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

Optimal. Leaf size=24 \[ \text {Int}\left (\sec ^n(e+f x) (a+b \sec (e+f x))^m,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx \]

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*}

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Mathematica [A]  time = 2.70, size = 0, normalized size = 0.00 \[ \int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx \]

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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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )^{n}, x\right ) \]

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

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)

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maple [A]  time = 3.00, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{n}\left (f x +e \right )\right ) \left (a +b \sec \left (f x +e \right )\right )^{m}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )^{n}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(e + f*x))^m*(1/cos(e + f*x))^n,x)

[Out]

int((a + b/cos(e + f*x))^m*(1/cos(e + f*x))^n, x)

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

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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