Optimal. Leaf size=34 \[ \frac{i a \tan (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}-i a x \]
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Rubi [A] time = 0.0175204, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3525, 3475} \[ \frac{i a \tan (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}-i a x \]
Antiderivative was successfully verified.
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Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \tan (c+d x) (a+i a \tan (c+d x)) \, dx &=-i a x+\frac{i a \tan (c+d x)}{d}+a \int \tan (c+d x) \, dx\\ &=-i a x-\frac{a \log (\cos (c+d x))}{d}+\frac{i a \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0173237, size = 43, normalized size = 1.26 \[ -\frac{i a \tan ^{-1}(\tan (c+d x))}{d}+\frac{i a \tan (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 46, normalized size = 1.4 \begin{align*}{\frac{ia\tan \left ( dx+c \right ) }{d}}+{\frac{a\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}-{\frac{ia\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 = 2.10057, size = 50, normalized size = 1.47 \begin{align*} -\frac{2 i \,{\left (d x + c\right )} a - a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 i \, 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 = 2.51703, size = 128, normalized size = 3.76 \begin{align*} -\frac{{\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2 \, 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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Sympy [A] time = 0.832894, size = 51, normalized size = 1.5 \begin{align*} - \frac{a \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} - \frac{2 a e^{- 2 i c}}{d \left (e^{2 i d x} + e^{- 2 i c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17392, size = 78, normalized size = 2.29 \begin{align*} -\frac{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 ) + a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2 \, 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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