Optimal. Leaf size=91 \[ \frac{a b \cos ^2(e+f x)^{\frac{1-m}{2}} (a \cos (e+f x))^{m-1} (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]
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Rubi [A] time = 0.0987331, antiderivative size = 91, 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 = {2587, 2577} \[ \frac{a b \cos ^2(e+f x)^{\frac{1-m}{2}} (a \cos (e+f x))^{m-1} (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]
Antiderivative was successfully verified.
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Rule 2587
Rule 2577
Rubi steps
\begin{align*} \int (a \cos (e+f x))^m (b \csc (e+f x))^n \, dx &=\left (b^2 (b \csc (e+f x))^{-1+n} (b \sin (e+f x))^{-1+n}\right ) \int (a \cos (e+f x))^m (b \sin (e+f x))^{-n} \, dx\\ &=\frac{a b (a \cos (e+f x))^{-1+m} \cos ^2(e+f x)^{\frac{1-m}{2}} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)}\\ \end{align*}
Mathematica [C] time = 0.275486, size = 316, normalized size = 3.47 \[ -\frac{2 (n-3) \sin \left (\frac{1}{2} (e+f x)\right ) \cos ^3\left (\frac{1}{2} (e+f x)\right ) (a \cos (e+f x))^m (b \csc (e+f x))^n F_1\left (\frac{1}{2}-\frac{n}{2};-m,m-n+1;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (n-1) \left (2 \sin ^2\left (\frac{1}{2} (e+f x)\right ) \left (m F_1\left (\frac{3}{2}-\frac{n}{2};1-m,m-n+1;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+(m-n+1) F_1\left (\frac{3}{2}-\frac{n}{2};-m,m-n+2;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )+(n-3) \cos ^2\left (\frac{1}{2} (e+f x)\right ) F_1\left (\frac{1}{2}-\frac{n}{2};-m,m-n+1;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.109, size = 0, normalized size = 0. \begin{align*} \int \left ( a\cos \left ( fx+e \right ) \right ) ^{m} \left ( b\csc \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 \csc \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 \csc \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 \csc{\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 \csc \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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