3.786 \(\int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx\)

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\sec ^n(e+f x) (a+b \sec (e+f x))^m,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.041781, 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 = {} \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx &=\int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx\\ \end{align*}

Mathematica [A]  time = 2.17769, size = 0, normalized size = 0. \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.816, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( fx+e \right ) \right ) ^{n} \left ( a+b\sec \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^n*(a+b*sec(f*x+e))^m,x)

[Out]

int(sec(f*x+e)^n*(a+b*sec(f*x+e))^m,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sec(f*x + e) + a)^m*sec(f*x + e)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*sec(f*x + e) + a)^m*sec(f*x + e)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)**n*(a+b*sec(f*x+e))**m,x)

[Out]

Integral((a + b*sec(e + f*x))**m*sec(e + f*x)**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sec(f*x + e) + a)^m*sec(f*x + e)^n, x)