\(\int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx\) [449]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (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 = {} \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx=\int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \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 = 16.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx=\int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.87 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{n} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 123.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx=\int \left (c \cos {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right ) \left (a + b \cos {\left (e + f x \right )}\right )^{n}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 3.95 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{n} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{n} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.73 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx=\int {\left (c\,\cos \left (e+f\,x\right )\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^n \,d x \]

[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)