Optimal. Leaf size=46 \[ -\frac {a (d+i c) \log (\cos (e+f x))}{f}+a x (c-i d)+\frac {i a d \tan (e+f x)}{f} \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3525, 3475} \[ -\frac {a (d+i c) \log (\cos (e+f x))}{f}+a x (c-i d)+\frac {i a d \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) (c+d \tan (e+f x)) \, dx &=a (c-i d) x+\frac {i a d \tan (e+f x)}{f}+(a (i c+d)) \int \tan (e+f x) \, dx\\ &=a (c-i d) x-\frac {a (i c+d) \log (\cos (e+f x))}{f}+\frac {i a d \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 1.43 \[ -\frac {i a c \log (\cos (e+f x))}{f}+a c x-\frac {i a d \tan ^{-1}(\tan (e+f x))}{f}+\frac {i a d \tan (e+f x)}{f}-\frac {a d \log (\cos (e+f x))}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 64, normalized size = 1.39 \[ -\frac {2 \, a d - {\left (-i \, a c - a d + {\left (-i \, a c - a d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 110, normalized size = 2.39 \[ \frac {-i \, a c e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - a d e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - i \, a c \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - a d \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 2 \, a d}{f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 81, normalized size = 1.76 \[ \frac {i a d \tan \left (f x +e \right )}{f}+\frac {i a \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c}{2 f}+\frac {a \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) d}{2 f}-\frac {i a \arctan \left (\tan \left (f x +e \right )\right ) d}{f}+\frac {a \arctan \left (\tan \left (f x +e \right )\right ) c}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 52, normalized size = 1.13 \[ -\frac {-2 i \, a d \tan \left (f x + e\right ) - 2 \, {\left (a c - i \, a d\right )} {\left (f x + e\right )} + {\left (-i \, a c - a d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.88, size = 38, normalized size = 0.83 \[ \frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (a\,d+a\,c\,1{}\mathrm {i}\right )}{f}+\frac {a\,d\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 53, normalized size = 1.15 \[ \frac {2 a d}{- f e^{2 i e} e^{2 i f x} - f} - \frac {i a \left (c - i d\right ) \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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