Integrand size = 35, antiderivative size = 215 \[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\frac {4 \sqrt {2} C \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}}+\frac {2 \sqrt {2} (A-C) \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}} \]
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Time = 0.31 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3096, 2834, 144, 143, 2863} \[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\frac {2 \sqrt {2} (A-C) \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{f \sqrt {\cos (e+f x)+1}}+\frac {4 \sqrt {2} C \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{f \sqrt {\cos (e+f x)+1}} \]
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Rule 143
Rule 144
Rule 2834
Rule 2863
Rule 3096
Rubi steps \begin{align*} \text {integral}& = (A-C) \int (1+\cos (e+f x)) (a+b \cos (e+f x))^m \, dx+C \int (1+\cos (e+f x))^2 (a+b \cos (e+f x))^m \, dx \\ & = -\frac {(C \sin (e+f x)) \text {Subst}\left (\int \frac {(1+x)^{3/2} (a+b x)^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}+\frac {((-A+C) \sin (e+f x)) \text {Subst}\left (\int \frac {\sqrt {1+x} (a+b x)^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}} \\ & = -\frac {\left (C (a+b \cos (e+f x))^m \left (-\frac {a+b \cos (e+f x)}{-a-b}\right )^{-m} \sin (e+f x)\right ) \text {Subst}\left (\int \frac {(1+x)^{3/2} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}+\frac {\left ((-A+C) (a+b \cos (e+f x))^m \left (-\frac {a+b \cos (e+f x)}{-a-b}\right )^{-m} \sin (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {1+x} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}} \\ & = \frac {4 \sqrt {2} C \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}}+\frac {2 \sqrt {2} (A-C) \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}} \\ \end{align*}
\[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx \]
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\[\int \left (a +b \cos \left (f x +e \right )\right )^{m} \left (A +\left (A +C \right ) \cos \left (f x +e \right )+C \left (\cos ^{2}\left (f x +e \right )\right )\right )d x\]
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\[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\int { {\left (C \cos \left (f x + e\right )^{2} + {\left (A + C\right )} \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\text {Timed out} \]
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\[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\int { {\left (C \cos \left (f x + e\right )^{2} + {\left (A + C\right )} \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\int { {\left (C \cos \left (f x + e\right )^{2} + {\left (A + C\right )} \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\int {\left (a+b\,\cos \left (e+f\,x\right )\right )}^m\,\left (C\,{\cos \left (e+f\,x\right )}^2+\left (A+C\right )\,\cos \left (e+f\,x\right )+A\right ) \,d x \]
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