\(\int (c \cos (e+f x))^m \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx\) [456]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.12 (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 = {} \[ \int (c \cos (e+f x))^m \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx=\int (c \cos (e+f x))^m \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \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 = 28.85 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int (c \cos (e+f x))^m \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx=\int (c \cos (e+f x))^m \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.80 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int (c \cos (e+f x))^m \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} \sqrt {b \cos \left (f x + e\right ) + a} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 7.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int (c \cos (e+f x))^m \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx=\int \left (c \cos {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right ) \sqrt {a + b \cos {\left (e + f x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 2.59 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int (c \cos (e+f x))^m \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} \sqrt {b \cos \left (f x + e\right ) + a} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.96 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int (c \cos (e+f x))^m \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} \sqrt {b \cos \left (f x + e\right ) + a} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.72 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int (c \cos (e+f x))^m \sqrt {a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx=\int {\left (c\,\cos \left (e+f\,x\right )\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,\sqrt {a+b\,\cos \left (e+f\,x\right )} \,d x \]

[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)