Optimal. Leaf size=66 \[ -\frac{a c^2 (-B+i A) \tan ^2(e+f x)}{2 f}+\frac{a A c^2 \tan (e+f x)}{f}-\frac{i a B c^2 \tan ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.0852537, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {3588, 43} \[ -\frac{a c^2 (-B+i A) \tan ^2(e+f x)}{2 f}+\frac{a A c^2 \tan (e+f x)}{f}-\frac{i a B c^2 \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 43
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))^2 \, dx &=\frac{(a c) \operatorname{Subst}(\int (A+B x) (c-i c x) \, dx,x,\tan (e+f x))}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (A c+(-i A+B) c x-i B c x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{a A c^2 \tan (e+f x)}{f}-\frac{a (i A-B) c^2 \tan ^2(e+f x)}{2 f}-\frac{i a B c^2 \tan ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 2.34587, size = 109, normalized size = 1.65 \[ \frac{a c^2 \sec (e) \sec ^3(e+f x) (3 (B-i A) \cos (2 e+f x)+3 (B-i A) \cos (f x)-3 A \sin (2 e+f x)+3 A \sin (2 e+3 f x)+6 A \sin (f x)-3 i B \sin (2 e+f x)+i B \sin (2 e+3 f x))}{12 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 53, normalized size = 0.8 \begin{align*}{\frac{a{c}^{2}}{f} \left ( -{\frac{i}{3}}B \left ( \tan \left ( fx+e \right ) \right ) ^{3}-{\frac{i}{2}}A \left ( \tan \left ( fx+e \right ) \right ) ^{2}+{\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 = 1.72229, size = 74, normalized size = 1.12 \begin{align*} \frac{-2 i \, B a c^{2} \tan \left (f x + e\right )^{3} - 3 \,{\left (i \, A - B\right )} a c^{2} \tan \left (f x + e\right )^{2} + 6 \, A a c^{2} \tan \left (f x + e\right )}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28905, size = 201, normalized size = 3.05 \begin{align*} \frac{{\left (6 i \, A + 6 \, B\right )} a c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (6 i \, A - 2 \, B\right )} a c^{2}}{3 \,{\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.57772, size = 114, normalized size = 1.73 \begin{align*} \frac{\frac{\left (2 i A a c^{2} + 2 B a c^{2}\right ) e^{- 4 i e} e^{2 i f x}}{f} + \frac{\left (6 i A a c^{2} - 2 B a c^{2}\right ) e^{- 6 i e}}{3 f}}{e^{6 i f x} + 3 e^{- 2 i e} e^{4 i f x} + 3 e^{- 4 i e} e^{2 i f x} + e^{- 6 i e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50044, size = 126, normalized size = 1.91 \begin{align*} \frac{6 i \, A a c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 6 \, B a c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i \, A a c^{2} - 2 \, B a c^{2}}{3 \,{\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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