\(\int (a+b (c \sec (e+f x))^n)^p \, dx\) [469]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\text {Int}\left (\left (a+b (c \sec (e+f x))^n\right )^p,x\right ) \]

[Out]

Unintegrable((a+b*(c*sec(f*x+e))^n)^p,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int \left (a+b (c \sec (e+f x))^n\right )^p \, dx \]

[In]

Int[(a + b*(c*Sec[e + f*x])^n)^p,x]

[Out]

Defer[Int][(a + b*(c*Sec[e + f*x])^n)^p, x]

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

Mathematica [N/A]

Not integrable

Time = 1.74 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int \left (a+b (c \sec (e+f x))^n\right )^p \, dx \]

[In]

Integrate[(a + b*(c*Sec[e + f*x])^n)^p,x]

[Out]

Integrate[(a + b*(c*Sec[e + f*x])^n)^p, x]

Maple [N/A] (verified)

Not integrable

Time = 0.48 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[In]

int((a+b*(c*sec(f*x+e))^n)^p,x)

[Out]

int((a+b*(c*sec(f*x+e))^n)^p,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \,d x } \]

[In]

integrate((a+b*(c*sec(f*x+e))^n)^p,x, algorithm="fricas")

[Out]

integral(((c*sec(f*x + e))^n*b + a)^p, x)

Sympy [N/A]

Not integrable

Time = 1.73 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int \left (a + b \left (c \sec {\left (e + f x \right )}\right )^{n}\right )^{p}\, dx \]

[In]

integrate((a+b*(c*sec(f*x+e))**n)**p,x)

[Out]

Integral((a + b*(c*sec(e + f*x))**n)**p, x)

Maxima [N/A]

Not integrable

Time = 3.95 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \,d x } \]

[In]

integrate((a+b*(c*sec(f*x+e))^n)^p,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \,d x } \]

[In]

integrate((a+b*(c*sec(f*x+e))^n)^p,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 20.78 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int {\left (a+b\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n\right )}^p \,d x \]

[In]

int((a + b*(c/cos(e + f*x))^n)^p,x)

[Out]

int((a + b*(c/cos(e + f*x))^n)^p, x)