3.81 \(\int \log (\cos (x)) \sec ^2(x) \, dx\)

Optimal. Leaf size=12 \[ -x+\tan (x)+\tan (x) \log (\cos (x)) \]

[Out]

-x + Tan[x] + Log[Cos[x]]*Tan[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0333726, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -x+\tan (x)+\tan (x) \log (\cos (x)) \]

Antiderivative was successfully verified.

[In]  Int[Log[Cos[x]]*Sec[x]^2,x]

[Out]

-x + Tan[x] + Log[Cos[x]]*Tan[x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.67949, size = 15, normalized size = 1.25 \[ - x + \frac{\log{\left (\cos{\left (x \right )} \right )} \sin{\left (x \right )}}{\cos{\left (x \right )}} + \tan{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(cos(x))*sec(x)**2,x)

[Out]

-x + log(cos(x))*sin(x)/cos(x) + tan(x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0108513, size = 12, normalized size = 1. \[ -x+\tan (x)+\tan (x) \log (\cos (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Log[Cos[x]]*Sec[x]^2,x]

[Out]

-x + Tan[x] + Log[Cos[x]]*Tan[x]

_______________________________________________________________________________________

Maple [C]  time = 0.062, size = 61, normalized size = 5.1 \[{\frac{-2\,i{{\rm e}^{2\,ix}}\ln \left ( 2\,\cos \left ( x \right ) \right ) }{1+{{\rm e}^{2\,ix}}}}+{\frac{2\,i}{1+{{\rm e}^{2\,ix}}}}+i\ln \left ( 1+{{\rm e}^{2\,ix}} \right ) -{\frac{2\,i\ln \left ( 2 \right ) }{1+{{\rm e}^{2\,ix}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(cos(x))*sec(x)^2,x)

[Out]

-2*I/(1+exp(2*I*x))*exp(2*I*x)*ln(2*cos(x))+2*I/(1+exp(2*I*x))+I*ln(1+exp(2*I*x)
)-2*I*ln(2)/(1+exp(2*I*x))

_______________________________________________________________________________________

Maxima [A]  time = 1.51629, size = 127, normalized size = 10.58 \[ -\frac{2 \, \log \left (-\frac{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1}{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1}\right ) \sin \left (x\right )}{{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )}{\left (\cos \left (x\right ) + 1\right )}} - \frac{2 \, \sin \left (x\right )}{{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )}{\left (\cos \left (x\right ) + 1\right )}} - 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(cos(x))*sec(x)^2,x, algorithm="maxima")

[Out]

-2*log(-(sin(x)^2/(cos(x) + 1)^2 - 1)/(sin(x)^2/(cos(x) + 1)^2 + 1))*sin(x)/((si
n(x)^2/(cos(x) + 1)^2 - 1)*(cos(x) + 1)) - 2*sin(x)/((sin(x)^2/(cos(x) + 1)^2 -
1)*(cos(x) + 1)) - 2*arctan(sin(x)/(cos(x) + 1))

_______________________________________________________________________________________

Fricas [A]  time = 0.223666, size = 30, normalized size = 2.5 \[ -\frac{x \cos \left (x\right ) - \log \left (\cos \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(cos(x))*sec(x)^2,x, algorithm="fricas")

[Out]

-(x*cos(x) - log(cos(x))*sin(x) - sin(x))/cos(x)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(cos(x))*sec(x)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212001, size = 16, normalized size = 1.33 \[{\rm ln}\left (\cos \left (x\right )\right ) \tan \left (x\right ) - x + \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(cos(x))*sec(x)^2,x, algorithm="giac")

[Out]

ln(cos(x))*tan(x) - x + tan(x)