Optimal. Leaf size=13 \[ 2 x \tan (2 x)+\log (\cos (2 x)) \]
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Rubi [A] time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 3475
Rule 4184
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*}
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Mathematica [A] time = 0.01, 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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fricas [B] time = 1.32, size = 27, normalized size = 2.08 \[ \frac {\cos \left (2 \, x\right ) \log \left (-\cos \left (2 \, x\right )\right ) + 2 \, x \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 81, normalized size = 6.23 \[ \frac {\log \left (\frac {4 \, {\left (\tan \relax (x)^{4} - 2 \, \tan \relax (x)^{2} + 1\right )}}{\tan \relax (x)^{4} + 2 \, \tan \relax (x)^{2} + 1}\right ) \tan \relax (x)^{2} - 8 \, x \tan \relax (x) - \log \left (\frac {4 \, {\left (\tan \relax (x)^{4} - 2 \, \tan \relax (x)^{2} + 1\right )}}{\tan \relax (x)^{4} + 2 \, \tan \relax (x)^{2} + 1}\right )}{2 \, {\left (\tan \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 1.08 \[ \ln \left (\cos \left (2 x \right )\right )+2 x \tan \left (2 x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 74, normalized size = 5.69 \[ \frac {{\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )} \log \left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right ) + 8 \, x \sin \left (4 \, x\right )}{2 \, {\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.99, size = 13, normalized size = 1.00 \[ \ln \left (\cos \left (2\,x\right )\right )+2\,x\,\mathrm {tan}\left (2\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ 4 \int x \sec ^{2}{\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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