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 = 0.12, antiderivative size = 45, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.19, 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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fricas [A] time = 0.47, size = 71, normalized size = 1.58 \[ \frac {4 \, {\left (\cos \relax (t) + \sin \relax (t)\right )} \log \left (\frac {3}{4} \, \cos \relax (t)^{2} + \cos \relax (t) \sin \relax (t) + \frac {1}{4}\right ) - 3 \, {\left (\cos \relax (t) + \sin \relax (t)\right )} \log \left (2 \, \cos \relax (t) \sin \relax (t) + 1\right ) - {\left (\cos \relax (t) + \sin \relax (t)\right )} \log \left (-2 \, \cos \relax (t) \sin \relax (t) + 1\right ) - 6 \, \cos \relax (t) + 6 \, \sin \relax (t)}{24 \, {\left (\cos \relax (t) + \sin \relax (t)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 33, normalized size = 0.73 \[ -\frac {1}{2 \, {\left (\tan \relax (t) + 1\right )}} + \frac {1}{3} \, \log \left ({\left | \tan \relax (t) + 2 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | \tan \relax (t) + 1 \right |}\right ) - \frac {1}{12} \, \log \left ({\left | \tan \relax (t) - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 31, normalized size = 0.69 \[ -\frac {\ln \left (\tan \relax (t )+1\right )}{4}-\frac {\ln \left (\tan \relax (t )-1\right )}{12}+\frac {\ln \left (\tan \relax (t )+2\right )}{3}-\frac {1}{2 \left (\tan \relax (t )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.98, size = 256, normalized size = 5.69 \[ \frac {3 \, {\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \sin \left (2 \, t\right ) + 1\right )} \log \left (953674316406250 \, {\left (3 \, \cos \left (2 \, t\right ) + \sin \left (2 \, t\right ) + 4\right )} \cos \left (4 \, t\right ) + 2384185791015625 \, \cos \left (4 \, t\right )^{2} + 953674316406250 \, \cos \left (2 \, t\right )^{2} - 953674316406250 \, {\left (\cos \left (2 \, t\right ) - 3 \, \sin \left (2 \, t\right ) + 3\right )} \sin \left (4 \, t\right ) + 2384185791015625 \, \sin \left (4 \, t\right )^{2} + 953674316406250 \, \sin \left (2 \, t\right )^{2} + 2861022949218750 \, \cos \left (2 \, t\right ) - 953674316406250 \, \sin \left (2 \, t\right ) + 2384185791015625\right ) - 6 \, {\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \sin \left (2 \, t\right ) + 1\right )} \log \left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \sin \left (2 \, t\right ) + 1\right ) + 5 \, {\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \sin \left (2 \, t\right ) + 1\right )} \log \left (\frac {5 \, \cos \left (2 \, t\right )^{2} + 5 \, \sin \left (2 \, t\right )^{2} + 6 \, \cos \left (2 \, t\right ) + 8 \, \sin \left (2 \, t\right ) + 5}{5 \, {\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} - 2 \, \sin \left (2 \, t\right ) + 1\right )}}\right ) - 24 \, \cos \left (2 \, t\right )}{48 \, {\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \sin \left (2 \, t\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.72, size = 32, normalized size = 0.71 \[ \frac {\ln \left (\mathrm {tan}\relax (t)+2\right )}{3}-\frac {\ln \left (\mathrm {tan}\relax (t)+1\right )}{4}-\frac {\ln \left (\mathrm {tan}\relax (t)-1\right )}{12}-\frac {1}{2\,\left (\mathrm {tan}\relax (t)+1\right )} \]
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 \[ \int \frac {\sec {\left (2 t \right )}}{3 \tan {\relax (t )} + \sec ^{2}{\relax (t )} + 1}\, dt \]
Verification of antiderivative is not currently implemented for this CAS.
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