Optimal. Leaf size=25 \[ -\frac {i c (a+i a \tan (e+f x))^2}{2 f} \]
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Rubi [A] time = 0.06, antiderivative size = 36, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3522, 3486, 3767, 8} \[ \frac {a^2 c \tan (e+f x)}{f}+\frac {i a^2 c \sec ^2(e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3486
Rule 3522
Rule 3767
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x)) \, dx &=(a c) \int \sec ^2(e+f x) (a+i a \tan (e+f x)) \, dx\\ &=\frac {i a^2 c \sec ^2(e+f x)}{2 f}+\left (a^2 c\right ) \int \sec ^2(e+f x) \, dx\\ &=\frac {i a^2 c \sec ^2(e+f x)}{2 f}-\frac {\left (a^2 c\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{f}\\ &=\frac {i a^2 c \sec ^2(e+f x)}{2 f}+\frac {a^2 c \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 45, normalized size = 1.80 \[ \frac {a^2 c \left (-2 \tan ^{-1}(\tan (e+f x))+i \tan ^2(e+f x)+2 \tan (e+f x)+2 f x\right )}{2 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 50, normalized size = 2.00 \[ \frac {4 i \, a^{2} c e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, a^{2} c}{f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.78, size = 53, normalized size = 2.12 \[ \frac {4 i \, a^{2} c e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, a^{2} c}{f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 27, normalized size = 1.08 \[ \frac {a^{2} c \left (\frac {i \left (\tan ^{2}\left (f x +e \right )\right )}{2}+\tan \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 32, normalized size = 1.28 \[ -\frac {-i \, a^{2} c \tan \left (f x + e\right )^{2} - 2 \, a^{2} c \tan \left (f x + e\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.73, size = 26, normalized size = 1.04 \[ \frac {a^2\,c\,\mathrm {tan}\left (e+f\,x\right )\,\left (2+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}{2\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.25, size = 68, normalized size = 2.72 \[ \frac {4 i a^{2} c e^{2 i e} e^{2 i f x} + 2 i a^{2} c}{f e^{4 i e} e^{4 i f x} + 2 f e^{2 i e} e^{2 i f x} + f} \]
Verification of antiderivative is not currently implemented for this CAS.
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