Optimal. Leaf size=38 \[ \frac{a^2 c^2 \tan ^3(e+f x)}{3 f}+\frac{a^2 c^2 \tan (e+f x)}{f} \]
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Rubi [A] time = 0.0623472, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {3522, 3767} \[ \frac{a^2 c^2 \tan ^3(e+f x)}{3 f}+\frac{a^2 c^2 \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3767
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \sec ^4(e+f x) \, dx\\ &=-\frac{\left (a^2 c^2\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{f}\\ &=\frac{a^2 c^2 \tan (e+f x)}{f}+\frac{a^2 c^2 \tan ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.0495802, size = 29, normalized size = 0.76 \[ \frac{a^2 c^2 \left (\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)\right )}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}{c}^{2}}{f} \left ({\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{3}}+\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 = 1.6742, size = 47, normalized size = 1.24 \begin{align*} \frac{a^{2} c^{2} \tan \left (f x + e\right )^{3} + 3 \, a^{2} c^{2} \tan \left (f x + e\right )}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.13645, size = 181, normalized size = 4.76 \begin{align*} \frac{12 i \, a^{2} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i \, a^{2} 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 [C] time = 1.68066, size = 97, normalized size = 2.55 \begin{align*} \frac{\frac{4 i a^{2} c^{2} e^{- 4 i e} e^{2 i f x}}{f} + \frac{4 i a^{2} c^{2} 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 [B] time = 1.6101, size = 224, normalized size = 5.89 \begin{align*} -\frac{3 \, a^{2} c^{2} \tan \left (f x\right )^{3} \tan \left (e\right )^{2} + 3 \, a^{2} c^{2} \tan \left (f x\right )^{2} \tan \left (e\right )^{3} + a^{2} c^{2} \tan \left (f x\right )^{3} - 3 \, a^{2} c^{2} \tan \left (f x\right )^{2} \tan \left (e\right ) - 3 \, a^{2} c^{2} \tan \left (f x\right ) \tan \left (e\right )^{2} + a^{2} c^{2} \tan \left (e\right )^{3} + 3 \, a^{2} c^{2} \tan \left (f x\right ) + 3 \, a^{2} c^{2} \tan \left (e\right )}{3 \,{\left (f \tan \left (f x\right )^{3} \tan \left (e\right )^{3} - 3 \, f \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + 3 \, 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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