3.642 \(\int \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx\)

Optimal. Leaf size=60 \[ (c \cos (e+f x))^m (c \sec (e+f x))^m \text{Unintegrable}\left (\sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \cos (e+f x))^{-m},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.238709, 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 = {} \[ \int \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

Rubi steps

\begin{align*} \int \sqrt{a+b \cos (e+f x)} (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} \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx\\ \end{align*}

Mathematica [A]  time = 12.3413, size = 0, normalized size = 0. \[ \int \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.773, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b\cos \left ( fx+e \right ) } \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))^(1/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x)

[Out]

int((a+b*cos(f*x+e))^(1/2)*(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 )} \sqrt{b \cos \left (f x + e\right ) + a} \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))^(1/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((B*cos(f*x + e) + A)*sqrt(b*cos(f*x + e) + a)*(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 )} \sqrt{b \cos \left (f x + e\right ) + a} \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))^(1/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(f*x+e))**(1/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))**m,x)

[Out]

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

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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 )} \sqrt{b \cos \left (f x + e\right ) + a} \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))^(1/2)*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorithm="giac")

[Out]

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