Optimal. Leaf size=91 \[ \frac {a \cos (e+f x) (a \csc (e+f x))^{m-1} (b \csc (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-n+1);\frac {1}{2} (-m-n+3);\sin ^2(e+f x)\right )}{f (-m-n+1) \sqrt {\cos ^2(e+f x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 91, normalized size of antiderivative = 1.00, 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 2643
Rule 3772
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*}
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Mathematica [A] time = 0.14, size = 77, normalized size = 0.85 \[ -\frac {\sin (e+f x) \cos (e+f x) (a \csc (e+f x))^m (b \csc (e+f x))^n \sin ^2(e+f x)^{\frac {1}{2} (m+n-1)} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+1);\frac {3}{2};\cos ^2(e+f x)\right )}{f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \csc \left (f x + e\right )\right )^{m} \left (b \csc \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int \left (a \csc \left (f x + e\right )\right )^{m} \left (b \csc \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.90, size = 0, normalized size = 0.00 \[ \int \left (a \csc \left (f x +e \right )\right )^{m} \left (b \csc \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int \left (a \csc \left (f x + e\right )\right )^{m} \left (b \csc \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {a}{\sin \left (e+f\,x\right )}\right )}^m\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n \,d x \]
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 \[ \int \left (a \csc {\left (e + f x \right )}\right )^{m} \left (b \csc {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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