Optimal. Leaf size=16 \[ \frac{1}{2} \tan (2 x)-\log (\cos (2 x)) \]
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Rubi [A] time = 0.0109919, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3477, 3475} \[ \frac{1}{2} \tan (2 x)-\log (\cos (2 x)) \]
Antiderivative was successfully verified.
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Rule 3477
Rule 3475
Rubi steps
\begin{align*} \int (1+\tan (2 x))^2 \, dx &=\frac{1}{2} \tan (2 x)+2 \int \tan (2 x) \, dx\\ &=-\log (\cos (2 x))+\frac{1}{2} \tan (2 x)\\ \end{align*}
Mathematica [A] time = 0.0107909, size = 26, normalized size = 1.62 \[ x-\frac{1}{2} \tan ^{-1}(\tan (2 x))+\frac{1}{2} \tan (2 x)-\log (\cos (2 x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 19, normalized size = 1.2 \begin{align*}{\frac{\tan \left ( 2\,x \right ) }{2}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( 2\,x \right ) \right ) ^{2} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41653, size = 16, normalized size = 1. \begin{align*} \log \left (\sec \left (2 \, x\right )\right ) + \frac{1}{2} \, \tan \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00127, size = 61, normalized size = 3.81 \begin{align*} -\frac{1}{2} \, \log \left (\frac{1}{\tan \left (2 \, x\right )^{2} + 1}\right ) + \frac{1}{2} \, \tan \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.141312, size = 17, normalized size = 1.06 \begin{align*} \frac{\log{\left (\tan ^{2}{\left (2 x \right )} + 1 \right )}}{2} + \frac{\tan{\left (2 x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09449, size = 30, normalized size = 1.88 \begin{align*} -\frac{1}{2} \, \log \left (\frac{4}{\tan \left (2 \, x\right )^{2} + 1}\right ) + \frac{1}{2} \, \tan \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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