Optimal. Leaf size=19 \[ a x-\frac{i a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0074034, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3475} \[ a x-\frac{i a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rubi steps
\begin{align*} \int (a+i a \tan (c+d x)) \, dx &=a x+(i a) \int \tan (c+d x) \, dx\\ &=a x-\frac{i a \log (\cos (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0058757, size = 19, normalized size = 1. \[ a x-\frac{i a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 23, normalized size = 1.2 \begin{align*} ax+{\frac{{\frac{i}{2}}a\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09955, size = 23, normalized size = 1.21 \begin{align*} a x + \frac{i \, a \log \left (\sec \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20903, size = 50, normalized size = 2.63 \begin{align*} -\frac{i \, a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.49256, size = 24, normalized size = 1.26 \begin{align*} - \frac{i a \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 [A] time = 1.12551, size = 24, normalized size = 1.26 \begin{align*} a x - \frac{i \, a \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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