Optimal. Leaf size=59 \[ \frac{a c^4 (B+i A) (1-i \tan (e+f x))^4}{4 f}-\frac{a B c^4 (1-i \tan (e+f x))^5}{5 f} \]
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Rubi [A] time = 0.0835138, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {3588, 43} \[ \frac{a c^4 (B+i A) (1-i \tan (e+f x))^4}{4 f}-\frac{a B c^4 (1-i \tan (e+f x))^5}{5 f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 43
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx &=\frac{(a c) \operatorname{Subst}\left (\int (A+B x) (c-i c x)^3 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left ((A-i B) (c-i c x)^3+\frac{i B (c-i c x)^4}{c}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{a (i A+B) c^4 (1-i \tan (e+f x))^4}{4 f}-\frac{a B c^4 (1-i \tan (e+f x))^5}{5 f}\\ \end{align*}
Mathematica [B] time = 3.47291, size = 226, normalized size = 3.83 \[ \frac{a c^4 \sec (e) \sec ^5(e+f x) (5 (3 B-5 i A) \cos (2 e+f x)+5 (3 B-5 i A) \cos (f x)-25 A \sin (2 e+f x)+15 A \sin (2 e+3 f x)-10 A \sin (4 e+3 f x)+5 A \sin (4 e+5 f x)-10 i A \cos (2 e+3 f x)-10 i A \cos (4 e+3 f x)+25 A \sin (f x)-15 i B \sin (2 e+f x)+5 i B \sin (2 e+3 f x)-10 i B \sin (4 e+3 f x)+3 i B \sin (4 e+5 f x)+10 B \cos (2 e+3 f x)+10 B \cos (4 e+3 f x)+15 i B \sin (f x))}{40 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 99, normalized size = 1.7 \begin{align*}{\frac{a{c}^{4}}{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}-iB \left ( \tan \left ( fx+e \right ) \right ) ^{3}-{\frac{3\,B \left ( \tan \left ( fx+e \right ) \right ) ^{4}}{4}}-{\frac{3\,i}{2}}A \left ( \tan \left ( fx+e \right ) \right ) ^{2}-A \left ( \tan \left ( fx+e \right ) \right ) ^{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 = 2.18342, size = 131, normalized size = 2.22 \begin{align*} \frac{12 i \, B a c^{4} \tan \left (f x + e\right )^{5} - 15 \,{\left (-i \, A + 3 \, B\right )} a c^{4} \tan \left (f x + e\right )^{4} -{\left (60 \, A + 60 i \, B\right )} a c^{4} \tan \left (f x + e\right )^{3} - 30 \,{\left (3 i \, A - B\right )} a c^{4} \tan \left (f x + e\right )^{2} + 60 \, A a c^{4} \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 [B] time = 1.29535, size = 282, normalized size = 4.78 \begin{align*} \frac{{\left (20 i \, A + 20 \, B\right )} a c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (20 i \, A - 12 \, B\right )} a 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 = 15.0631, size = 148, normalized size = 2.51 \begin{align*} \frac{\frac{\left (4 i A a c^{4} + 4 B a c^{4}\right ) e^{- 8 i e} e^{2 i f x}}{f} + \frac{\left (20 i A a c^{4} - 12 B a c^{4}\right ) 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 = 1.72624, size = 161, normalized size = 2.73 \begin{align*} \frac{20 i \, A a c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 20 \, B a c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 20 i \, A a c^{4} - 12 \, B a 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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