Optimal. Leaf size=8 \[ t \tan (t)+\log (\cos (t)) \]
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Rubi [A] time = 0.0275211, 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 \[ t \tan (t)+\log (\cos (t)) \]
Antiderivative was successfully verified.
[In] Int[t*Sec[t]^2,t]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{t}{\cos ^{2}{\left (t \right )}}\, dt \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(t*sec(t)**2,t)
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Mathematica [A] time = 0.0114151, size = 8, normalized size = 1. \[ t \tan (t)+\log (\cos (t)) \]
Antiderivative was successfully verified.
[In] Integrate[t*Sec[t]^2,t]
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Maple [A] time = 0.007, size = 9, normalized size = 1.1 \[ \ln \left ( \cos \left ( t \right ) \right ) +t\tan \left ( t \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(t*sec(t)^2,t)
[Out]
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Maxima [A] time = 1.52218, size = 100, normalized size = 12.5 \[ \frac{{\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \cos \left (2 \, t\right ) + 1\right )} \log \left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \cos \left (2 \, t\right ) + 1\right ) + 4 \, t \sin \left (2 \, t\right )}{2 \,{\left (\cos \left (2 \, t\right )^{2} + \sin \left (2 \, t\right )^{2} + 2 \, \cos \left (2 \, t\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(t*sec(t)^2,t, algorithm="maxima")
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Fricas [A] time = 0.243203, size = 24, normalized size = 3. \[ \frac{\cos \left (t\right ) \log \left (-\cos \left (t\right )\right ) + t \sin \left (t\right )}{\cos \left (t\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(t*sec(t)^2,t, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int t \sec ^{2}{\left (t \right )}\, dt \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(t*sec(t)**2,t)
[Out]
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GIAC/XCAS [A] time = 0.22179, size = 139, normalized size = 17.38 \[ \frac{{\rm ln}\left (\frac{4 \,{\left (\tan \left (\frac{1}{2} \, t\right )^{4} - 2 \, \tan \left (\frac{1}{2} \, t\right )^{2} + 1\right )}}{\tan \left (\frac{1}{2} \, t\right )^{4} + 2 \, \tan \left (\frac{1}{2} \, t\right )^{2} + 1}\right ) \tan \left (\frac{1}{2} \, t\right )^{2} - 4 \, t \tan \left (\frac{1}{2} \, t\right ) -{\rm ln}\left (\frac{4 \,{\left (\tan \left (\frac{1}{2} \, t\right )^{4} - 2 \, \tan \left (\frac{1}{2} \, t\right )^{2} + 1\right )}}{\tan \left (\frac{1}{2} \, t\right )^{4} + 2 \, \tan \left (\frac{1}{2} \, t\right )^{2} + 1}\right )}{2 \,{\left (\tan \left (\frac{1}{2} \, t\right )^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(t*sec(t)^2,t, algorithm="giac")
[Out]