3.29 \(\int t \sec ^2(t) \, dt\)

Optimal. Leaf size=8 \[ t \tan (t)+\log (\cos (t)) \]

[Out]

Log[Cos[t]] + t*Tan[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]

[Out]

Log[Cos[t]] + t*Tan[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)

[Out]

Integral(t/cos(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]

[Out]

Log[Cos[t]] + t*Tan[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]

ln(cos(t))+t*tan(t)

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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")

[Out]

1/2*((cos(2*t)^2 + sin(2*t)^2 + 2*cos(2*t) + 1)*log(cos(2*t)^2 + sin(2*t)^2 + 2*
cos(2*t) + 1) + 4*t*sin(2*t))/(cos(2*t)^2 + sin(2*t)^2 + 2*cos(2*t) + 1)

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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]

(cos(t)*log(-cos(t)) + t*sin(t))/cos(t)

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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]

Integral(t*sec(t)**2, t)

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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]

1/2*(ln(4*(tan(1/2*t)^4 - 2*tan(1/2*t)^2 + 1)/(tan(1/2*t)^4 + 2*tan(1/2*t)^2 + 1
))*tan(1/2*t)^2 - 4*t*tan(1/2*t) - ln(4*(tan(1/2*t)^4 - 2*tan(1/2*t)^2 + 1)/(tan
(1/2*t)^4 + 2*tan(1/2*t)^2 + 1)))/(tan(1/2*t)^2 - 1)