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

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\tan ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.055236, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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 = 1.94417, size = 0, normalized size = 0. \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^2(e+f x) \, dx \]

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]

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Maple [A]  time = 0.493, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( c\sec \left ( fx+e \right ) \right ) ^{n} \right ) ^{p} \left ( \tan \left ( fx+e \right ) \right ) ^{2}\, dx \end{align*}

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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \left (c \sec{\left (e + f x \right )}\right )^{n}\right )^{p} \tan ^{2}{\left (e + f x \right )}\, dx \end{align*}

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)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)