Optimal. Leaf size=12 \[ \frac{a c \tan (e+f x)}{f} \]
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Rubi [A] time = 0.0239701, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3522, 3767, 8} \[ \frac{a c \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x)) \, dx &=(a c) \int \sec ^2(e+f x) \, dx\\ &=-\frac{(a c) \operatorname{Subst}(\int 1 \, dx,x,-\tan (e+f x))}{f}\\ &=\frac{a c \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0091411, size = 12, normalized size = 1. \[ \frac{a c \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 13, normalized size = 1.1 \begin{align*}{\frac{ac\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 = 1.57676, size = 16, normalized size = 1.33 \begin{align*} \frac{a c \tan \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.06589, size = 50, normalized size = 4.17 \begin{align*} \frac{2 i \, a c}{f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.368875, size = 31, normalized size = 2.58 \begin{align*} \frac{2 i a c e^{- 2 i e}}{f \left (e^{2 i f x} + e^{- 2 i e}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36223, size = 18, normalized size = 1.5 \begin{align*} \frac{a c \tan \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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