Optimal. Leaf size=91 \[ \frac{a \cos (e+f x) (a \csc (e+f x))^{m-1} (b \csc (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (-m-n+1),\frac{1}{2} (-m-n+3),\sin ^2(e+f x)\right )}{f (-m-n+1) \sqrt{\cos ^2(e+f x)}} \]
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Rubi [A] time = 0.0448884, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {20, 3772, 2643} \[ \frac{a \cos (e+f x) (a \csc (e+f x))^{m-1} (b \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-m-n+1);\frac{1}{2} (-m-n+3);\sin ^2(e+f x)\right )}{f (-m-n+1) \sqrt{\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int (a \csc (e+f x))^m (b \csc (e+f x))^n \, dx &=\left ((a \csc (e+f x))^{-n} (b \csc (e+f x))^n\right ) \int (a \csc (e+f x))^{m+n} \, dx\\ &=\left ((a \csc (e+f x))^m (b \csc (e+f x))^n \left (\frac{\sin (e+f x)}{a}\right )^{m+n}\right ) \int \left (\frac{\sin (e+f x)}{a}\right )^{-m-n} \, dx\\ &=\frac{\cos (e+f x) (a \csc (e+f x))^m (b \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1-m-n);\frac{1}{2} (3-m-n);\sin ^2(e+f x)\right ) \sin (e+f x)}{f (1-m-n) \sqrt{\cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.140602, size = 77, normalized size = 0.85 \[ -\frac{\sin (e+f x) \cos (e+f x) (a \csc (e+f x))^m (b \csc (e+f x))^n \sin ^2(e+f x)^{\frac{1}{2} (m+n-1)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (m+n+1),\frac{3}{2},\cos ^2(e+f x)\right )}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.082, size = 0, normalized size = 0. \begin{align*} \int \left ( a\csc \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 \csc \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 \csc \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 \csc{\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 \csc \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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