Optimal. Leaf size=217 \[ -\frac{c^3 \sin (e+f x) (a A (2-m)+b B (1-m)) (c \sec (e+f x))^{m-3} \, _2F_1\left (\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(e+f x)\right )}{f (1-m) (3-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^2 (a B+A b) \sin (e+f x) (c \sec (e+f x))^{m-2} \, _2F_1\left (\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(e+f x)\right )}{f (2-m) \sqrt{\sin ^2(e+f x)}}-\frac{a A c^2 \tan (e+f x) (c \sec (e+f x))^{m-2}}{f (1-m)} \]
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Rubi [A] time = 0.359662, antiderivative size = 217, normalized size of antiderivative = 1., 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 = {2960, 3997, 3787, 3772, 2643} \[ -\frac{c^3 \sin (e+f x) (a A (2-m)+b B (1-m)) (c \sec (e+f x))^{m-3} \, _2F_1\left (\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(e+f x)\right )}{f (1-m) (3-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^2 (a B+A b) \sin (e+f x) (c \sec (e+f x))^{m-2} \, _2F_1\left (\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(e+f x)\right )}{f (2-m) \sqrt{\sin ^2(e+f x)}}-\frac{a A c^2 \tan (e+f x) (c \sec (e+f x))^{m-2}}{f (1-m)} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 3997
Rule 3787
Rule 3772
Rule 2643
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*}
Mathematica [A] time = 0.361708, size = 163, normalized size = 0.75 \[ \frac{\sqrt{-\tan ^2(e+f x)} \cot (e+f x) (c \sec (e+f x))^m \left ((m-2) \left (m (a B+A b) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sec ^2(e+f x)\right )+a A (m-1) \, _2F_1\left (\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right )\right )+b B (m-1) m \cos ^2(e+f x) \, _2F_1\left (\frac{1}{2},\frac{m-2}{2};\frac{m}{2};\sec ^2(e+f x)\right )\right )}{f (m-2) (m-1) m} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 2.181, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\cos \left ( fx+e \right ) \right ) \left ( A+B\cos \left ( fx+e \right ) \right ) \left ( c\sec \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (f x + e\right ) + A\right )}{\left (b \cos \left (f x + e\right ) + a\right )} \left (c \sec \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 \sec \left (f x + e\right )\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (f x + e\right ) + A\right )}{\left (b \cos \left (f x + e\right ) + a\right )} \left (c \sec \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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