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

Optimal. Leaf size=28 \[ \text {Int}\left (\tan ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p,x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 2.44, size = 0, normalized size = 0.00 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx \]

Verification is Not applicable to the result.

[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]

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 1.64, size = 0, normalized size = 0.00 \[ \int \left (a +b \left (c \sec \left (f x +e \right )\right )^{n}\right )^{p} \left (\tan ^{2}\left (f x +e \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (a+b\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \left (c \sec {\left (e + f x \right )}\right )^{n}\right )^{p} \tan ^{2}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________