3.637 \(\int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx\)

Optimal. Leaf size=455 \[ -\frac{c^5 \sin (e+f x) \left (a^3 A \left (m^2-6 m+8\right )+3 a^2 b B \left (m^2-5 m+4\right )+3 a A b^2 \left (m^2-5 m+4\right )+b^3 B \left (m^2-4 m+3\right )\right ) (c \sec (e+f x))^{m-5} \, _2F_1\left (\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right )}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^4 \sin (e+f x) \left (3 a^2 A b (3-m)+a^3 B (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) (c \sec (e+f x))^{m-4} \, _2F_1\left (\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(e+f x)\right )}{f (2-m) (4-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^4 \tan (e+f x) \left (a^2 A (2-m)+3 a b B (1-m)-2 A b^2 m\right ) (c \sec (e+f x))^{m-4}}{f (1-m) (3-m)}-\frac{a^2 c^4 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (1-m) (2-m)}-\frac{a A c^4 \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)} \]

[Out]

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

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Rubi [A]  time = 1.14548, antiderivative size = 455, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {2960, 4026, 4076, 4047, 3772, 2643, 4046} \[ -\frac{c^5 \sin (e+f x) \left (a^3 A \left (m^2-6 m+8\right )+3 a^2 b B \left (m^2-5 m+4\right )+3 a A b^2 \left (m^2-5 m+4\right )+b^3 B \left (m^2-4 m+3\right )\right ) (c \sec (e+f x))^{m-5} \, _2F_1\left (\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right )}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^4 \sin (e+f x) \left (3 a^2 A b (3-m)+a^3 B (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) (c \sec (e+f x))^{m-4} \, _2F_1\left (\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(e+f x)\right )}{f (2-m) (4-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^4 \tan (e+f x) \left (a^2 A (2-m)+3 a b B (1-m)-2 A b^2 m\right ) (c \sec (e+f x))^{m-4}}{f (1-m) (3-m)}-\frac{a^2 c^4 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (1-m) (2-m)}-\frac{a A c^4 \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[e + f*x])^3*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]

[Out]

-((c^5*(a^3*A*(8 - 6*m + m^2) + 3*a*A*b^2*(4 - 5*m + m^2) + 3*a^2*b*B*(4 - 5*m + m^2) + b^3*B*(3 - 4*m + m^2))
*Hypergeometric2F1[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])) - (c^4*(A*b^3*(2 - m) + 3*a*b^2*B*(2 - m) + 3*a^2*A*b*(3 - m) + a^3*
B*(3 - m))*Hypergeometric2F1[1/2, (4 - m)/2, (6 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-4 + m)*Sin[e + f*x]
)/(f*(2 - m)*(4 - m)*Sqrt[Sin[e + f*x]^2]) - (a*c^4*(3*a*b*B*(1 - m) + a^2*A*(2 - m) - 2*A*b^2*m)*(c*Sec[e + f
*x])^(-4 + m)*Tan[e + f*x])/(f*(1 - m)*(3 - m)) - (a^2*c^4*(a*B*(1 - m) - A*b*(1 + m))*Sec[e + f*x]*(c*Sec[e +
f*x])^(-4 + m)*Tan[e + f*x])/(f*(1 - m)*(2 - m)) - (a*A*c^4*(c*Sec[e + f*x])^(-4 + m)*(b + a*Sec[e + f*x])^2*
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 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))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx &=c^4 \int (c \sec (e+f x))^{-4+m} (b+a \sec (e+f x))^3 (B+A \sec (e+f x)) \, dx\\ &=-\frac{a A c^4 (c \sec (e+f x))^{-4+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m)}-\frac{c^4 \int (c \sec (e+f x))^{-4+m} (b+a \sec (e+f x)) \left (-b (b B (1-m)+a A (4-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 (1+m)) \sec ^2(e+f x)\right ) \, dx}{1-m}\\ &=-\frac{a^2 c^4 (a B (1-m)-A b (1+m)) \sec (e+f x) (c \sec (e+f x))^{-4+m} \tan (e+f x)}{f (1-m) (2-m)}-\frac{a A c^4 (c \sec (e+f x))^{-4+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m)}+\frac{c^4 \int (c \sec (e+f x))^{-4+m} \left (b^2 (b B (1-m)+a A (4-m)) (2-m)+\left (A b^3 (2-m)+3 a b^2 B (2-m)+3 a^2 A b (3-m)+a^3 B (3-m)\right ) (1-m) \sec (e+f x)+a (2-m) \left (3 a b B (1-m)+a^2 A (2-m)-2 A b^2 m\right ) \sec ^2(e+f x)\right ) \, dx}{2-3 m+m^2}\\ &=-\frac{a^2 c^4 (a B (1-m)-A b (1+m)) \sec (e+f x) (c \sec (e+f x))^{-4+m} \tan (e+f x)}{f (1-m) (2-m)}-\frac{a A c^4 (c \sec (e+f x))^{-4+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m)}+\frac{\left (c^3 \left (A b^3 (2-m)+3 a b^2 B (2-m)+3 a^2 A b (3-m)+a^3 B (3-m)\right )\right ) \int (c \sec (e+f x))^{-3+m} \, dx}{2-m}+\frac{c^4 \int (c \sec (e+f x))^{-4+m} \left (b^2 (b B (1-m)+a A (4-m)) (2-m)+a (2-m) \left (3 a b B (1-m)+a^2 A (2-m)-2 A b^2 m\right ) \sec ^2(e+f x)\right ) \, dx}{2-3 m+m^2}\\ &=-\frac{a c^4 \left (3 a b B (1-m)+a^2 A (2-m)-2 A b^2 m\right ) (c \sec (e+f x))^{-4+m} \tan (e+f x)}{f (1-m) (3-m)}-\frac{a^2 c^4 (a B (1-m)-A b (1+m)) \sec (e+f x) (c \sec (e+f x))^{-4+m} \tan (e+f x)}{f (1-m) (2-m)}-\frac{a A c^4 (c \sec (e+f x))^{-4+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m)}+\frac{\left (c^4 \left (a^3 A \left (8-6 m+m^2\right )+3 a A b^2 \left (4-5 m+m^2\right )+3 a^2 b B \left (4-5 m+m^2\right )+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{\left (c^3 \left (A b^3 (2-m)+3 a b^2 B (2-m)+3 a^2 A b (3-m)+a^3 B (3-m)\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 )^{3-m} \, dx}{2-m}\\ &=-\frac{\left (A b^3 (2-m)+3 a b^2 B (2-m)+3 a^2 A b (3-m)+a^3 B (3-m)\right ) \cos ^4(e+f x) \, _2F_1\left (\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^m \sin (e+f x)}{f (2-m) (4-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^4 \left (3 a b B (1-m)+a^2 A (2-m)-2 A b^2 m\right ) (c \sec (e+f x))^{-4+m} \tan (e+f x)}{f (1-m) (3-m)}-\frac{a^2 c^4 (a B (1-m)-A b (1+m)) \sec (e+f x) (c \sec (e+f x))^{-4+m} \tan (e+f x)}{f (1-m) (2-m)}-\frac{a A c^4 (c \sec (e+f x))^{-4+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m)}+\frac{\left (c^4 \left (a^3 A \left (8-6 m+m^2\right )+3 a A b^2 \left (4-5 m+m^2\right )+3 a^2 b B \left (4-5 m+m^2\right )+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 b^3 (2-m)+3 a b^2 B (2-m)+3 a^2 A b (3-m)+a^3 B (3-m)\right ) \cos ^4(e+f x) \, _2F_1\left (\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^m \sin (e+f x)}{f (2-m) (4-m) \sqrt{\sin ^2(e+f x)}}-\frac{\left (a^3 A \left (8-6 m+m^2\right )+3 a A b^2 \left (4-5 m+m^2\right )+3 a^2 b B \left (4-5 m+m^2\right )+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^4 \left (3 a b B (1-m)+a^2 A (2-m)-2 A b^2 m\right ) (c \sec (e+f x))^{-4+m} \tan (e+f x)}{f (1-m) (3-m)}-\frac{a^2 c^4 (a B (1-m)-A b (1+m)) \sec (e+f x) (c \sec (e+f x))^{-4+m} \tan (e+f x)}{f (1-m) (2-m)}-\frac{a A c^4 (c \sec (e+f x))^{-4+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m)}\\ \end{align*}

Mathematica [A]  time = 2.16282, size = 259, normalized size = 0.57 \[ \frac{\sqrt{-\tan ^2(e+f x)} \cot (e+f x) (c \sec (e+f x))^m \left (\frac{b^2 (3 a B+A b) \cos ^3(e+f x) \, _2F_1\left (\frac{1}{2},\frac{m-3}{2};\frac{m-1}{2};\sec ^2(e+f x)\right )}{m-3}+a \left (\frac{3 b (a B+A b) \cos ^2(e+f x) \, _2F_1\left (\frac{1}{2},\frac{m-2}{2};\frac{m}{2};\sec ^2(e+f x)\right )}{m-2}+a \left (\frac{(a B+3 A b) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sec ^2(e+f x)\right )}{m-1}+\frac{a A \, _2F_1\left (\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right )}{m}\right )\right )+\frac{b^3 B \cos ^4(e+f x) \, _2F_1\left (\frac{1}{2},\frac{m-4}{2};\frac{m-2}{2};\sec ^2(e+f x)\right )}{m-4}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[e + f*x])^3*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]

[Out]

(Cot[e + f*x]*((b^3*B*Cos[e + f*x]^4*Hypergeometric2F1[1/2, (-4 + m)/2, (-2 + m)/2, Sec[e + f*x]^2])/(-4 + m)
+ (b^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*((3*b*(A*b + a*B)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (-2 + m)/2, m/2, Sec[e + f*x]^2])/(-2 + m) + a*(((3
*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*Hyperg
eometric2F1[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 = 2.351, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\cos \left ( fx+e \right ) \right ) ^{3} \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))^3*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x)

[Out]

int((a+b*cos(f*x+e))^3*(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 )}^{3} \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))^3*(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)^3*(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^{3} \cos \left (f x + e\right )^{4} + A a^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \cos \left (f x + e\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} 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))^3*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="fricas")

[Out]

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