Optimal. Leaf size=19 \[ -\frac{\tan (e+f x) \sec ^3(e+f x)}{f} \]
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Rubi [A] time = 0.0232353, 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} \[ -\frac{\tan (e+f x) \sec ^3(e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 4043
Rubi steps
\begin{align*} \int \sec ^3(e+f x) \left (3-4 \sec ^2(e+f x)\right ) \, dx &=-\frac{\sec ^3(e+f x) \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0313312, size = 19, normalized size = 1. \[ -\frac{\tan (e+f x) \sec ^3(e+f x)}{f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 47, normalized size = 2.5 \begin{align*}{\frac{1}{f} \left ({\frac{3\,\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{2}}+4\, \left ( -1/4\, \left ( \sec \left ( fx+e \right ) \right ) ^{3}-3/8\,\sec \left ( fx+e \right ) \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 = 0.921748, size = 45, normalized size = 2.37 \begin{align*} -\frac{\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.455161, size = 46, normalized size = 2.42 \begin{align*} -\frac{\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{4}} \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 - 3 \sec ^{3}{\left (e + f x \right )}\, dx - \int 4 \sec ^{5}{\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 = 1.19714, size = 34, normalized size = 1.79 \begin{align*} -\frac{\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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