\(\int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx\) [786]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx=\text {Int}\left (\sec ^n(e+f x) (a+b \sec (e+f x))^m,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 21, 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 = {} \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx=\int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 8.39 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx=\int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.62 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )^{n} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 5.62 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx=\int \left (a + b \sec {\left (e + f x \right )}\right )^{m} \sec ^{n}{\left (e + f x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )^{n} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )^{n} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 15.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx=\int {\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]

[In]

int((a + b/cos(e + f*x))^m*(1/cos(e + f*x))^n,x)

[Out]

int((a + b/cos(e + f*x))^m*(1/cos(e + f*x))^n, x)