\(\int (d \sec (e+f x))^m (a+b (c \tan (e+f x))^n)^p \, dx\) [486]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int (d \sec (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\text {Int}\left ((d \sec (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 27, 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 = {} \[ \int (d \sec (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int (d \sec (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.88 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \sec (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int (d \sec (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.49 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \sec (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 93.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int (d \sec (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int \left (d \sec {\left (e + f x \right )}\right )^{m} \left (a + b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 7.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \sec (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \sec (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \sec \left (f x + e\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int (d \sec (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int {\left (a+b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]

[In]

int((a + b*(c*tan(e + f*x))^n)^p*(d/cos(e + f*x))^m,x)

[Out]

int((a + b*(c*tan(e + f*x))^n)^p*(d/cos(e + f*x))^m, x)