Optimal. Leaf size=88 \[ -\frac{\sin (e+f x) (a \cos (e+f x))^{m+1} (b \sec (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m-n+1);\frac{1}{2} (m-n+3);\cos ^2(e+f x)\right )}{a f (m-n+1) \sqrt{\sin ^2(e+f x)}} \]
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Rubi [A] time = 0.070436, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2588, 2643} \[ -\frac{\sin (e+f x) (a \cos (e+f x))^{m+1} (b \sec (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m-n+1);\frac{1}{2} (m-n+3);\cos ^2(e+f x)\right )}{a f (m-n+1) \sqrt{\sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2588
Rule 2643
Rubi steps
\begin{align*} \int (a \cos (e+f x))^m (b \sec (e+f x))^n \, dx &=\left ((a \cos (e+f x))^n (b \sec (e+f x))^n\right ) \int (a \cos (e+f x))^{m-n} \, dx\\ &=-\frac{(a \cos (e+f x))^{1+m} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1+m-n);\frac{1}{2} (3+m-n);\cos ^2(e+f x)\right ) (b \sec (e+f x))^n \sin (e+f x)}{a f (1+m-n) \sqrt{\sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 10.4935, size = 89, normalized size = 1.01 \[ -\frac{\sin (e+f x) \cos (e+f x) (a \cos (e+f x))^m (b \sec (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m-n+1);\frac{1}{2} (m-n+3);\cos ^2(e+f x)\right )}{f (m-n+1) \sqrt{\sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.065, size = 0, normalized size = 0. \begin{align*} \int \left ( a\cos \left ( fx+e \right ) \right ) ^{m} \left ( b\sec \left ( fx+e \right ) \right ) ^{n}\, 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 (a \cos \left (f x + e\right )\right )^{m} \left (b \sec \left (f x + e\right )\right )^{n}\,{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 (a \cos \left (f x + e\right )\right )^{m} \left (b \sec \left (f x + e\right )\right )^{n}, 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 (a \cos{\left (e + f x \right )}\right )^{m} \left (b \sec{\left (e + f x \right )}\right )^{n}\, 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 (a \cos \left (f x + e\right )\right )^{m} \left (b \sec \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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