Optimal. Leaf size=32 \[ \frac {a A c \tan (e+f x)}{f}+\frac {a B c \tan ^2(e+f x)}{2 f} \]
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Rubi [A] time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {3588} \[ \frac {a A c \tan (e+f x)}{f}+\frac {a B c \tan ^2(e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Rule 3588
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)) \, dx &=\frac {(a c) \operatorname {Subst}(\int (A+B x) \, dx,x,\tan (e+f x))}{f}\\ &=\frac {a A c \tan (e+f x)}{f}+\frac {a B c \tan ^2(e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 32, normalized size = 1.00 \[ \frac {a A c \tan (e+f x)}{f}+\frac {a B c \sec ^2(e+f x)}{2 f} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.84, size = 53, normalized size = 1.66 \[ \frac {{\left (2 i \, A + 2 \, B\right )} a c e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, A a c}{f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.97, size = 113, normalized size = 3.53 \[ \frac {B a c \tan \left (f x\right )^{2} \tan \relax (e)^{2} - 2 \, A a c \tan \left (f x\right )^{2} \tan \relax (e) - 2 \, A a c \tan \left (f x\right ) \tan \relax (e)^{2} + B a c \tan \left (f x\right )^{2} + B a c \tan \relax (e)^{2} + 2 \, A a c \tan \left (f x\right ) + 2 \, A a c \tan \relax (e) + B a c}{2 \, {\left (f \tan \left (f x\right )^{2} \tan \relax (e)^{2} - 2 \, f \tan \left (f x\right ) \tan \relax (e) + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 27, normalized size = 0.84 \[ \frac {c a \left (\frac {B \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \tan \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 29, normalized size = 0.91 \[ \frac {B a c \tan \left (f x + e\right )^{2} + 2 \, A a c \tan \left (f x + e\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.42, size = 25, normalized size = 0.78 \[ \frac {a\,c\,\mathrm {tan}\left (e+f\,x\right )\,\left (2\,A+B\,\mathrm {tan}\left (e+f\,x\right )\right )}{2\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.31, size = 82, normalized size = 2.56 \[ \frac {2 i A a c + \left (2 i A a c e^{2 i e} + 2 B a c e^{2 i e}\right ) e^{2 i f x}}{f e^{4 i e} e^{4 i f x} + 2 f e^{2 i e} e^{2 i f x} + f} \]
Verification of antiderivative is not currently implemented for this CAS.
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