Integrand size = 25, antiderivative size = 170 \[ \int (a+a \cos (e+f x))^m \left (A+C \cos ^2(e+f x)\right ) \, dx=-\frac {C (a+a \cos (e+f x))^m \sin (e+f x)}{f \left (2+3 m+m^2\right )}+\frac {C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac {2^{\frac {1}{2}+m} \left (C \left (1+m+m^2\right )+A \left (2+3 m+m^2\right )\right ) (1+\cos (e+f x))^{-\frac {1}{2}-m} (a+a \cos (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x))\right ) \sin (e+f x)}{f (1+m) (2+m)} \]
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Time = 0.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3103, 2830, 2731, 2730} \[ \int (a+a \cos (e+f x))^m \left (A+C \cos ^2(e+f x)\right ) \, dx=\frac {2^{m+\frac {1}{2}} \left (A \left (m^2+3 m+2\right )+C \left (m^2+m+1\right )\right ) \sin (e+f x) (\cos (e+f x)+1)^{-m-\frac {1}{2}} (a \cos (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x))\right )}{f (m+1) (m+2)}-\frac {C \sin (e+f x) (a \cos (e+f x)+a)^m}{f \left (m^2+3 m+2\right )}+\frac {C \sin (e+f x) (a \cos (e+f x)+a)^{m+1}}{a f (m+2)} \]
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Rule 2730
Rule 2731
Rule 2830
Rule 3103
Rubi steps \begin{align*} \text {integral}& = \frac {C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac {\int (a+a \cos (e+f x))^m (a (C (1+m)+A (2+m))-a C \cos (e+f x)) \, dx}{a (2+m)} \\ & = -\frac {C (a+a \cos (e+f x))^m \sin (e+f x)}{f \left (2+3 m+m^2\right )}+\frac {C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac {\left (C \left (1+m+m^2\right )+A \left (2+3 m+m^2\right )\right ) \int (a+a \cos (e+f x))^m \, dx}{(1+m) (2+m)} \\ & = -\frac {C (a+a \cos (e+f x))^m \sin (e+f x)}{f \left (2+3 m+m^2\right )}+\frac {C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac {\left (\left (C \left (1+m+m^2\right )+A \left (2+3 m+m^2\right )\right ) (1+\cos (e+f x))^{-m} (a+a \cos (e+f x))^m\right ) \int (1+\cos (e+f x))^m \, dx}{(1+m) (2+m)} \\ & = -\frac {C (a+a \cos (e+f x))^m \sin (e+f x)}{f \left (2+3 m+m^2\right )}+\frac {C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac {2^{\frac {1}{2}+m} \left (C \left (1+m+m^2\right )+A \left (2+3 m+m^2\right )\right ) (1+\cos (e+f x))^{-\frac {1}{2}-m} (a+a \cos (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x))\right ) \sin (e+f x)}{f (1+m) (2+m)} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.29 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.42 \[ \int (a+a \cos (e+f x))^m \left (A+C \cos ^2(e+f x)\right ) \, dx=\frac {i 4^{-1-m} e^{-i (2+m) (e+f x)} \left (1+e^{i (e+f x)}\right ) \left (e^{-\frac {1}{2} i (e+f x)} \left (1+e^{i (e+f x)}\right )\right )^{2 m} \cos ^{-2 m}\left (\frac {1}{2} (e+f x)\right ) (a (1+\cos (e+f x)))^m \left (C e^{i m (e+f x)} (-2+m) m \operatorname {Hypergeometric2F1}\left (1,-1+m,-1-m,-e^{i (e+f x)}\right )+e^{i (2+m) (e+f x)} (2+m) \left (2 (2 A+C) (-2+m) \operatorname {Hypergeometric2F1}\left (1,1+m,1-m,-e^{i (e+f x)}\right )+C e^{2 i (e+f x)} m \operatorname {Hypergeometric2F1}\left (1,3+m,3-m,-e^{i (e+f x)}\right )\right )\right )}{f (-2+m) m (2+m)} \]
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\[\int \left (a +\cos \left (f x +e \right ) a \right )^{m} \left (A +C \left (\cos ^{2}\left (f x +e \right )\right )\right )d x\]
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\[ \int (a+a \cos (e+f x))^m \left (A+C \cos ^2(e+f x)\right ) \, dx=\int { {\left (C \cos \left (f x + e\right )^{2} + A\right )} {\left (a \cos \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+a \cos (e+f x))^m \left (A+C \cos ^2(e+f x)\right ) \, dx=\int \left (a \left (\cos {\left (e + f x \right )} + 1\right )\right )^{m} \left (A + C \cos ^{2}{\left (e + f x \right )}\right )\, dx \]
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\[ \int (a+a \cos (e+f x))^m \left (A+C \cos ^2(e+f x)\right ) \, dx=\int { {\left (C \cos \left (f x + e\right )^{2} + A\right )} {\left (a \cos \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+a \cos (e+f x))^m \left (A+C \cos ^2(e+f x)\right ) \, dx=\int { {\left (C \cos \left (f x + e\right )^{2} + A\right )} {\left (a \cos \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (a+a \cos (e+f x))^m \left (A+C \cos ^2(e+f x)\right ) \, dx=\int \left (C\,{\cos \left (e+f\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (e+f\,x\right )\right )}^m \,d x \]
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