Optimal. Leaf size=45 \[ -\frac{1}{2 (\tan (t)+1)}-\frac{1}{12} \log (\cos (t)-\sin (t))-\frac{1}{4} \log (\sin (t)+\cos (t))+\frac{1}{3} \log (\sin (t)+2 \cos (t)) \]
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Rubi [A] time = 0.123094, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {709, 800} \[ -\frac{1}{2 (\tan (t)+1)}-\frac{1}{12} \log (\cos (t)-\sin (t))-\frac{1}{4} \log (\sin (t)+\cos (t))+\frac{1}{3} \log (\sin (t)+2 \cos (t)) \]
Antiderivative was successfully verified.
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Rule 709
Rule 800
Rubi steps
\begin{align*} \int \frac{\sec (2 t)}{1+\sec ^2(t)+3 \tan (t)} \, dt &=\operatorname{Subst}\left (\int \frac{1}{(1+t)^2 \left (2-t-t^2\right )} \, dt,t,\tan (t)\right )\\ &=-\frac{1}{2 (1+\tan (t))}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{t}{(1+t) \left (2-t-t^2\right )} \, dt,t,\tan (t)\right )\\ &=-\frac{1}{2 (1+\tan (t))}+\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{6 (-1+t)}-\frac{1}{2 (1+t)}+\frac{2}{3 (2+t)}\right ) \, dt,t,\tan (t)\right )\\ &=-\frac{1}{12} \log (\cos (t)-\sin (t))-\frac{1}{4} \log (\cos (t)+\sin (t))+\frac{1}{3} \log (2 \cos (t)+\sin (t))-\frac{1}{2 (1+\tan (t))}\\ \end{align*}
Mathematica [A] time = 0.180955, size = 73, normalized size = 1.62 \[ -\frac{\cos (t) (\log (\cos (t)-\sin (t))+3 \log (\sin (t)+\cos (t))-4 \log (\sin (t)+2 \cos (t)))+\sin (t) (\log (\cos (t)-\sin (t))+3 \log (\sin (t)+\cos (t))-4 \log (\sin (t)+2 \cos (t))-6)}{12 (\sin (t)+\cos (t))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.142, size = 31, normalized size = 0.7 \begin{align*}{\frac{\ln \left ( \tan \left ( t \right ) +2 \right ) }{3}}-{\frac{1}{2+2\,\tan \left ( t \right ) }}-{\frac{\ln \left ( 1+\tan \left ( t \right ) \right ) }{4}}-{\frac{\ln \left ( -1+\tan \left ( t \right ) \right ) }{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.78015, size = 346, normalized size = 7.69 \begin{align*} \frac{3 \,{\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \sin \left (2 \, t\right ) + 1\right )} \log \left (953674316406250 \,{\left (3 \, \cos \left (2 \, t\right ) + \sin \left (2 \, t\right ) + 4\right )} \cos \left (4 \, t\right ) + 2384185791015625 \, \cos \left (4 \, t\right )^{2} + 953674316406250 \, \cos \left (2 \, t\right )^{2} - 953674316406250 \,{\left (\cos \left (2 \, t\right ) - 3 \, \sin \left (2 \, t\right ) + 3\right )} \sin \left (4 \, t\right ) + 2384185791015625 \, \sin \left (4 \, t\right )^{2} + 953674316406250 \, \sin \left (2 \, t\right )^{2} + 2861022949218750 \, \cos \left (2 \, t\right ) - 953674316406250 \, \sin \left (2 \, t\right ) + 2384185791015625\right ) - 6 \,{\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \sin \left (2 \, t\right ) + 1\right )} \log \left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \sin \left (2 \, t\right ) + 1\right ) + 5 \,{\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \sin \left (2 \, t\right ) + 1\right )} \log \left (\frac{5 \, \cos \left (2 \, t\right )^{2} + 5 \, \sin \left (2 \, t\right )^{2} + 6 \, \cos \left (2 \, t\right ) + 8 \, \sin \left (2 \, t\right ) + 5}{5 \,{\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} - 2 \, \sin \left (2 \, t\right ) + 1\right )}}\right ) - 24 \, \cos \left (2 \, t\right )}{48 \,{\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \sin \left (2 \, t\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.61796, size = 271, normalized size = 6.02 \begin{align*} \frac{4 \,{\left (\cos \left (t\right ) + \sin \left (t\right )\right )} \log \left (\frac{3}{4} \, \cos \left (t\right )^{2} + \cos \left (t\right ) \sin \left (t\right ) + \frac{1}{4}\right ) - 3 \,{\left (\cos \left (t\right ) + \sin \left (t\right )\right )} \log \left (2 \, \cos \left (t\right ) \sin \left (t\right ) + 1\right ) -{\left (\cos \left (t\right ) + \sin \left (t\right )\right )} \log \left (-2 \, \cos \left (t\right ) \sin \left (t\right ) + 1\right ) - 6 \, \cos \left (t\right ) + 6 \, \sin \left (t\right )}{24 \,{\left (\cos \left (t\right ) + \sin \left (t\right )\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 \frac{\sec{\left (2 t \right )}}{3 \tan{\left (t \right )} + \sec ^{2}{\left (t \right )} + 1}\, dt \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10377, size = 45, normalized size = 1. \begin{align*} -\frac{1}{2 \,{\left (\tan \left (t\right ) + 1\right )}} + \frac{1}{3} \, \log \left ({\left | \tan \left (t\right ) + 2 \right |}\right ) - \frac{1}{4} \, \log \left ({\left | \tan \left (t\right ) + 1 \right |}\right ) - \frac{1}{12} \, \log \left ({\left | \tan \left (t\right ) - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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