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