Optimal. Leaf size=187 \[ -\frac {B \sin (e+f x) (a \cos (e+f x))^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(e+f x)\right )}{a^2 f (m+2) \sqrt {\sin ^2(e+f x)}}-\frac {(A (m+2)+C (m+1)) \sin (e+f x) (a \cos (e+f x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{a f (m+1) (m+2) \sqrt {\sin ^2(e+f x)}}+\frac {C \sin (e+f x) (a \cos (e+f x))^{m+1}}{a f (m+2)} \]
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Rubi [A] time = 0.16, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3023, 2748, 2643} \[ -\frac {B \sin (e+f x) (a \cos (e+f x))^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(e+f x)\right )}{a^2 f (m+2) \sqrt {\sin ^2(e+f x)}}-\frac {(A (m+2)+C (m+1)) \sin (e+f x) (a \cos (e+f x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{a f (m+1) (m+2) \sqrt {\sin ^2(e+f x)}}+\frac {C \sin (e+f x) (a \cos (e+f x))^{m+1}}{a f (m+2)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 3023
Rubi steps
\begin {align*} \int (a \cos (e+f x))^m \left (A+B \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx &=\frac {C (a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac {\int (a \cos (e+f x))^m (a (C (1+m)+A (2+m))+a B (2+m) \cos (e+f x)) \, dx}{a (2+m)}\\ &=\frac {C (a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac {B \int (a \cos (e+f x))^{1+m} \, dx}{a}+\left (A+\frac {C (1+m)}{2+m}\right ) \int (a \cos (e+f x))^m \, dx\\ &=\frac {C (a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}-\frac {\left (A+\frac {C (1+m)}{2+m}\right ) (a \cos (e+f x))^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{a f (1+m) \sqrt {\sin ^2(e+f x)}}-\frac {B (a \cos (e+f x))^{2+m} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{a^2 f (2+m) \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 142, normalized size = 0.76 \[ -\frac {\sin (e+f x) \cos (e+f x) (a \cos (e+f x))^m \left ((A (m+2)+C (m+1)) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )+(m+1) \left (B \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(e+f x)\right )-C \sqrt {\sin ^2(e+f x)}\right )\right )}{f (m+1) (m+2) \sqrt {\sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (f x + e\right )^{2} + B \cos \left (f x + e\right ) + A\right )} \left (a \cos \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int {\left (C \cos \left (f x + e\right )^{2} + B \cos \left (f x + e\right ) + A\right )} \left (a \cos \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.52, size = 0, normalized size = 0.00 \[ \int \left (a \cos \left (f x +e \right )\right )^{m} \left (A +B \cos \left (f x +e \right )+C \left (\cos ^{2}\left (f x +e \right )\right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (f x + e\right )^{2} + B \cos \left (f x + e\right ) + A\right )} \left (a \cos \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,\left (C\,{\cos \left (e+f\,x\right )}^2+B\,\cos \left (e+f\,x\right )+A\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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