Integrand size = 35, antiderivative size = 35 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx=\text {Int}\left (\frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}},x\right ) \]
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Not integrable
Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx=\int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx \\ \end{align*}
Not integrable
Time = 34.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx=\int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx \]
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Not integrable
Time = 0.92 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94
\[\int \frac {\left (c \cos \left (f x +e \right )\right )^{m} \left (A +\cos \left (f x +e \right ) B \right )}{\sqrt {a +b \cos \left (f x +e \right )}}d x\]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \cos \left (f x + e\right )\right )^{m}}{\sqrt {b \cos \left (f x + e\right ) + a}} \,d x } \]
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Not integrable
Time = 3.76 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx=\int \frac {\left (c \cos {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right )}{\sqrt {a + b \cos {\left (e + f x \right )}}}\, dx \]
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Not integrable
Time = 2.42 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \cos \left (f x + e\right )\right )^{m}}{\sqrt {b \cos \left (f x + e\right ) + a}} \,d x } \]
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Not integrable
Time = 0.74 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx=\int { \frac {{\left (B \cos \left (f x + e\right ) + A\right )} \left (c \cos \left (f x + e\right )\right )^{m}}{\sqrt {b \cos \left (f x + e\right ) + a}} \,d x } \]
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Not integrable
Time = 4.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt {a+b \cos (e+f x)}} \, dx=\int \frac {{\left (c\,\cos \left (e+f\,x\right )\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )}{\sqrt {a+b\,\cos \left (e+f\,x\right )}} \,d x \]
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