Optimal. Leaf size=83 \[ -\frac{(a \csc (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac{1}{2} (m+n+1)} \text{Hypergeometric2F1}\left (\frac{n+1}{2},\frac{1}{2} (m+n+1),\frac{n+3}{2},\cos ^2(e+f x)\right )}{b f (n+1)} \]
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Rubi [A] time = 0.0457312, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2617} \[ -\frac{(a \csc (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac{1}{2} (m+n+1)} \, _2F_1\left (\frac{n+1}{2},\frac{1}{2} (m+n+1);\frac{n+3}{2};\cos ^2(e+f x)\right )}{b f (n+1)} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx &=-\frac{(b \cot (e+f x))^{1+n} (a \csc (e+f x))^m \, _2F_1\left (\frac{1+n}{2},\frac{1}{2} (1+m+n);\frac{3+n}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac{1}{2} (1+m+n)}}{b f (1+n)}\\ \end{align*}
Mathematica [C] time = 1.79764, size = 306, normalized size = 3.69 \[ -\frac{a (m+n-3) (a \csc (e+f x))^{m-1} (b \cot (e+f x))^n F_1\left (\frac{1}{2} (-m-n+1);-n,1-m;\frac{1}{2} (-m-n+3);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (m+n-1) \left (2 \tan ^2\left (\frac{1}{2} (e+f x)\right ) \left (n F_1\left (\frac{1}{2} (-m-n+3);1-n,1-m;\frac{1}{2} (-m-n+5);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-(m-1) F_1\left (\frac{1}{2} (-m-n+3);-n,2-m;\frac{1}{2} (-m-n+5);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )+(m+n-3) F_1\left (\frac{1}{2} (-m-n+1);-n,1-m;\frac{1}{2} (-m-n+3);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.115, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cot \left ( fx+e \right ) \right ) ^{n} \left ( a\csc \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 \cot \left (f x + e\right )\right )^{n} \left (a \csc \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 \cot \left (f x + e\right )\right )^{n} \left (a \csc \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 (a \csc{\left (e + f x \right )}\right )^{m} \left (b \cot{\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 (b \cot \left (f x + e\right )\right )^{n} \left (a \csc \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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