\(\int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx\) [635]

Optimal result

Integrand size = 33, antiderivative size = 33 \[ \int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=(c \cos (e+f x))^m (c \sec (e+f x))^m \text {Int}\left ((c \cos (e+f x))^{-m} (a+b \cos (e+f x))^n (A+B \cos (e+f x)),x\right ) \]

[Out]

(c*cos(f*x+e))^m*(c*sec(f*x+e))^m*Unintegrable((a+b*cos(f*x+e))^n*(A+B*cos(f*x+e))/((c*cos(f*x+e))^m),x)

Rubi [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \left ((c \cos (e+f x))^m (c \sec (e+f x))^m\right ) \int (c \cos (e+f x))^{-m} (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 20.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.88 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00

\[\int \left (a +b \cos \left (f x +e \right )\right )^{n} \left (A +\cos \left (f x +e \right ) B \right ) \left (c \sec \left (f x +e \right )\right )^{m}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{n} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 165.51 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int \left (c \sec {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right ) \left (a + b \cos {\left (e + f x \right )}\right )^{n}\, dx \]

[In]

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

[Out]

Integral((c*sec(e + f*x))**m*(A + B*cos(e + f*x))*(a + b*cos(e + f*x))**n, x)

Maxima [N/A]

Not integrable

Time = 5.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{n} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

Giac [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{n} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

Mupad [N/A]

Not integrable

Time = 2.85 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int {\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^n \,d x \]

[In]

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

[Out]

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