Optimal. Leaf size=99 \[ -\frac{a^2 c^3 (3 B+i A) (1-i \tan (e+f x))^4}{4 f}+\frac{2 a^2 c^3 (B+i A) (1-i \tan (e+f x))^3}{3 f}+\frac{a^2 B c^3 (1-i \tan (e+f x))^5}{5 f} \]
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Rubi [A] time = 0.150024, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049, Rules used = {3588, 77} \[ -\frac{a^2 c^3 (3 B+i A) (1-i \tan (e+f x))^4}{4 f}+\frac{2 a^2 c^3 (B+i A) (1-i \tan (e+f x))^3}{3 f}+\frac{a^2 B c^3 (1-i \tan (e+f x))^5}{5 f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 77
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^3 \, dx &=\frac{(a c) \operatorname{Subst}\left (\int (a+i a x) (A+B x) (c-i c x)^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (2 a (A-i B) (c-i c x)^2-\frac{a (A-3 i B) (c-i c x)^3}{c}-\frac{i a B (c-i c x)^4}{c^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 a^2 (i A+B) c^3 (1-i \tan (e+f x))^3}{3 f}-\frac{a^2 (i A+3 B) c^3 (1-i \tan (e+f x))^4}{4 f}+\frac{a^2 B c^3 (1-i \tan (e+f x))^5}{5 f}\\ \end{align*}
Mathematica [A] time = 5.69938, size = 146, normalized size = 1.47 \[ \frac{a^2 c^3 \sec (e) \sec ^5(e+f x) (15 (B-i A) \cos (2 e+f x)+15 (B-i A) \cos (f x)-15 A \sin (2 e+f x)+25 A \sin (2 e+3 f x)+5 A \sin (4 e+5 f x)+35 A \sin (f x)-15 i B \sin (2 e+f x)+5 i B \sin (2 e+3 f x)+i B \sin (4 e+5 f x)-5 i B \sin (f x))}{120 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 101, normalized size = 1. \begin{align*}{\frac{{c}^{3}{a}^{2}}{f} \left ( -{\frac{i}{5}}B \left ( \tan \left ( fx+e \right ) \right ) ^{5}-{\frac{i}{4}}A \left ( \tan \left ( fx+e \right ) \right ) ^{4}-{\frac{i}{3}}B \left ( \tan \left ( fx+e \right ) \right ) ^{3}+{\frac{B \left ( \tan \left ( fx+e \right ) \right ) ^{4}}{4}}-{\frac{i}{2}}A \left ( \tan \left ( fx+e \right ) \right ) ^{2}+{\frac{A \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{3}}+{\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 = 1.90972, size = 139, normalized size = 1.4 \begin{align*} -\frac{12 i \, B a^{2} c^{3} \tan \left (f x + e\right )^{5} - 15 \,{\left (-i \, A + B\right )} a^{2} c^{3} \tan \left (f x + e\right )^{4} -{\left (20 \, A - 20 i \, B\right )} a^{2} c^{3} \tan \left (f x + e\right )^{3} - 30 \,{\left (-i \, A + B\right )} a^{2} c^{3} \tan \left (f x + e\right )^{2} - 60 \, A a^{2} c^{3} \tan \left (f x + e\right )}{60 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38537, size = 351, normalized size = 3.55 \begin{align*} \frac{{\left (80 i \, A + 80 \, B\right )} a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (100 i \, A - 20 \, B\right )} a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (20 i \, A - 4 \, B\right )} a^{2} 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 [B] time = 18.0525, size = 197, normalized size = 1.99 \begin{align*} \frac{\frac{\left (16 i A a^{2} c^{3} + 16 B a^{2} c^{3}\right ) e^{- 6 i e} e^{4 i f x}}{3 f} + \frac{\left (20 i A a^{2} c^{3} - 4 B a^{2} c^{3}\right ) e^{- 8 i e} e^{2 i f x}}{3 f} + \frac{\left (20 i A a^{2} c^{3} - 4 B a^{2} c^{3}\right ) 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 [A] time = 1.69948, size = 223, normalized size = 2.25 \begin{align*} \frac{80 i \, A a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 80 \, B a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 100 i \, A a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 20 \, B a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 20 i \, A a^{2} c^{3} - 4 \, B a^{2} 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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