\)	\(1648\)

[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,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 450, normalized size of antiderivative = 0.99 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {105 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 8 \, {\left (3 \, A + 4 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 8 \, {\left (3 \, A + 4 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (8 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 224 \, B a^{3} b + 84 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 280 \, B a b^{3} + 35 \, {\left (2 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} + 105 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 8 \, {\left (3 \, A + 4 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} + 240 \, A a^{4} + 16 \, {\left (4 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 112 \, B a^{3} b + 42 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 140 \, B a b^{3} + 35 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 48 \, {\left ({\left (6 \, A + 7 \, C\right )} a^{4} + 28 \, B a^{3} b + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 280 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\text {Timed out} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.64 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\text {Too large to display} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1888 vs. \(2 (439) = 878\).

Time = 0.43 (sec) , antiderivative size = 1888, normalized size of antiderivative = 4.16 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\text {Too large to display} \]

[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

Mupad [B] (verification not implemented)

Time = 5.58 (sec) , antiderivative size = 1044, normalized size of antiderivative = 2.30 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atanh((4*tan(c/2 + (d*x)/2)*((5*B*a^4)/16 + (B*b^4)/2 + (9*B*a^2*b^2)/4 + (3*A*a*b^3)/2 + (5*A*a^3*b)/4 + 2*C
*a*b^3 + (3*C*a^3*b)/2))/((5*B*a^4)/4 + 2*B*b^4 + 9*B*a^2*b^2 + 6*A*a*b^3 + 5*A*a^3*b + 8*C*a*b^3 + 6*C*a^3*b)
)*((5*B*a^4)/8 + B*b^4 + (9*B*a^2*b^2)/2 + 3*A*a*b^3 + (5*A*a^3*b)/2 + 4*C*a*b^3 + 3*C*a^3*b))/d - (tan(c/2 +
(d*x)/2)^13*(2*A*a^4 + 2*A*b^4 - (11*B*a^4)/8 - B*b^4 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 - (15*B*a^2*b^2)/2 +
12*C*a^2*b^2 - 5*A*a*b^3 - (11*A*a^3*b)/2 + 8*B*a*b^3 + 8*B*a^3*b - 4*C*a*b^3 - 5*C*a^3*b) - tan(c/2 + (d*x)/2
)^3*(4*A*a^4 + (28*A*b^4)/3 + (7*B*a^4)/6 + 4*B*b^4 + (20*C*a^4)/3 + 12*C*b^4 + 40*A*a^2*b^2 + 18*B*a^2*b^2 +
56*C*a^2*b^2 + 12*A*a*b^3 + (14*A*a^3*b)/3 + (112*B*a*b^3)/3 + (80*B*a^3*b)/3 + 16*C*a*b^3 + 12*C*a^3*b) - tan
(c/2 + (d*x)/2)^11*(4*A*a^4 + (28*A*b^4)/3 - (7*B*a^4)/6 - 4*B*b^4 + (20*C*a^4)/3 + 12*C*b^4 + 40*A*a^2*b^2 -
18*B*a^2*b^2 + 56*C*a^2*b^2 - 12*A*a*b^3 - (14*A*a^3*b)/3 + (112*B*a*b^3)/3 + (80*B*a^3*b)/3 - 16*C*a*b^3 - 12
*C*a^3*b) + tan(c/2 + (d*x)/2)^5*((86*A*a^4)/5 + (58*A*b^4)/3 + (85*B*a^4)/24 + 5*B*b^4 + (226*C*a^4)/15 + 30*
C*b^4 + (452*A*a^2*b^2)/5 + (27*B*a^2*b^2)/2 + 116*C*a^2*b^2 + 9*A*a*b^3 + (85*A*a^3*b)/6 + (232*B*a*b^3)/3 +
(904*B*a^3*b)/15 + 20*C*a*b^3 + 9*C*a^3*b) + tan(c/2 + (d*x)/2)^9*((86*A*a^4)/5 + (58*A*b^4)/3 - (85*B*a^4)/24
- 5*B*b^4 + (226*C*a^4)/15 + 30*C*b^4 + (452*A*a^2*b^2)/5 - (27*B*a^2*b^2)/2 + 116*C*a^2*b^2 - 9*A*a*b^3 - (8
5*A*a^3*b)/6 + (232*B*a*b^3)/3 + (904*B*a^3*b)/15 - 20*C*a*b^3 - 9*C*a^3*b) - tan(c/2 + (d*x)/2)^7*((424*A*a^4
)/35 + 24*A*b^4 + (104*C*a^4)/5 + 40*C*b^4 + (624*A*a^2*b^2)/5 + 144*C*a^2*b^2 + 96*B*a*b^3 + (416*B*a^3*b)/5)
+ tan(c/2 + (d*x)/2)*(2*A*a^4 + 2*A*b^4 + (11*B*a^4)/8 + B*b^4 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 + (15*B*a^2
*b^2)/2 + 12*C*a^2*b^2 + 5*A*a*b^3 + (11*A*a^3*b)/2 + 8*B*a*b^3 + 8*B*a^3*b + 4*C*a*b^3 + 5*C*a^3*b))/(d*(7*ta
n(c/2 + (d*x)/2)^2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2
+ (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 - 1))