Optimal. Leaf size=183 \[ \frac {2^{m+\frac {1}{2}} \left (A \left (m^2+3 m+2\right )+B m (m+2)+C \left (m^2+m+1\right )\right ) \sin (e+f x) (\cos (e+f x)+1)^{-m-\frac {1}{2}} (a \cos (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x))\right )}{f (m+1) (m+2)}-\frac {(C-B (m+2)) \sin (e+f x) (a \cos (e+f x)+a)^m}{f (m+1) (m+2)}+\frac {C \sin (e+f x) (a \cos (e+f x)+a)^{m+1}}{a f (m+2)} \]
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Rubi [A] time = 0.25, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3023, 2751, 2652, 2651} \[ \frac {2^{m+\frac {1}{2}} \left (A \left (m^2+3 m+2\right )+B m (m+2)+C \left (m^2+m+1\right )\right ) \sin (e+f x) (\cos (e+f x)+1)^{-m-\frac {1}{2}} (a \cos (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x))\right )}{f (m+1) (m+2)}-\frac {(C-B (m+2)) \sin (e+f x) (a \cos (e+f x)+a)^m}{f (m+1) (m+2)}+\frac {C \sin (e+f x) (a \cos (e+f x)+a)^{m+1}}{a f (m+2)} \]
Antiderivative was successfully verified.
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Rule 2651
Rule 2652
Rule 2751
Rule 3023
Rubi steps
\begin {align*} \int (a+a \cos (e+f x))^m \left (A+B \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx &=\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-B (2+m)) \cos (e+f x)) \, dx}{a (2+m)}\\ &=-\frac {(C-B (2+m)) (a+a \cos (e+f x))^m \sin (e+f x)}{f (1+m) (2+m)}+\frac {C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac {\left (B m (2+m)+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-B (2+m)) (a+a \cos (e+f x))^m \sin (e+f x)}{f (1+m) (2+m)}+\frac {C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac {\left (\left (B m (2+m)+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-B (2+m)) (a+a \cos (e+f x))^m \sin (e+f x)}{f (1+m) (2+m)}+\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 (B m (2+m)+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 \, _2F_1\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*}
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Mathematica [C] time = 3.82, size = 557, normalized size = 3.04 \[ \frac {\cos ^{-2 m}\left (\frac {1}{2} (e+f x)\right ) (a (\cos (e+f x)+1))^m \left (\frac {i A 4^{1-m} \left (1+e^{i (e+f x)}\right ) \left (e^{-\frac {1}{2} i (e+f x)} \left (1+e^{i (e+f x)}\right )\right )^{2 m} \, _2F_1\left (1,m+1;1-m;-e^{i (e+f x)}\right )}{m}+\frac {2 i B e^{-i f x} (\cos (f x)+i \sin (f x)) \cos ^{2 m}\left (\frac {1}{2} (e+f x)\right ) \left (i \sin (e) e^{i f x}+\cos (e) e^{i f x}+1\right )^{-2 m} \left ((m-1) (\cos (e+f x)-i \sin (e+f x)) \, _2F_1\left (-m-1,-2 m;-m;-e^{i f x} (\cos (e)+i \sin (e))\right )+(m+1) (\cos (e+f x)+i \sin (e+f x)) \, _2F_1\left (1-m,-2 m;2-m;-e^{i f x} (\cos (e)+i \sin (e))\right )\right )}{m^2-1}+\frac {C e^{-2 i f x} \cos ^{2 m}\left (\frac {1}{2} (e+f x)\right ) \left (i \sin (e) e^{i f x}+\cos (e) e^{i f x}+1\right )^{-2 m} \left (i (m+2) e^{4 i f x} (\cos (2 e)+i \sin (2 e)) \, _2F_1\left (2-m,-2 m;3-m;-e^{i f x} (\cos (e)+i \sin (e))\right )+(m-2) (\sin (2 e)+i \cos (2 e)) \, _2F_1\left (-m-2,-2 m;-m-1;-e^{i f x} (\cos (e)+i \sin (e))\right )\right )}{m^2-4}+\frac {i C 2^{1-2 m} \left (1+e^{i (e+f x)}\right ) \left (e^{-\frac {1}{2} i (e+f x)} \left (1+e^{i (e+f x)}\right )\right )^{2 m} \, _2F_1\left (1,m+1;1-m;-e^{i (e+f x)}\right )}{m}\right )}{4 f} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (f x + e\right )^{2} + B \cos \left (f x + e\right ) + A\right )} {\left (a \cos \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (f x + e\right )^{2} + B \cos \left (f x + e\right ) + A\right )} {\left (a \cos \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.66, size = 0, normalized size = 0.00 \[ \int \left (a +a \cos \left (f x +e \right )\right )^{m} \left (A +B \cos \left (f x +e \right )+C \left (\cos ^{2}\left (f x +e \right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (f x + e\right )^{2} + B \cos \left (f x + e\right ) + A\right )} {\left (a \cos \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+a\,\cos \left (e+f\,x\right )\right )}^m\,\left (C\,{\cos \left (e+f\,x\right )}^2+B\,\cos \left (e+f\,x\right )+A\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\cos {\left (e + f x \right )} + 1\right )\right )^{m} \left (A + B \cos {\left (e + f x \right )} + C \cos ^{2}{\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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