Optimal. Leaf size=77 \[ \frac{a^4 c^4 \tan ^7(e+f x)}{7 f}+\frac{3 a^4 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^4 c^4 \tan ^3(e+f x)}{f}+\frac{a^4 c^4 \tan (e+f x)}{f} \]
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Rubi [A] time = 0.0707983, antiderivative size = 77, 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} \[ \frac{a^4 c^4 \tan ^7(e+f x)}{7 f}+\frac{3 a^4 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^4 c^4 \tan ^3(e+f x)}{f}+\frac{a^4 c^4 \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3767
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx &=\left (a^4 c^4\right ) \int \sec ^8(e+f x) \, dx\\ &=-\frac{\left (a^4 c^4\right ) \operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (e+f x)\right )}{f}\\ &=\frac{a^4 c^4 \tan (e+f x)}{f}+\frac{a^4 c^4 \tan ^3(e+f x)}{f}+\frac{3 a^4 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^4 c^4 \tan ^7(e+f x)}{7 f}\\ \end{align*}
Mathematica [A] time = 0.219376, size = 49, normalized size = 0.64 \[ \frac{a^4 c^4 \left (\frac{1}{7} \tan ^7(e+f x)+\frac{3}{5} \tan ^5(e+f x)+\tan ^3(e+f x)+\tan (e+f x)\right )}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 46, normalized size = 0.6 \begin{align*}{\frac{{a}^{4}{c}^{4}}{f} \left ({\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{7}}{7}}+{\frac{3\, \left ( \tan \left ( fx+e \right ) \right ) ^{5}}{5}}+ \left ( \tan \left ( fx+e \right ) \right ) ^{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 = 1.7413, size = 92, normalized size = 1.19 \begin{align*} \frac{5 \, a^{4} c^{4} \tan \left (f x + e\right )^{7} + 21 \, a^{4} c^{4} \tan \left (f x + e\right )^{5} + 35 \, a^{4} c^{4} \tan \left (f x + e\right )^{3} + 35 \, a^{4} c^{4} \tan \left (f x + e\right )}{35 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.27123, size = 437, normalized size = 5.68 \begin{align*} \frac{1120 i \, a^{4} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 672 i \, a^{4} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 224 i \, a^{4} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 32 i \, a^{4} c^{4}}{35 \,{\left (f e^{\left (14 i \, f x + 14 i \, e\right )} + 7 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 21 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 35 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 35 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 21 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 7 \, 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 = 15.8109, size = 223, normalized size = 2.9 \begin{align*} \frac{\frac{32 i a^{4} c^{4} e^{- 8 i e} e^{6 i f x}}{f} + \frac{96 i a^{4} c^{4} e^{- 10 i e} e^{4 i f x}}{5 f} + \frac{32 i a^{4} c^{4} e^{- 12 i e} e^{2 i f x}}{5 f} + \frac{32 i a^{4} c^{4} e^{- 14 i e}}{35 f}}{e^{14 i f x} + 7 e^{- 2 i e} e^{12 i f x} + 21 e^{- 4 i e} e^{10 i f x} + 35 e^{- 6 i e} e^{8 i f x} + 35 e^{- 8 i e} e^{6 i f x} + 21 e^{- 10 i e} e^{4 i f x} + 7 e^{- 12 i e} e^{2 i f x} + e^{- 14 i e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.50071, size = 878, normalized size = 11.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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