Optimal. Leaf size=46 \[ -\frac{a (B+i A) \log (\cos (c+d x))}{d}+a x (A-i B)+\frac{i a B \tan (c+d x)}{d} \]
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Rubi [A] time = 0.0278612, 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} \[ -\frac{a (B+i A) \log (\cos (c+d x))}{d}+a x (A-i B)+\frac{i a B \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx &=a (A-i B) x+\frac{i a B \tan (c+d x)}{d}+(a (i A+B)) \int \tan (c+d x) \, dx\\ &=a (A-i B) x-\frac{a (i A+B) \log (\cos (c+d x))}{d}+\frac{i a B \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0263979, size = 66, normalized size = 1.43 \[ -\frac{i a A \log (\cos (c+d x))}{d}+a A x-\frac{i a B \tan ^{-1}(\tan (c+d x))}{d}+\frac{i a B \tan (c+d x)}{d}-\frac{a B \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 81, normalized size = 1.8 \begin{align*}{\frac{iaB\tan \left ( dx+c \right ) }{d}}+{\frac{{\frac{i}{2}}a\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) A}{d}}+{\frac{a\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) B}{2\,d}}-{\frac{iaB\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}}+{\frac{aA\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66479, size = 68, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (d x + c\right )}{\left (A - i \, B\right )} a -{\left (-i \, A - B\right )} a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 i \, B a \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36108, size = 161, normalized size = 3.5 \begin{align*} -\frac{2 \, B a -{\left ({\left (-i \, A - B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (-i \, A - B\right )} a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.38147, size = 58, normalized size = 1.26 \begin{align*} - \frac{2 B a e^{- 2 i c}}{d \left (e^{2 i d x} + e^{- 2 i c}\right )} - \frac{a \left (i A + B\right ) \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34306, size = 139, normalized size = 3.02 \begin{align*} \frac{-i \, A a e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - B a e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - i \, A a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - B a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 2 \, B a}{d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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