\(\int (a+b (c \sec (e+f x))^n)^p \tan ^2(e+f x) \, dx\) [468]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx=\text {Int}\left (\left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x),x\right ) \]

[Out]

Unintegrable((a+b*(c*sec(f*x+e))^n)^p*tan(f*x+e)^2,x)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 25, 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 = {} \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx=\int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx \]

[In]

Int[(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^2,x]

[Out]

Defer[Int][(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx=\int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx \]

[In]

Integrate[(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^2,x]

[Out]

Integrate[(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

\[\int \left (a +b \left (c \sec \left (f x +e \right )\right )^{n}\right )^{p} \tan \left (f x +e \right )^{2}d x\]

[In]

int((a+b*(c*sec(f*x+e))^n)^p*tan(f*x+e)^2,x)

[Out]

int((a+b*(c*sec(f*x+e))^n)^p*tan(f*x+e)^2,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{2} \,d x } \]

[In]

integrate((a+b*(c*sec(f*x+e))^n)^p*tan(f*x+e)^2,x, algorithm="fricas")

[Out]

integral(((c*sec(f*x + e))^n*b + a)^p*tan(f*x + e)^2, x)

Sympy [N/A]

Not integrable

Time = 19.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx=\int \left (a + b \left (c \sec {\left (e + f x \right )}\right )^{n}\right )^{p} \tan ^{2}{\left (e + f x \right )}\, dx \]

[In]

integrate((a+b*(c*sec(f*x+e))**n)**p*tan(f*x+e)**2,x)

[Out]

Integral((a + b*(c*sec(e + f*x))**n)**p*tan(e + f*x)**2, x)

Maxima [N/A]

Not integrable

Time = 5.86 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{2} \,d x } \]

[In]

integrate((a+b*(c*sec(f*x+e))^n)^p*tan(f*x+e)^2,x, algorithm="maxima")

[Out]

integrate(((c*sec(f*x + e))^n*b + a)^p*tan(f*x + e)^2, x)

Giac [N/A]

Not integrable

Time = 1.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{2} \,d x } \]

[In]

integrate((a+b*(c*sec(f*x+e))^n)^p*tan(f*x+e)^2,x, algorithm="giac")

[Out]

integrate(((c*sec(f*x + e))^n*b + a)^p*tan(f*x + e)^2, x)

Mupad [N/A]

Not integrable

Time = 21.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (a+b\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n\right )}^p \,d x \]

[In]

int(tan(e + f*x)^2*(a + b*(c/cos(e + f*x))^n)^p,x)

[Out]

int(tan(e + f*x)^2*(a + b*(c/cos(e + f*x))^n)^p, x)