Optimal. Leaf size=32 \[ \frac{a A c \tan (e+f x)}{f}+\frac{a B c \tan ^2(e+f x)}{2 f} \]
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Rubi [A] time = 0.0398157, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027, Rules used = {3588} \[ \frac{a A c \tan (e+f x)}{f}+\frac{a B c \tan ^2(e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Rule 3588
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx &=\frac{(a c) \operatorname{Subst}(\int (A+B x) \, dx,x,\tan (e+f x))}{f}\\ &=\frac{a A c \tan (e+f x)}{f}+\frac{a B c \tan ^2(e+f x)}{2 f}\\ \end{align*}
Mathematica [A] time = 0.0414586, size = 32, normalized size = 1. \[ \frac{a A c \tan (e+f x)}{f}+\frac{a B c \sec ^2(e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 27, normalized size = 0.8 \begin{align*}{\frac{ac}{f} \left ({\frac{B \left ( \tan \left ( fx+e \right ) \right ) ^{2}}{2}}+A\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 = 1.74679, size = 39, normalized size = 1.22 \begin{align*} \frac{B a c \tan \left (f x + e\right )^{2} + 2 \, A a c \tan \left (f x + e\right )}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.31214, size = 144, normalized size = 4.5 \begin{align*} \frac{{\left (2 i \, A + 2 \, B\right )} a c e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, A a c}{f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, 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 = 2.80183, size = 82, normalized size = 2.56 \begin{align*} \frac{\frac{2 i A a c e^{- 4 i e}}{f} + \frac{\left (2 i A a c + 2 B a c\right ) e^{- 2 i e} e^{2 i f x}}{f}}{e^{4 i f x} + 2 e^{- 2 i e} e^{2 i f x} + e^{- 4 i e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.42013, size = 153, normalized size = 4.78 \begin{align*} \frac{B a c \tan \left (f x\right )^{2} \tan \left (e\right )^{2} - 2 \, A a c \tan \left (f x\right )^{2} \tan \left (e\right ) - 2 \, A a c \tan \left (f x\right ) \tan \left (e\right )^{2} + B a c \tan \left (f x\right )^{2} + B a c \tan \left (e\right )^{2} + 2 \, A a c \tan \left (f x\right ) + 2 \, A a c \tan \left (e\right ) + B a c}{2 \,{\left (f \tan \left (f x\right )^{2} \tan \left (e\right )^{2} - 2 \, f \tan \left (f x\right ) \tan \left (e\right ) + f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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