\(\int \cos (e+f x) (a+b \sec (e+f x))^m \, dx\) [792]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \cos (e+f x) (a+b \sec (e+f x))^m \, dx=\text {Int}\left (\cos (e+f x) (a+b \sec (e+f x))^m,x\right ) \]

[Out]

Unintegrable(cos(f*x+e)*(a+b*sec(f*x+e))^m,x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 19, 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 = {} \[ \int \cos (e+f x) (a+b \sec (e+f x))^m \, dx=\int \cos (e+f x) (a+b \sec (e+f x))^m \, dx \]

[In]

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

[Out]

Defer[Int][Cos[e + f*x]*(a + b*Sec[e + f*x])^m, x]

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

Mathematica [N/A]

Not integrable

Time = 13.48 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \cos (e+f x) (a+b \sec (e+f x))^m \, dx=\int \cos (e+f x) (a+b \sec (e+f x))^m \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.56 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00

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

[In]

int(cos(f*x+e)*(a+b*sec(f*x+e))^m,x)

[Out]

int(cos(f*x+e)*(a+b*sec(f*x+e))^m,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \cos (e+f x) (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right ) \,d x } \]

[In]

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

[Out]

integral((b*sec(f*x + e) + a)^m*cos(f*x + e), x)

Sympy [N/A]

Not integrable

Time = 7.94 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \cos (e+f x) (a+b \sec (e+f x))^m \, dx=\int \left (a + b \sec {\left (e + f x \right )}\right )^{m} \cos {\left (e + f x \right )}\, dx \]

[In]

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

[Out]

Integral((a + b*sec(e + f*x))**m*cos(e + f*x), x)

Maxima [N/A]

Not integrable

Time = 1.56 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \cos (e+f x) (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right ) \,d x } \]

[In]

integrate(cos(f*x+e)*(a+b*sec(f*x+e))^m,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \cos (e+f x) (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right ) \,d x } \]

[In]

integrate(cos(f*x+e)*(a+b*sec(f*x+e))^m,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 14.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \cos (e+f x) (a+b \sec (e+f x))^m \, dx=\int \cos \left (e+f\,x\right )\,{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]

[In]

int(cos(e + f*x)*(a + b/cos(e + f*x))^m,x)

[Out]

int(cos(e + f*x)*(a + b/cos(e + f*x))^m, x)