Optimal. Leaf size=38 \[ -\frac{a^2 \tan (c+d x)}{d}-\frac{2 i a^2 \log (\cos (c+d x))}{d}+2 a^2 x \]
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Rubi [A] time = 0.0169765, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3477, 3475} \[ -\frac{a^2 \tan (c+d x)}{d}-\frac{2 i a^2 \log (\cos (c+d x))}{d}+2 a^2 x \]
Antiderivative was successfully verified.
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Rule 3477
Rule 3475
Rubi steps
\begin{align*} \int (a+i a \tan (c+d x))^2 \, dx &=2 a^2 x-\frac{a^2 \tan (c+d x)}{d}+\left (2 i a^2\right ) \int \tan (c+d x) \, dx\\ &=2 a^2 x-\frac{2 i a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [B] time = 0.632605, size = 100, normalized size = 2.63 \[ -\frac{a^2 \sec (c) \sec (c+d x) \left (-4 d x \cos (2 c+d x)+\cos (d x) \left (-4 d x+i \log \left (\cos ^2(c+d x)\right )\right )+i \cos (2 c+d x) \log \left (\cos ^2(c+d x)\right )+4 \cos (c) \cos (c+d x) \tan ^{-1}(\tan (3 c+d x))+2 \sin (d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 51, normalized size = 1.3 \begin{align*}{\frac{i{a}^{2}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}+2\,{\frac{{a}^{2}\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2}\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66464, size = 55, normalized size = 1.45 \begin{align*} a^{2} x + \frac{{\left (d x + c - \tan \left (d x + c\right )\right )} a^{2}}{d} + \frac{2 i \, a^{2} \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 = 1.06008, size = 151, normalized size = 3.97 \begin{align*} \frac{-2 i \, a^{2} +{\left (-2 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, a^{2}\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 = 0.937538, size = 60, normalized size = 1.58 \begin{align*} - \frac{2 i a^{2} \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} - \frac{2 i a^{2} 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 = 1.13516, size = 88, normalized size = 2.32 \begin{align*} \frac{-2 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 2 i \, a^{2} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 2 i \, a^{2}}{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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