Optimal. Leaf size=59 \[ \frac{a^3 c^3 \tan ^5(e+f x)}{5 f}+\frac{2 a^3 c^3 \tan ^3(e+f x)}{3 f}+\frac{a^3 c^3 \tan (e+f x)}{f} \]
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Rubi [A] time = 0.0681824, antiderivative size = 59, 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} \[ \frac{a^3 c^3 \tan ^5(e+f x)}{5 f}+\frac{2 a^3 c^3 \tan ^3(e+f x)}{3 f}+\frac{a^3 c^3 \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3767
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^3 \, dx &=\left (a^3 c^3\right ) \int \sec ^6(e+f x) \, dx\\ &=-\frac{\left (a^3 c^3\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{f}\\ &=\frac{a^3 c^3 \tan (e+f x)}{f}+\frac{2 a^3 c^3 \tan ^3(e+f x)}{3 f}+\frac{a^3 c^3 \tan ^5(e+f x)}{5 f}\\ \end{align*}
Mathematica [A] time = 0.118951, size = 41, normalized size = 0.69 \[ \frac{a^3 c^3 \left (\frac{1}{5} \tan ^5(e+f x)+\frac{2}{3} \tan ^3(e+f x)+\tan (e+f x)\right )}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 38, normalized size = 0.6 \begin{align*}{\frac{{a}^{3}{c}^{3}}{f} \left ({\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{5}}{5}}+{\frac{2\, \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 = 1.7079, size = 70, normalized size = 1.19 \begin{align*} \frac{3 \, a^{3} c^{3} \tan \left (f x + e\right )^{5} + 10 \, a^{3} c^{3} \tan \left (f x + e\right )^{3} + 15 \, a^{3} c^{3} \tan \left (f x + e\right )}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.44973, size = 308, normalized size = 5.22 \begin{align*} \frac{160 i \, a^{3} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 80 i \, a^{3} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, a^{3} c^{3}}{15 \,{\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, 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 = 6.60708, size = 162, normalized size = 2.75 \begin{align*} \frac{\frac{32 i a^{3} c^{3} e^{- 6 i e} e^{4 i f x}}{3 f} + \frac{16 i a^{3} c^{3} e^{- 8 i e} e^{2 i f x}}{3 f} + \frac{16 i a^{3} c^{3} e^{- 10 i e}}{15 f}}{e^{10 i f x} + 5 e^{- 2 i e} e^{8 i f x} + 10 e^{- 4 i e} e^{6 i f x} + 10 e^{- 6 i e} e^{4 i f x} + 5 e^{- 8 i e} e^{2 i f x} + e^{- 10 i e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.85655, size = 501, normalized size = 8.49 \begin{align*} -\frac{15 \, a^{3} c^{3} \tan \left (f x\right )^{5} \tan \left (e\right )^{4} + 15 \, a^{3} c^{3} \tan \left (f x\right )^{4} \tan \left (e\right )^{5} + 10 \, a^{3} c^{3} \tan \left (f x\right )^{5} \tan \left (e\right )^{2} - 30 \, a^{3} c^{3} \tan \left (f x\right )^{4} \tan \left (e\right )^{3} - 30 \, a^{3} c^{3} \tan \left (f x\right )^{3} \tan \left (e\right )^{4} + 10 \, a^{3} c^{3} \tan \left (f x\right )^{2} \tan \left (e\right )^{5} + 3 \, a^{3} c^{3} \tan \left (f x\right )^{5} - 5 \, a^{3} c^{3} \tan \left (f x\right )^{4} \tan \left (e\right ) + 60 \, a^{3} c^{3} \tan \left (f x\right )^{3} \tan \left (e\right )^{2} + 60 \, a^{3} c^{3} \tan \left (f x\right )^{2} \tan \left (e\right )^{3} - 5 \, a^{3} c^{3} \tan \left (f x\right ) \tan \left (e\right )^{4} + 3 \, a^{3} c^{3} \tan \left (e\right )^{5} + 10 \, a^{3} c^{3} \tan \left (f x\right )^{3} - 30 \, a^{3} c^{3} \tan \left (f x\right )^{2} \tan \left (e\right ) - 30 \, a^{3} c^{3} \tan \left (f x\right ) \tan \left (e\right )^{2} + 10 \, a^{3} c^{3} \tan \left (e\right )^{3} + 15 \, a^{3} c^{3} \tan \left (f x\right ) + 15 \, a^{3} c^{3} \tan \left (e\right )}{15 \,{\left (f \tan \left (f x\right )^{5} \tan \left (e\right )^{5} - 5 \, f \tan \left (f x\right )^{4} \tan \left (e\right )^{4} + 10 \, f \tan \left (f x\right )^{3} \tan \left (e\right )^{3} - 10 \, f \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + 5 \, 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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