Optimal. Leaf size=196 \[ -\frac {(a B+A b) \sin (e+f x) (c \cos (e+f x))^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(e+f x)\right )}{c^2 f (m+2) \sqrt {\sin ^2(e+f x)}}-\frac {\sin (e+f x) (a A (m+2)+b B (m+1)) (c \cos (e+f x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{c f (m+1) (m+2) \sqrt {\sin ^2(e+f x)}}+\frac {b B \sin (e+f x) (c \cos (e+f x))^{m+1}}{c f (m+2)} \]
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Rubi [A] time = 0.25, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2968, 3023, 2748, 2643} \[ -\frac {(a B+A b) \sin (e+f x) (c \cos (e+f x))^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(e+f x)\right )}{c^2 f (m+2) \sqrt {\sin ^2(e+f x)}}-\frac {\sin (e+f x) (a A (m+2)+b B (m+1)) (c \cos (e+f x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{c f (m+1) (m+2) \sqrt {\sin ^2(e+f x)}}+\frac {b B \sin (e+f x) (c \cos (e+f x))^{m+1}}{c f (m+2)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int (c \cos (e+f x))^m (a+b \cos (e+f x)) (A+B \cos (e+f x)) \, dx &=\int (c \cos (e+f x))^m \left (a A+(A b+a B) \cos (e+f x)+b B \cos ^2(e+f x)\right ) \, dx\\ &=\frac {b B (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (2+m)}+\frac {\int (c \cos (e+f x))^m (c (b B (1+m)+a A (2+m))+(A b+a B) c (2+m) \cos (e+f x)) \, dx}{c (2+m)}\\ &=\frac {b B (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (2+m)}+\frac {(A b+a B) \int (c \cos (e+f x))^{1+m} \, dx}{c}+\left (a A+\frac {b B (1+m)}{2+m}\right ) \int (c \cos (e+f x))^m \, dx\\ &=\frac {b B (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (2+m)}-\frac {\left (a A+\frac {b B (1+m)}{2+m}\right ) (c \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)}{c f (1+m) \sqrt {\sin ^2(e+f x)}}-\frac {(A b+a B) (c \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)}{c^2 f (2+m) \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 151, normalized size = 0.77 \[ -\frac {\sin (e+f x) \cos (e+f x) (c \cos (e+f x))^m \left ((a A (m+2)+b B (m+1)) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )+(m+1) \left ((a B+A b) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(e+f x)\right )-b B \sqrt {\sin ^2(e+f x)}\right )\right )}{f (m+1) (m+2) \sqrt {\sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b \cos \left (f x + e\right )^{2} + A a + {\left (B a + A b\right )} \cos \left (f x + e\right )\right )} \left (c \cos \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )} \left (c \cos \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.99, size = 0, normalized size = 0.00 \[ \int \left (c \cos \left (f x +e \right )\right )^{m} \left (a +b \cos \left (f x +e \right )\right ) \left (A +B \cos \left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )} \left (c \cos \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (c\,\cos \left (e+f\,x\right )\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,\left (a+b\,\cos \left (e+f\,x\right )\right ) \,d x \]
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 \[ \int \left (c \cos {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right ) \left (a + b \cos {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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