3.635 \(\int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx\)

Optimal. Leaf size=58 \[ (c \cos (e+f x))^m (c \sec (e+f x))^m \text{Unintegrable}\left ((A+B \cos (e+f x)) (c \cos (e+f x))^{-m} (a+b \cos (e+f x))^n,x\right ) \]

[Out]

(c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Unintegrable[((a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x]))/(c*Cos[e + f*
x])^m, x]

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Rubi [A]  time = 0.18063, 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 = {} \[ \int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Defer[Int][((a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x]))/(c*Cos[e + f*x]
)^m, x]

Rubi steps

\begin{align*} \int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx &=\left ((c \cos (e+f x))^m (c \sec (e+f x))^m\right ) \int (c \cos (e+f x))^{-m} (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx\\ \end{align*}

Mathematica [A]  time = 8.45503, size = 0, normalized size = 0. \[ \int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 3.207, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\cos \left ( fx+e \right ) \right ) ^{n} \left ( A+B\cos \left ( fx+e \right ) \right ) \left ( c\sec \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(f*x+e))^n*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x)

[Out]

int((a+b*cos(f*x+e))^n*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^n*(c*sec(f*x + e))^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^n*(c*sec(f*x + e))^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(f*x+e))**n*(A+B*cos(f*x+e))*(c*sec(f*x+e))**m,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^n*(c*sec(f*x + e))^m, x)