\(\int \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx\) [642]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=(c \cos (e+f x))^m (c \sec (e+f x))^m \text {Int}\left ((c \cos (e+f x))^{-m} \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)),x\right ) \]

[Out]

(c*cos(f*x+e))^m*(c*sec(f*x+e))^m*Unintegrable((A+B*cos(f*x+e))*(a+b*cos(f*x+e))^(1/2)/((c*cos(f*x+e))^m),x)

Rubi [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 35, 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 = {} \[ \int \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]

[In]

Int[Sqrt[a + b*Cos[e + f*x]]*(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][(Sqrt[a + b*Cos[e + f*x]]*(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} \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 33.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.77 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94

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

[In]

int((a+b*cos(f*x+e))^(1/2)*(A+cos(f*x+e)*B)*(c*sec(f*x+e))^m,x)

[Out]

int((a+b*cos(f*x+e))^(1/2)*(A+cos(f*x+e)*B)*(c*sec(f*x+e))^m,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} \sqrt {b \cos \left (f x + e\right ) + a} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((a+b*cos(f*x+e))^(1/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 8.46 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int \left (c \sec {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right ) \sqrt {a + b \cos {\left (e + f x \right )}}\, dx \]

[In]

integrate((a+b*cos(f*x+e))**(1/2)*(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))*sqrt(a + b*cos(e + f*x)), x)

Maxima [N/A]

Not integrable

Time = 2.71 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} \sqrt {b \cos \left (f x + e\right ) + a} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((a+b*cos(f*x+e))^(1/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} \sqrt {b \cos \left (f x + e\right ) + a} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((a+b*cos(f*x+e))^(1/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int {\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,\sqrt {a+b\,\cos \left (e+f\,x\right )} \,d x \]

[In]

int((c/cos(e + f*x))^m*(A + B*cos(e + f*x))*(a + b*cos(e + f*x))^(1/2),x)

[Out]

int((c/cos(e + f*x))^m*(A + B*cos(e + f*x))*(a + b*cos(e + f*x))^(1/2), x)