Optimal. Leaf size=8 \[ t \tan (t)+\log (\cos (t)) \]
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Rubi [A] time = 0.0180556, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4184, 3475} \[ t \tan (t)+\log (\cos (t)) \]
Antiderivative was successfully verified.
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Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int t \sec ^2(t) \, dt &=t \tan (t)-\int \tan (t) \, dt\\ &=\log (\cos (t))+t \tan (t)\\ \end{align*}
Mathematica [A] time = 0.004618, size = 8, normalized size = 1. \[ t \tan (t)+\log (\cos (t)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 9, normalized size = 1.1 \begin{align*} \ln \left ( \cos \left ( t \right ) \right ) +t\tan \left ( t \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.42639, size = 100, normalized size = 12.5 \begin{align*} \frac{{\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \cos \left (2 \, t\right ) + 1\right )} \log \left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \cos \left (2 \, t\right ) + 1\right ) + 4 \, t \sin \left (2 \, t\right )}{2 \,{\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \cos \left (2 \, t\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.08662, size = 55, normalized size = 6.88 \begin{align*} \frac{\cos \left (t\right ) \log \left (-\cos \left (t\right )\right ) + t \sin \left (t\right )}{\cos \left (t\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*} \int t \sec ^{2}{\left (t \right )}\, dt \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10245, size = 139, normalized size = 17.38 \begin{align*} \frac{\log \left (\frac{4 \,{\left (\tan \left (\frac{1}{2} \, t\right )^{4} - 2 \, \tan \left (\frac{1}{2} \, t\right )^{2} + 1\right )}}{\tan \left (\frac{1}{2} \, t\right )^{4} + 2 \, \tan \left (\frac{1}{2} \, t\right )^{2} + 1}\right ) \tan \left (\frac{1}{2} \, t\right )^{2} - 4 \, t \tan \left (\frac{1}{2} \, t\right ) - \log \left (\frac{4 \,{\left (\tan \left (\frac{1}{2} \, t\right )^{4} - 2 \, \tan \left (\frac{1}{2} \, t\right )^{2} + 1\right )}}{\tan \left (\frac{1}{2} \, t\right )^{4} + 2 \, \tan \left (\frac{1}{2} \, t\right )^{2} + 1}\right )}{2 \,{\left (\tan \left (\frac{1}{2} \, t\right )^{2} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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