Optimal. Leaf size=63 \[ \frac{\tan ^3(e+f x) \sin ^2(e+f x)^{\frac{m-3}{2}} (b \csc (e+f x))^m \, _2F_1\left (-\frac{3}{2},\frac{m-3}{2};-\frac{1}{2};\cos ^2(e+f x)\right )}{3 f} \]
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Rubi [A] time = 0.0361783, 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} \[ \frac{\tan ^3(e+f x) \sin ^2(e+f x)^{\frac{m-3}{2}} (b \csc (e+f x))^m \, _2F_1\left (-\frac{3}{2},\frac{m-3}{2};-\frac{1}{2};\cos ^2(e+f x)\right )}{3 f} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int (b \csc (e+f x))^m \tan ^4(e+f x) \, dx &=\frac{(b \csc (e+f x))^m \, _2F_1\left (-\frac{3}{2},\frac{1}{2} (-3+m);-\frac{1}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac{1}{2} (-3+m)} \tan ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.699332, size = 79, normalized size = 1.25 \[ \frac{\tan ^5(e+f x) \sec ^2(e+f x)^{-m/2} (b \csc (e+f x))^m \, _2F_1\left (1-\frac{m}{2},\frac{5}{2}-\frac{m}{2};\frac{7}{2}-\frac{m}{2};-\tan ^2(e+f x)\right )}{f (5-m)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.221, size = 0, normalized size = 0. \begin{align*} \int \left ( b\csc \left ( fx+e \right ) \right ) ^{m} \left ( \tan \left ( fx+e \right ) \right ) ^{4}\, 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} \tan \left (f x + e\right )^{4}\,{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} \tan \left (f x + e\right )^{4}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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} \tan \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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