3.301 \(\int \csc (x) \log (\tan (x)) \sec (x) \, dx\)

Optimal. Leaf size=9 \[ \frac{1}{2} \log ^2(\tan (x)) \]

[Out]

Log[Tan[x]]^2/2

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Rubi [A]  time = 0.0329528, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{1}{2} \log ^2(\tan (x)) \]

Antiderivative was successfully verified.

[In]  Int[Csc[x]*Log[Tan[x]]*Sec[x],x]

[Out]

Log[Tan[x]]^2/2

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Rubi in Sympy [A]  time = 9.2886, size = 48, normalized size = 5.33 \[ - \frac{\left (\log{\left (- \cos{\left (2 x \right )} + 1 \right )} - \log{\left (\cos{\left (2 x \right )} + 1 \right )}\right )^{2}}{8} - \frac{\log{\left (- \sin ^{2}{\left (x \right )} + 1 \right )} \log{\left (\tan{\left (x \right )} \right )}}{2} + \frac{\log{\left (\sin ^{2}{\left (x \right )} \right )} \log{\left (\tan{\left (x \right )} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(tan(x))/cos(x)/sin(x),x)

[Out]

-(log(-cos(2*x) + 1) - log(cos(2*x) + 1))**2/8 - log(-sin(x)**2 + 1)*log(tan(x))
/2 + log(sin(x)**2)*log(tan(x))/2

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Mathematica [A]  time = 0.00588257, size = 9, normalized size = 1. \[ \frac{1}{2} \log ^2(\tan (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[x]*Log[Tan[x]]*Sec[x],x]

[Out]

Log[Tan[x]]^2/2

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Maple [A]  time = 0.024, size = 8, normalized size = 0.9 \[{\frac{ \left ( \ln \left ( \tan \left ( x \right ) \right ) \right ) ^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(tan(x))/cos(x)/sin(x),x)

[Out]

1/2*ln(tan(x))^2

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Maxima [A]  time = 1.34224, size = 9, normalized size = 1. \[ \frac{1}{2} \, \log \left (\tan \left (x\right )\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(tan(x))/(cos(x)*sin(x)),x, algorithm="maxima")

[Out]

1/2*log(tan(x))^2

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Fricas [A]  time = 0.223696, size = 16, normalized size = 1.78 \[ \frac{1}{2} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(tan(x))/(cos(x)*sin(x)),x, algorithm="fricas")

[Out]

1/2*log(sin(x)/cos(x))^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\log{\left (\tan{\left (x \right )} \right )}}{\sin{\left (x \right )} \cos{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(tan(x))/cos(x)/sin(x),x)

[Out]

Integral(log(tan(x))/(sin(x)*cos(x)), x)

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GIAC/XCAS [A]  time = 0.205264, size = 9, normalized size = 1. \[ \frac{1}{2} \,{\rm ln}\left (\tan \left (x\right )\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(tan(x))/(cos(x)*sin(x)),x, algorithm="giac")

[Out]

1/2*ln(tan(x))^2