Optimal. Leaf size=18 \[ \frac{\cot (e+f x) \csc ^2(e+f x)}{f} \]
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Rubi [A] time = 0.0215144, antiderivative size = 18, 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 = {3011} \[ \frac{\cot (e+f x) \csc ^2(e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3011
Rubi steps
\begin{align*} \int \csc ^4(e+f x) \left (-3+2 \sin ^2(e+f x)\right ) \, dx &=\frac{\cot (e+f x) \csc ^2(e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0445803, size = 18, normalized size = 1. \[ \frac{\cot (e+f x) \csc ^2(e+f x)}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 34, normalized size = 1.9 \begin{align*}{\frac{1}{f} \left ( -3\, \left ( -2/3-1/3\, \left ( \csc \left ( fx+e \right ) \right ) ^{2} \right ) \cot \left ( fx+e \right ) -2\,\cot \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966618, size = 30, normalized size = 1.67 \begin{align*} \frac{\tan \left (f x + e\right )^{2} + 1}{f \tan \left (f x + e\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60481, size = 72, normalized size = 4. \begin{align*} -\frac{\cos \left (f x + e\right )}{{\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\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 [A] time = 1.12219, size = 32, normalized size = 1.78 \begin{align*} \frac{\tan \left (f x + e\right )^{2} + 1}{f \tan \left (f x + e\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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