\(\int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx\) [478]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.13 (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 = {} \[ \int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx=\int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 18.65 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx=\int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx=\int { {\left (B \sec \left (f x + e\right ) + A\right )} {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \left (c \sec \left (f x + e\right )\right )^{n} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 42.34 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx=\int \left (c \sec {\left (e + f x \right )}\right )^{n} \left (A + B \sec {\left (e + f x \right )}\right ) \left (a + b \sec {\left (e + f x \right )}\right )^{m}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 8.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx=\int { {\left (B \sec \left (f x + e\right ) + A\right )} {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \left (c \sec \left (f x + e\right )\right )^{n} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.89 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx=\int { {\left (B \sec \left (f x + e\right ) + A\right )} {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \left (c \sec \left (f x + e\right )\right )^{n} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 19.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx=\int \left (A+\frac {B}{\cos \left (e+f\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]

[In]

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

[Out]

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