Optimal. Leaf size=9 \[ -\frac{1}{2} \log ^2(\cos (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0148105, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3475, 4339, 2301} \[ -\frac{1}{2} \log ^2(\cos (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 4339
Rule 2301
Rubi steps
\begin{align*} \int \log (\cos (x)) \tan (x) \, dx &=-\operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{2} \log ^2(\cos (x))\\ \end{align*}
Mathematica [A] time = 0.003192, size = 9, normalized size = 1. \[ -\frac{1}{2} \log ^2(\cos (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 8, normalized size = 0.9 \begin{align*} -{\frac{ \left ( \ln \left ( \cos \left ( x \right ) \right ) \right ) ^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.994473, size = 9, normalized size = 1. \begin{align*} -\frac{1}{2} \, \log \left (\cos \left (x\right )\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.15888, size = 27, normalized size = 3. \begin{align*} -\frac{1}{2} \, \log \left (\cos \left (x\right )\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 7.45885, size = 8, normalized size = 0.89 \begin{align*} - \frac{\log{\left (\cos{\left (x \right )} \right )}^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25589, size = 9, normalized size = 1. \begin{align*} -\frac{1}{2} \, \log \left (\cos \left (x\right )\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]