3.792 \(\int \cos (e+f x) (a+b \sec (e+f x))^m \, dx\)

Optimal. Leaf size=21 \[ \text{Unintegrable}\left (\cos (e+f x) (a+b \sec (e+f x))^m,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0328224, 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 = {} \[ \int \cos (e+f x) (a+b \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 5.55849, size = 0, normalized size = 0. \[ \int \cos (e+f x) (a+b \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.335, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( fx+e \right ) \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(cos(f*x+e)*(a+b*sec(f*x+e))^m,x)

[Out]

int(cos(f*x+e)*(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} \cos \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sec(f*x + e) + a)^m*cos(f*x + e), 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} \cos \left (f x + e\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*sec(f*x + e) + a)^m*cos(f*x + e), 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} \cos{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*sec(e + f*x))**m*cos(e + f*x), 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} \cos \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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