3.636 \(\int (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx\)

Optimal. Leaf size=644 \[ -\frac{c^6 \sin (e+f x) \left (4 a^3 A b \left (m^2-8 m+15\right )+6 a^2 b^2 B \left (m^2-7 m+10\right )+a^4 B \left (m^2-8 m+15\right )+4 a A b^3 \left (m^2-7 m+10\right )+b^4 B \left (m^2-6 m+8\right )\right ) (c \sec (e+f x))^{m-6} \, _2F_1\left (\frac{1}{2},\frac{6-m}{2};\frac{8-m}{2};\cos ^2(e+f x)\right )}{f (2-m) (4-m) (6-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^5 \sin (e+f x) \left (6 a^2 A b^2 \left (m^2-5 m+4\right )+a^4 A \left (m^2-6 m+8\right )+4 a^3 b B \left (m^2-5 m+4\right )+4 a b^3 B \left (m^2-4 m+3\right )+A b^4 \left (m^2-4 m+3\right )\right ) (c \sec (e+f x))^{m-5} \, _2F_1\left (\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right )}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^5 \tan (e+f x) \left (4 a^2 A b \left (m^2-4 m+3\right )+a^3 B \left (m^2-4 m+3\right )+a b^2 B \left (5 m^2-13 m+8\right )+2 A b^3 \left (m^2-2 m+4\right )\right ) (c \sec (e+f x))^{m-5}}{f (1-m) (2-m) (4-m)}-\frac{a^2 c^5 \tan (e+f x) \sec (e+f x) \left (a^2 A (2-m)^2+2 a b B (1-m)^2+A b^2 \left (m^2-m+6\right )\right ) (c \sec (e+f x))^{m-5}}{f (1-m) (2-m) (3-m)}-\frac{a c^5 \tan (e+f x) (a B (1-m)-A b (m+2)) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-5}}{f (1-m) (2-m)}-\frac{a A c^5 \tan (e+f x) (a \sec (e+f x)+b)^3 (c \sec (e+f x))^{m-5}}{f (1-m)} \]

[Out]

-((c^6*(4*a^3*A*b*(15 - 8*m + m^2) + a^4*B*(15 - 8*m + m^2) + 4*a*A*b^3*(10 - 7*m + m^2) + 6*a^2*b^2*B*(10 - 7
*m + m^2) + b^4*B*(8 - 6*m + m^2))*Hypergeometric2F1[1/2, (6 - m)/2, (8 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x
])^(-6 + m)*Sin[e + f*x])/(f*(2 - m)*(4 - m)*(6 - m)*Sqrt[Sin[e + f*x]^2])) - (c^5*(a^4*A*(8 - 6*m + m^2) + 6*
a^2*A*b^2*(4 - 5*m + m^2) + 4*a^3*b*B*(4 - 5*m + m^2) + A*b^4*(3 - 4*m + m^2) + 4*a*b^3*B*(3 - 4*m + m^2))*Hyp
ergeometric2F1[1/2, (5 - m)/2, (7 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-5 + m)*Sin[e + f*x])/(f*(1 - m)*(
3 - m)*(5 - m)*Sqrt[Sin[e + f*x]^2]) - (a*c^5*(4*a^2*A*b*(3 - 4*m + m^2) + a^3*B*(3 - 4*m + m^2) + 2*A*b^3*(4
- 2*m + m^2) + a*b^2*B*(8 - 13*m + 5*m^2))*(c*Sec[e + f*x])^(-5 + m)*Tan[e + f*x])/(f*(1 - m)*(2 - m)*(4 - m))
- (a^2*c^5*(2*a*b*B*(1 - m)^2 + a^2*A*(2 - m)^2 + A*b^2*(6 - m + m^2))*Sec[e + f*x]*(c*Sec[e + f*x])^(-5 + m)
*Tan[e + f*x])/(f*(1 - m)*(2 - m)*(3 - m)) - (a*c^5*(a*B*(1 - m) - A*b*(2 + m))*(c*Sec[e + f*x])^(-5 + m)*(b +
a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(1 - m)*(2 - m)) - (a*A*c^5*(c*Sec[e + f*x])^(-5 + m)*(b + a*Sec[e + f*x])
^3*Tan[e + f*x])/(f*(1 - m))

________________________________________________________________________________________

Rubi [A]  time = 2.04236, antiderivative size = 644, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {2960, 4026, 4096, 4076, 4047, 3772, 2643, 4046} \[ -\frac{c^6 \sin (e+f x) \left (4 a^3 A b \left (m^2-8 m+15\right )+6 a^2 b^2 B \left (m^2-7 m+10\right )+a^4 B \left (m^2-8 m+15\right )+4 a A b^3 \left (m^2-7 m+10\right )+b^4 B \left (m^2-6 m+8\right )\right ) (c \sec (e+f x))^{m-6} \, _2F_1\left (\frac{1}{2},\frac{6-m}{2};\frac{8-m}{2};\cos ^2(e+f x)\right )}{f (2-m) (4-m) (6-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^5 \sin (e+f x) \left (6 a^2 A b^2 \left (m^2-5 m+4\right )+a^4 A \left (m^2-6 m+8\right )+4 a^3 b B \left (m^2-5 m+4\right )+4 a b^3 B \left (m^2-4 m+3\right )+A b^4 \left (m^2-4 m+3\right )\right ) (c \sec (e+f x))^{m-5} \, _2F_1\left (\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right )}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^5 \tan (e+f x) \left (4 a^2 A b \left (m^2-4 m+3\right )+a^3 B \left (m^2-4 m+3\right )+a b^2 B \left (5 m^2-13 m+8\right )+2 A b^3 \left (m^2-2 m+4\right )\right ) (c \sec (e+f x))^{m-5}}{f (1-m) (2-m) (4-m)}-\frac{a^2 c^5 \tan (e+f x) \sec (e+f x) \left (a^2 A (2-m)^2+2 a b B (1-m)^2+A b^2 \left (m^2-m+6\right )\right ) (c \sec (e+f x))^{m-5}}{f (1-m) (2-m) (3-m)}-\frac{a c^5 \tan (e+f x) (a B (1-m)-A b (m+2)) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-5}}{f (1-m) (2-m)}-\frac{a A c^5 \tan (e+f x) (a \sec (e+f x)+b)^3 (c \sec (e+f x))^{m-5}}{f (1-m)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[e + f*x])^4*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]

[Out]

-((c^6*(4*a^3*A*b*(15 - 8*m + m^2) + a^4*B*(15 - 8*m + m^2) + 4*a*A*b^3*(10 - 7*m + m^2) + 6*a^2*b^2*B*(10 - 7
*m + m^2) + b^4*B*(8 - 6*m + m^2))*Hypergeometric2F1[1/2, (6 - m)/2, (8 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x
])^(-6 + m)*Sin[e + f*x])/(f*(2 - m)*(4 - m)*(6 - m)*Sqrt[Sin[e + f*x]^2])) - (c^5*(a^4*A*(8 - 6*m + m^2) + 6*
a^2*A*b^2*(4 - 5*m + m^2) + 4*a^3*b*B*(4 - 5*m + m^2) + A*b^4*(3 - 4*m + m^2) + 4*a*b^3*B*(3 - 4*m + m^2))*Hyp
ergeometric2F1[1/2, (5 - m)/2, (7 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-5 + m)*Sin[e + f*x])/(f*(1 - m)*(
3 - m)*(5 - m)*Sqrt[Sin[e + f*x]^2]) - (a*c^5*(4*a^2*A*b*(3 - 4*m + m^2) + a^3*B*(3 - 4*m + m^2) + 2*A*b^3*(4
- 2*m + m^2) + a*b^2*B*(8 - 13*m + 5*m^2))*(c*Sec[e + f*x])^(-5 + m)*Tan[e + f*x])/(f*(1 - m)*(2 - m)*(4 - m))
- (a^2*c^5*(2*a*b*B*(1 - m)^2 + a^2*A*(2 - m)^2 + A*b^2*(6 - m + m^2))*Sec[e + f*x]*(c*Sec[e + f*x])^(-5 + m)
*Tan[e + f*x])/(f*(1 - m)*(2 - m)*(3 - m)) - (a*c^5*(a*B*(1 - m) - A*b*(2 + m))*(c*Sec[e + f*x])^(-5 + m)*(b +
a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(1 - m)*(2 - m)) - (a*A*c^5*(c*Sec[e + f*x])^(-5 + m)*(b + a*Sec[e + f*x])
^3*Tan[e + f*x])/(f*(1 - m))

Rule 2960

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[g^(m + n), Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^m*(
d + c*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[p] && I
ntegerQ[m] && IntegerQ[n]

Rule 4026

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> -Simp[(b*B*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*(m + n
)), x] + Dist[1/(m + n), Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n*Simp[a^2*A*(m + n) + a*b*B*n + (a
*(2*A*b + a*B)*(m + n) + b^2*B*(m + n - 1))*Csc[e + f*x] + b*(A*b*(m + n) + a*B*(2*m + n - 1))*Csc[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] &&  !
(IGtQ[n, 1] &&  !IntegerQ[m])

Rule 4096

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[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d
*Csc[e + f*x])^n)/(f*(m + n + 1)), x] + Dist[1/(m + n + 1), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^
n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a*B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a*C
*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] &&
!LeQ[n, -1]

Rule 4076

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[(b*C*Csc[e + f*x]*Cot[e + f*x]*(d*Csc[e + f*x
])^n)/(f*(n + 2)), x] + Dist[1/(n + 2), Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1)
+ A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*(n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C,
n}, 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 3772

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

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 4046

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> -Simp[(C*Cot[
e + f*x]*(b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[(C*m + A*(m + 1))/(m + 1), Int[(b*Csc[e + f*x])^m, x], x]
/; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] &&  !LeQ[m, -1]

Rubi steps

\begin{align*} \int (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx &=c^5 \int (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^4 (B+A \sec (e+f x)) \, dx\\ &=-\frac{a A c^5 (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^3 \tan (e+f x)}{f (1-m)}-\frac{c^5 \int (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^2 \left (-b (b B (1-m)+a A (5-m))-\left (b (A b+2 a B) (1-m)+a^2 A (2-m)\right ) \sec (e+f x)-a (a B (1-m)-A b (2+m)) \sec ^2(e+f x)\right ) \, dx}{1-m}\\ &=-\frac{a c^5 (a B (1-m)-A b (2+m)) (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m) (2-m)}-\frac{a A c^5 (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^3 \tan (e+f x)}{f (1-m)}+\frac{c^5 \int (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x)) \left (-b \left (2 a A b (5-m) m-a^2 B \left (5-6 m+m^2\right )-b^2 B \left (2-3 m+m^2\right )\right )+\left (b \left (a^2 A (7-2 m)+A b^2 (1-m)+3 a b B (1-m)\right ) (2-m)+a^2 (3-m) (a B (1-m)-A b (2+m))\right ) \sec (e+f x)+a \left (2 a b B (1-m)^2+a^2 A (2-m)^2+A b^2 \left (6-m+m^2\right )\right ) \sec ^2(e+f x)\right ) \, dx}{2-3 m+m^2}\\ &=-\frac{a^2 c^5 \left (2 a b B (1-m)^2+a^2 A (2-m)^2+A b^2 \left (6-m+m^2\right )\right ) \sec (e+f x) (c \sec (e+f x))^{-5+m} \tan (e+f x)}{f (3-m) \left (2-3 m+m^2\right )}-\frac{a c^5 (a B (1-m)-A b (2+m)) (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m) (2-m)}-\frac{a A c^5 (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^3 \tan (e+f x)}{f (1-m)}+\frac{c^5 \int (c \sec (e+f x))^{-5+m} \left (b^2 (3-m) \left (2 a A b (5-m) m-a^2 B \left (5-6 m+m^2\right )-b^2 B \left (2-3 m+m^2\right )\right )-(2-m) \left (a^4 A \left (8-6 m+m^2\right )+6 a^2 A b^2 \left (4-5 m+m^2\right )+4 a^3 b B \left (4-5 m+m^2\right )+A b^4 \left (3-4 m+m^2\right )+4 a b^3 B \left (3-4 m+m^2\right )\right ) \sec (e+f x)-a (3-m) \left (4 a^2 A b \left (3-4 m+m^2\right )+a^3 B \left (3-4 m+m^2\right )+2 A b^3 \left (4-2 m+m^2\right )+a b^2 B \left (8-13 m+5 m^2\right )\right ) \sec ^2(e+f x)\right ) \, dx}{(-3+m) \left (2-3 m+m^2\right )}\\ &=-\frac{a^2 c^5 \left (2 a b B (1-m)^2+a^2 A (2-m)^2+A b^2 \left (6-m+m^2\right )\right ) \sec (e+f x) (c \sec (e+f x))^{-5+m} \tan (e+f x)}{f (3-m) \left (2-3 m+m^2\right )}-\frac{a c^5 (a B (1-m)-A b (2+m)) (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m) (2-m)}-\frac{a A c^5 (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^3 \tan (e+f x)}{f (1-m)}+\frac{c^5 \int (c \sec (e+f x))^{-5+m} \left (b^2 (3-m) \left (2 a A b (5-m) m-a^2 B \left (5-6 m+m^2\right )-b^2 B \left (2-3 m+m^2\right )\right )-a (3-m) \left (4 a^2 A b \left (3-4 m+m^2\right )+a^3 B \left (3-4 m+m^2\right )+2 A b^3 \left (4-2 m+m^2\right )+a b^2 B \left (8-13 m+5 m^2\right )\right ) \sec ^2(e+f x)\right ) \, dx}{(-3+m) \left (2-3 m+m^2\right )}+\frac{\left (c^4 \left (a^4 A \left (8-6 m+m^2\right )+6 a^2 A b^2 \left (4-5 m+m^2\right )+4 a^3 b B \left (4-5 m+m^2\right )+A b^4 \left (3-4 m+m^2\right )+4 a b^3 B \left (3-4 m+m^2\right )\right )\right ) \int (c \sec (e+f x))^{-4+m} \, dx}{(1-m) (3-m)}\\ &=-\frac{a c^5 \left (4 a^2 A b \left (3-4 m+m^2\right )+a^3 B \left (3-4 m+m^2\right )+2 A b^3 \left (4-2 m+m^2\right )+a b^2 B \left (8-13 m+5 m^2\right )\right ) (c \sec (e+f x))^{-5+m} \tan (e+f x)}{f (4-m) \left (2-3 m+m^2\right )}-\frac{a^2 c^5 \left (2 a b B (1-m)^2+a^2 A (2-m)^2+A b^2 \left (6-m+m^2\right )\right ) \sec (e+f x) (c \sec (e+f x))^{-5+m} \tan (e+f x)}{f (3-m) \left (2-3 m+m^2\right )}-\frac{a c^5 (a B (1-m)-A b (2+m)) (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m) (2-m)}-\frac{a A c^5 (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^3 \tan (e+f x)}{f (1-m)}+\frac{\left (c^5 \left (4 a^3 A b \left (15-8 m+m^2\right )+a^4 B \left (15-8 m+m^2\right )+4 a A b^3 \left (10-7 m+m^2\right )+6 a^2 b^2 B \left (10-7 m+m^2\right )+b^4 B \left (8-6 m+m^2\right )\right )\right ) \int (c \sec (e+f x))^{-5+m} \, dx}{(2-m) (4-m)}+\frac{\left (c^4 \left (a^4 A \left (8-6 m+m^2\right )+6 a^2 A b^2 \left (4-5 m+m^2\right )+4 a^3 b B \left (4-5 m+m^2\right )+A b^4 \left (3-4 m+m^2\right )+4 a b^3 B \left (3-4 m+m^2\right )\right ) \left (\frac{\cos (e+f x)}{c}\right )^m (c \sec (e+f x))^m\right ) \int \left (\frac{\cos (e+f x)}{c}\right )^{4-m} \, dx}{(1-m) (3-m)}\\ &=-\frac{\left (a^4 A \left (8-6 m+m^2\right )+6 a^2 A b^2 \left (4-5 m+m^2\right )+4 a^3 b B \left (4-5 m+m^2\right )+A b^4 \left (3-4 m+m^2\right )+4 a b^3 B \left (3-4 m+m^2\right )\right ) \cos ^5(e+f x) \, _2F_1\left (\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^m \sin (e+f x)}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^5 \left (4 a^2 A b \left (3-4 m+m^2\right )+a^3 B \left (3-4 m+m^2\right )+2 A b^3 \left (4-2 m+m^2\right )+a b^2 B \left (8-13 m+5 m^2\right )\right ) (c \sec (e+f x))^{-5+m} \tan (e+f x)}{f (4-m) \left (2-3 m+m^2\right )}-\frac{a^2 c^5 \left (2 a b B (1-m)^2+a^2 A (2-m)^2+A b^2 \left (6-m+m^2\right )\right ) \sec (e+f x) (c \sec (e+f x))^{-5+m} \tan (e+f x)}{f (3-m) \left (2-3 m+m^2\right )}-\frac{a c^5 (a B (1-m)-A b (2+m)) (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m) (2-m)}-\frac{a A c^5 (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^3 \tan (e+f x)}{f (1-m)}+\frac{\left (c^5 \left (4 a^3 A b \left (15-8 m+m^2\right )+a^4 B \left (15-8 m+m^2\right )+4 a A b^3 \left (10-7 m+m^2\right )+6 a^2 b^2 B \left (10-7 m+m^2\right )+b^4 B \left (8-6 m+m^2\right )\right ) \left (\frac{\cos (e+f x)}{c}\right )^m (c \sec (e+f x))^m\right ) \int \left (\frac{\cos (e+f x)}{c}\right )^{5-m} \, dx}{(2-m) (4-m)}\\ &=-\frac{\left (a^4 A \left (8-6 m+m^2\right )+6 a^2 A b^2 \left (4-5 m+m^2\right )+4 a^3 b B \left (4-5 m+m^2\right )+A b^4 \left (3-4 m+m^2\right )+4 a b^3 B \left (3-4 m+m^2\right )\right ) \cos ^5(e+f x) \, _2F_1\left (\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^m \sin (e+f x)}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{\left (4 a^3 A b \left (15-8 m+m^2\right )+a^4 B \left (15-8 m+m^2\right )+4 a A b^3 \left (10-7 m+m^2\right )+6 a^2 b^2 B \left (10-7 m+m^2\right )+b^4 B \left (8-6 m+m^2\right )\right ) \cos ^6(e+f x) \, _2F_1\left (\frac{1}{2},\frac{6-m}{2};\frac{8-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^m \sin (e+f x)}{f (2-m) (4-m) (6-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^5 \left (4 a^2 A b \left (3-4 m+m^2\right )+a^3 B \left (3-4 m+m^2\right )+2 A b^3 \left (4-2 m+m^2\right )+a b^2 B \left (8-13 m+5 m^2\right )\right ) (c \sec (e+f x))^{-5+m} \tan (e+f x)}{f (4-m) \left (2-3 m+m^2\right )}-\frac{a^2 c^5 \left (2 a b B (1-m)^2+a^2 A (2-m)^2+A b^2 \left (6-m+m^2\right )\right ) \sec (e+f x) (c \sec (e+f x))^{-5+m} \tan (e+f x)}{f (3-m) \left (2-3 m+m^2\right )}-\frac{a c^5 (a B (1-m)-A b (2+m)) (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m) (2-m)}-\frac{a A c^5 (c \sec (e+f x))^{-5+m} (b+a \sec (e+f x))^3 \tan (e+f x)}{f (1-m)}\\ \end{align*}

Mathematica [A]  time = 3.95873, size = 317, normalized size = 0.49 \[ \frac{\sqrt{-\tan ^2(e+f x)} \cot (e+f x) (c \sec (e+f x))^m \left (\frac{b^3 (4 a B+A b) \cos ^4(e+f x) \, _2F_1\left (\frac{1}{2},\frac{m-4}{2};\frac{m-2}{2};\sec ^2(e+f x)\right )}{m-4}+a \left (\frac{2 b^2 (3 a B+2 A b) \cos ^3(e+f x) \, _2F_1\left (\frac{1}{2},\frac{m-3}{2};\frac{m-1}{2};\sec ^2(e+f x)\right )}{m-3}+a \left (\frac{2 b (2 a B+3 A b) \cos ^2(e+f x) \, _2F_1\left (\frac{1}{2},\frac{m-2}{2};\frac{m}{2};\sec ^2(e+f x)\right )}{m-2}+a \left (\frac{(a B+4 A b) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sec ^2(e+f x)\right )}{m-1}+\frac{a A \, _2F_1\left (\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right )}{m}\right )\right )\right )+\frac{b^4 B \cos ^5(e+f x) \, _2F_1\left (\frac{1}{2},\frac{m-5}{2};\frac{m-3}{2};\sec ^2(e+f x)\right )}{m-5}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[e + f*x])^4*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]

[Out]

(Cot[e + f*x]*((b^4*B*Cos[e + f*x]^5*Hypergeometric2F1[1/2, (-5 + m)/2, (-3 + m)/2, Sec[e + f*x]^2])/(-5 + m)
+ (b^3*(A*b + 4*a*B)*Cos[e + f*x]^4*Hypergeometric2F1[1/2, (-4 + m)/2, (-2 + m)/2, Sec[e + f*x]^2])/(-4 + m) +
a*((2*b^2*(2*A*b + 3*a*B)*Cos[e + f*x]^3*Hypergeometric2F1[1/2, (-3 + m)/2, (-1 + m)/2, Sec[e + f*x]^2])/(-3
+ m) + a*((2*b*(3*A*b + 2*a*B)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (-2 + m)/2, m/2, Sec[e + f*x]^2])/(-2 + m
) + a*(((4*A*b + a*B)*Cos[e + f*x]*Hypergeometric2F1[1/2, (-1 + m)/2, (1 + m)/2, Sec[e + f*x]^2])/(-1 + m) + (
a*A*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[e + f*x]^2])/m))))*(c*Sec[e + f*x])^m*Sqrt[-Tan[e + f*x]^2])/f

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Maple [F]  time = 2.862, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\cos \left ( fx+e \right ) \right ) ^{4} \left ( A+B\cos \left ( fx+e \right ) \right ) \left ( c\sec \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(f*x+e))^4*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x)

[Out]

int((a+b*cos(f*x+e))^4*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (f x + e\right ) + A\right )}{\left (b \cos \left (f x + e\right ) + a\right )}^{4} \left (c \sec \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(f*x+e))^4*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^4*(c*sec(f*x + e))^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{4} \cos \left (f x + e\right )^{5} + A a^{4} +{\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (f x + e\right )^{4} + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (f x + e\right )^{3} + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} \cos \left (f x + e\right )^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (f x + e\right )\right )} \left (c \sec \left (f x + e\right )\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(f*x+e))^4*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((B*b^4*cos(f*x + e)^5 + A*a^4 + (4*B*a*b^3 + A*b^4)*cos(f*x + e)^4 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*cos(
f*x + e)^3 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*cos(f*x + e)^2 + (B*a^4 + 4*A*a^3*b)*cos(f*x + e))*(c*sec(f*x + e))^m
, x)

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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(f*x+e))**4*(A+B*cos(f*x+e))*(c*sec(f*x+e))**m,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (f x + e\right ) + A\right )}{\left (b \cos \left (f x + e\right ) + a\right )}^{4} \left (c \sec \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(f*x+e))^4*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^4*(c*sec(f*x + e))^m, x)