Optimal. Leaf size=13 \[ 2 x \tan (2 x)+\log (\cos (2 x)) \]
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Rubi [A] time = 0.0200766, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {12, 4184, 3475} \[ 2 x \tan (2 x)+\log (\cos (2 x)) \]
Antiderivative was successfully verified.
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Rule 12
Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int 4 x \sec ^2(2 x) \, dx &=4 \int x \sec ^2(2 x) \, dx\\ &=2 x \tan (2 x)-2 \int \tan (2 x) \, dx\\ &=\log (\cos (2 x))+2 x \tan (2 x)\\ \end{align*}
Mathematica [A] time = 0.0071375, size = 21, normalized size = 1.62 \[ 4 \left (\frac{1}{2} x \tan (2 x)+\frac{1}{4} \log (\cos (2 x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 14, normalized size = 1.1 \begin{align*} \ln \left ( \cos \left ( 2\,x \right ) \right ) +2\,x\tan \left ( 2\,x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.47006, size = 100, normalized size = 7.69 \begin{align*} \frac{{\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )} \log \left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right ) + 8 \, x \sin \left (4 \, x\right )}{2 \,{\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.50494, size = 69, normalized size = 5.31 \begin{align*} \frac{\cos \left (2 \, x\right ) \log \left (-\cos \left (2 \, x\right )\right ) + 2 \, x \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} 4 \int x \sec ^{2}{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10374, size = 109, normalized size = 8.38 \begin{align*} \frac{\log \left (\frac{4 \,{\left (\tan \left (x\right )^{4} - 2 \, \tan \left (x\right )^{2} + 1\right )}}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) \tan \left (x\right )^{2} - 8 \, x \tan \left (x\right ) - \log \left (\frac{4 \,{\left (\tan \left (x\right )^{4} - 2 \, \tan \left (x\right )^{2} + 1\right )}}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right )}{2 \,{\left (\tan \left (x\right )^{2} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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