Optimal. Leaf size=11 \[ -\frac{\tan (e+f x)}{f} \]
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Rubi [A] time = 0.0126015, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3767, 8} \[ -\frac{\tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 8
Rubi steps
\begin{align*} \int -\sec ^2(e+f x) \, dx &=\frac{\operatorname{Subst}(\int 1 \, dx,x,-\tan (e+f x))}{f}\\ &=-\frac{\tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0040374, size = 11, normalized size = 1. \[ -\frac{\tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 12, normalized size = 1.1 \begin{align*} -{\frac{\tan \left ( fx+e \right ) }{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.917863, size = 15, normalized size = 1.36 \begin{align*} -\frac{\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 = 0.445583, size = 43, normalized size = 3.91 \begin{align*} -\frac{\sin \left (f x + e\right )}{f \cos \left (f x + e\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 \sec ^{2}{\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 = 1.19418, size = 16, normalized size = 1.45 \begin{align*} -\frac{\tan \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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