Optimal. Leaf size=58 \[ \frac{i a^2 (c-i c \tan (e+f x))^4}{2 f}-\frac{i a^2 (c-i c \tan (e+f x))^5}{5 c f} \]
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Rubi [A] time = 0.10237, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 3487, 43} \[ \frac{i a^2 (c-i c \tan (e+f x))^4}{2 f}-\frac{i a^2 (c-i c \tan (e+f x))^5}{5 c f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^4 \, dx &=\left (a^2 c^2\right ) \int \sec ^4(e+f x) (c-i c \tan (e+f x))^2 \, dx\\ &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int (c-x) (c+x)^3 \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \left (2 c (c+x)^3-(c+x)^4\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac{i a^2 (c-i c \tan (e+f x))^4}{2 f}-\frac{i a^2 (c-i c \tan (e+f x))^5}{5 c f}\\ \end{align*}
Mathematica [A] time = 3.10456, size = 80, normalized size = 1.38 \[ \frac{a^2 c^4 \sec (e) \sec ^5(e+f x) (-5 \sin (2 e+f x)+5 \sin (2 e+3 f x)+\sin (4 e+5 f x)-5 i \cos (2 e+f x)+5 \sin (f x)-5 i \cos (f x))}{20 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 50, normalized size = 0.9 \begin{align*}{\frac{{a}^{2}{c}^{4}}{f} \left ( \tan \left ( fx+e \right ) -{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{5}}{5}}-{\frac{i}{2}} \left ( \tan \left ( fx+e \right ) \right ) ^{4}-i \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70275, size = 92, normalized size = 1.59 \begin{align*} -\frac{6 \, a^{2} c^{4} \tan \left (f x + e\right )^{5} + 15 i \, a^{2} c^{4} \tan \left (f x + e\right )^{4} + 30 i \, a^{2} c^{4} \tan \left (f x + e\right )^{2} - 30 \, a^{2} c^{4} \tan \left (f x + e\right )}{30 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30037, size = 257, normalized size = 4.43 \begin{align*} \frac{40 i \, a^{2} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, a^{2} c^{4}}{5 \,{\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 [B] time = 4.58181, size = 131, normalized size = 2.26 \begin{align*} \frac{\frac{8 i a^{2} c^{4} e^{- 8 i e} e^{2 i f x}}{f} + \frac{8 i a^{2} c^{4} e^{- 10 i e}}{5 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 = 1.76477, size = 131, normalized size = 2.26 \begin{align*} \frac{40 i \, a^{2} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, a^{2} c^{4}}{5 \,{\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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