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 = 0.0296109, antiderivative size = 46, normalized size of antiderivative = 1., 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 3525
Rule 3475
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*}
Mathematica [A] time = 0.0332174, 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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Maple [A] time = 0.004, size = 81, normalized size = 1.8 \begin{align*}{\frac{iad\tan \left ( fx+e \right ) }{f}}+{\frac{{\frac{i}{2}}a\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) c}{f}}+{\frac{a\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) d}{2\,f}}-{\frac{ia\arctan \left ( \tan \left ( fx+e \right ) \right ) d}{f}}+{\frac{a\arctan \left ( \tan \left ( fx+e \right ) \right ) c}{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5285, size = 70, normalized size = 1.52 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66419, size = 163, normalized size = 3.54 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.22202, size = 58, normalized size = 1.26 \begin{align*} - \frac{2 a d e^{- 2 i e}}{f \left (e^{2 i f x} + e^{- 2 i e}\right )} - \frac{a \left (i c + d\right ) \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31877, size = 149, normalized size = 3.24 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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