Optimal. Leaf size=33 \[ -\frac {1}{3 (\cos (3 x)+1)}-\frac {1}{3} \log (\cos (3 x))+\frac {1}{3} \log (\cos (3 x)+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2707, 44} \[ -\frac {1}{3 (\cos (3 x)+1)}-\frac {1}{3} \log (\cos (3 x))+\frac {1}{3} \log (\cos (3 x)+1) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 2707
Rubi steps
\begin {align*} \int \frac {\tan (3 x)}{(1+\cos (3 x))^2} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^2} \, dx,x,\cos (3 x)\right )\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^2}\right ) \, dx,x,\cos (3 x)\right )\right )\\ &=-\frac {1}{3 (1+\cos (3 x))}-\frac {1}{3} \log (\cos (3 x))+\frac {1}{3} \log (1+\cos (3 x))\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 49, normalized size = 1.48 \[ \frac {\cos ^4\left (\frac {3 x}{2}\right ) \left (8 \log \left (\cos \left (\frac {3 x}{2}\right )\right )-4 \log (\cos (3 x))\right )-2 \cos ^2\left (\frac {3 x}{2}\right )}{3 (\cos (3 x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.71, size = 43, normalized size = 1.30 \[ -\frac {{\left (\cos \left (3 \, x\right ) + 1\right )} \log \left (-\cos \left (3 \, x\right )\right ) - {\left (\cos \left (3 \, x\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (3 \, x\right ) + \frac {1}{2}\right ) + 1}{3 \, {\left (\cos \left (3 \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.57, size = 28, normalized size = 0.85 \[ -\frac {1}{3 \, {\left (\cos \left (3 \, x\right ) + 1\right )}} + \frac {1}{3} \, \log \left (\cos \left (3 \, x\right ) + 1\right ) - \frac {1}{3} \, \log \left ({\left | \cos \left (3 \, x\right ) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 28, normalized size = 0.85 \[ -\frac {1}{3 \left (1+\cos \left (3 x \right )\right )}-\frac {\ln \left (\cos \left (3 x \right )\right )}{3}+\frac {\ln \left (1+\cos \left (3 x \right )\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.57, size = 27, normalized size = 0.82 \[ -\frac {1}{3 \, {\left (\cos \left (3 \, x\right ) + 1\right )}} + \frac {1}{3} \, \log \left (\cos \left (3 \, x\right ) + 1\right ) - \frac {1}{3} \, \log \left (\cos \left (3 \, x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.37, size = 20, normalized size = 0.61 \[ -\frac {\ln \left ({\mathrm {tan}\left (\frac {3\,x}{2}\right )}^2-1\right )}{3}-\frac {{\mathrm {tan}\left (\frac {3\,x}{2}\right )}^2}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan {\left (3 x \right )}}{\left (\cos {\left (3 x \right )} + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________