Optimal. Leaf size=19 \[ -\frac{\tan (e+f x) \sec ^2(e+f x)}{f} \]
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Rubi [A] time = 0.0233235, antiderivative size = 19, 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 = {4043} \[ -\frac{\tan (e+f x) \sec ^2(e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 4043
Rubi steps
\begin{align*} \int \sec ^2(e+f x) \left (2-3 \sec ^2(e+f x)\right ) \, dx &=-\frac{\sec ^2(e+f x) \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.056089, size = 19, normalized size = 1. \[ -\frac{\tan (e+f x) \sec ^2(e+f x)}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 34, normalized size = 1.8 \begin{align*}{\frac{1}{f} \left ( 2\,\tan \left ( fx+e \right ) +3\, \left ( -2/3-1/3\, \left ( \sec \left ( fx+e \right ) \right ) ^{2} \right ) \tan \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.925501, size = 27, normalized size = 1.42 \begin{align*} -\frac{\tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.4483, size = 46, normalized size = 2.42 \begin{align*} -\frac{\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{3}} \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 - 2 \sec ^{2}{\left (e + f x \right )}\, dx - \int 3 \sec ^{4}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14055, size = 30, normalized size = 1.58 \begin{align*} -\frac{\tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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