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

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\sec (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0275891, 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 = {} \[ \int \sec (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.74632, size = 0, normalized size = 0. \[ \int \sec (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.305, size = 0, normalized size = 0. \begin{align*} \int \sec \left ( fx+e \right ) \left ( a+b \left ( c\tan \left ( fx+e \right ) \right ) ^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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