Optimal. Leaf size=16 \[ \frac {1}{2} \tan (2 x)-\log (\cos (2 x)) \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3477, 3475} \[ \frac {1}{2} \tan (2 x)-\log (\cos (2 x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3477
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*}
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Mathematica [A] time = 0.01, 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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int (1+\tan (2 x))^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.80, size = 20, normalized size = 1.25 \[ -\frac {1}{2} \, \log \left (\frac {1}{\tan \left (2 \, x\right )^{2} + 1}\right ) + \frac {1}{2} \, \tan \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 22, normalized size = 1.38 \[ -\frac {1}{2} \, \log \left (\frac {4}{\tan \left (2 \, x\right )^{2} + 1}\right ) + \frac {1}{2} \, \tan \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 19, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {\tan \left (2 x \right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (2 x \right )\right )}{2}\) | \(19\) |
default | \(\frac {\tan \left (2 x \right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (2 x \right )\right )}{2}\) | \(19\) |
norman | \(\frac {\tan \left (2 x \right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (2 x \right )\right )}{2}\) | \(19\) |
risch | \(2 i x +\frac {i}{{\mathrm e}^{4 i x}+1}-\ln \left ({\mathrm e}^{4 i x}+1\right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 12, normalized size = 0.75 \[ \log \left (\sec \left (2 \, x\right )\right ) + \frac {1}{2} \, \tan \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 18, normalized size = 1.12 \[ \frac {\mathrm {tan}\left (2\,x\right )}{2}+\frac {\ln \left ({\mathrm {tan}\left (2\,x\right )}^2+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 17, normalized size = 1.06 \[ \frac {\log {\left (\tan ^{2}{\left (2 x \right )} + 1 \right )}}{2} + \frac {\tan {\left (2 x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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