Optimal. Leaf size=12 \[ \frac {\tan ^2(x)}{2}+\log (\cos (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 3475} \[ \frac {\tan ^2(x)}{2}+\log (\cos (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3473
Rule 3475
Rubi steps
\begin {align*} \int \tan ^3(x) \, dx &=\frac {\tan ^2(x)}{2}-\int \tan (x) \, dx\\ &=\log (\cos (x))+\frac {\tan ^2(x)}{2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 12, normalized size = 1.00 \[ \frac {\sec ^2(x)}{2}+\log (\cos (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 18, normalized size = 1.50 \[ \frac {1}{2} \, \tan \relax (x)^{2} + \frac {1}{2} \, \log \left (\frac {1}{\tan \relax (x)^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.03, size = 16, normalized size = 1.33 \[ \frac {1}{2} \, \tan \relax (x)^{2} - \frac {1}{2} \, \log \left (\tan \relax (x)^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 17, normalized size = 1.42 \[ \frac {\left (\tan ^{2}\relax (x )\right )}{2}-\frac {\ln \left (\tan ^{2}\relax (x )+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 20, normalized size = 1.67 \[ -\frac {1}{2 \, {\left (\sin \relax (x)^{2} - 1\right )}} + \frac {1}{2} \, \log \left (\sin \relax (x)^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.02, size = 16, normalized size = 1.33 \[ \ln \left (\cos \relax (x)\right )-\frac {{\cos \relax (x)}^2-1}{2\,{\cos \relax (x)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 12, normalized size = 1.00 \[ \log {\left (\cos {\relax (x )} \right )} + \frac {1}{2 \cos ^{2}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________