Optimal. Leaf size=217 \[ -\frac {c^3 (b B (1-m)+a A (2-m)) \, _2F_1\left (\frac {1}{2},\frac {3-m}{2};\frac {5-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^{-3+m} \sin (e+f x)}{f (1-m) (3-m) \sqrt {\sin ^2(e+f x)}}-\frac {(A b+a B) c^2 \, _2F_1\left (\frac {1}{2},\frac {2-m}{2};\frac {4-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^{-2+m} \sin (e+f x)}{f (2-m) \sqrt {\sin ^2(e+f x)}}-\frac {a A c^2 (c \sec (e+f x))^{-2+m} \tan (e+f x)}{f (1-m)} \]
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Rubi [A]
time = 0.24, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3039, 4082,
3872, 3857, 2722} \begin {gather*} -\frac {c^3 \sin (e+f x) (a A (2-m)+b B (1-m)) (c \sec (e+f x))^{m-3} \, _2F_1\left (\frac {1}{2},\frac {3-m}{2};\frac {5-m}{2};\cos ^2(e+f x)\right )}{f (1-m) (3-m) \sqrt {\sin ^2(e+f x)}}-\frac {c^2 (a B+A b) \sin (e+f x) (c \sec (e+f x))^{m-2} \, _2F_1\left (\frac {1}{2},\frac {2-m}{2};\frac {4-m}{2};\cos ^2(e+f x)\right )}{f (2-m) \sqrt {\sin ^2(e+f x)}}-\frac {a A c^2 \tan (e+f x) (c \sec (e+f x))^{m-2}}{f (1-m)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3039
Rule 3857
Rule 3872
Rule 4082
Rubi steps
\begin {align*} \int (a+b \cos (e+f x)) (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx &=c^2 \int (c \sec (e+f x))^{-2+m} (b+a \sec (e+f x)) (B+A \sec (e+f x)) \, dx\\ &=-\frac {a A c^2 (c \sec (e+f x))^{-2+m} \tan (e+f x)}{f (1-m)}-\frac {c^2 \int (c \sec (e+f x))^{-2+m} (-b B (1-m)-a A (2-m)-(A b+a B) (1-m) \sec (e+f x)) \, dx}{1-m}\\ &=-\frac {a A c^2 (c \sec (e+f x))^{-2+m} \tan (e+f x)}{f (1-m)}+((A b+a B) c) \int (c \sec (e+f x))^{-1+m} \, dx+\frac {\left (c^2 (b B (1-m)+a A (2-m))\right ) \int (c \sec (e+f x))^{-2+m} \, dx}{1-m}\\ &=-\frac {a A c^2 (c \sec (e+f x))^{-2+m} \tan (e+f x)}{f (1-m)}+\left ((A b+a B) c \left (\frac {\cos (e+f x)}{c}\right )^m (c \sec (e+f x))^m\right ) \int \left (\frac {\cos (e+f x)}{c}\right )^{1-m} \, dx+\frac {\left (c^2 (b B (1-m)+a A (2-m)) \left (\frac {\cos (e+f x)}{c}\right )^m (c \sec (e+f x))^m\right ) \int \left (\frac {\cos (e+f x)}{c}\right )^{2-m} \, dx}{1-m}\\ &=-\frac {(A b+a B) \cos ^2(e+f x) \, _2F_1\left (\frac {1}{2},\frac {2-m}{2};\frac {4-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^m \sin (e+f x)}{f (2-m) \sqrt {\sin ^2(e+f x)}}-\frac {(b B (1-m)+a A (2-m)) \cos ^3(e+f x) \, _2F_1\left (\frac {1}{2},\frac {3-m}{2};\frac {5-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^m \sin (e+f x)}{f (1-m) (3-m) \sqrt {\sin ^2(e+f x)}}-\frac {a A c^2 (c \sec (e+f x))^{-2+m} \tan (e+f x)}{f (1-m)}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 163, normalized size = 0.75 \begin {gather*} \frac {\cot (e+f x) \left (b B (-1+m) m \cos ^2(e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2+m);\frac {m}{2};\sec ^2(e+f x)\right )+(-2+m) \left ((A b+a B) m \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1+m);\frac {1+m}{2};\sec ^2(e+f x)\right )+a A (-1+m) \, _2F_1\left (\frac {1}{2},\frac {m}{2};\frac {2+m}{2};\sec ^2(e+f x)\right )\right )\right ) (c \sec (e+f x))^m \sqrt {-\tan ^2(e+f x)}}{f (-2+m) (-1+m) m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.51, size = 0, normalized size = 0.00 \[\int \left (a +b \cos \left (f x +e \right )\right ) \left (A +B \cos \left (f x +e \right )\right ) \left (c \sec \left (f x +e \right )\right )^{m}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (c \sec {\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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (\frac {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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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