Optimal. Leaf size=63 \[ -\frac{\cot ^3(e+f x) \sin ^2(e+f x)^{\frac{m+3}{2}} (b \csc (e+f x))^m \, _2F_1\left (\frac{3}{2},\frac{m+3}{2};\frac{5}{2};\cos ^2(e+f x)\right )}{3 f} \]
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Rubi [A] time = 0.0356907, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2617} \[ -\frac{\cot ^3(e+f x) \sin ^2(e+f x)^{\frac{m+3}{2}} (b \csc (e+f x))^m \, _2F_1\left (\frac{3}{2},\frac{m+3}{2};\frac{5}{2};\cos ^2(e+f x)\right )}{3 f} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int \cot ^2(e+f x) (b \csc (e+f x))^m \, dx &=-\frac{\cot ^3(e+f x) (b \csc (e+f x))^m \, _2F_1\left (\frac{3}{2},\frac{3+m}{2};\frac{5}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac{3+m}{2}}}{3 f}\\ \end{align*}
Mathematica [B] time = 1.05563, size = 186, normalized size = 2.95 \[ -\frac{\tan \left (\frac{1}{2} (e+f x)\right ) \sec ^2\left (\frac{1}{2} (e+f x)\right )^{-m} (b \csc (e+f x))^m \left (-4 (m+1) \, _2F_1\left (1-m,\frac{1}{2}-\frac{m}{2};\frac{3}{2}-\frac{m}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+(m+1) \, _2F_1\left (\frac{1}{2}-\frac{m}{2},-m;\frac{3}{2}-\frac{m}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+(m-1) \cot ^2\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (-\frac{m}{2}-\frac{1}{2},-m;\frac{1}{2}-\frac{m}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )}{2 f \left (m^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.201, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( fx+e \right ) \right ) ^{2} \left ( b\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 \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2}\,{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 \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2}, 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 (b \csc{\left (e + f x \right )}\right )^{m} \cot ^{2}{\left (e + f x \right )}\, 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 \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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