Optimal. Leaf size=33 \[ -\frac{1}{3 (\cos (3 x)+1)}-\frac{1}{3} \log (\cos (3 x))+\frac{1}{3} \log (\cos (3 x)+1) \]
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Rubi [A] time = 0.0359734, antiderivative size = 33, normalized size of antiderivative = 1., 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.
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Rule 2707
Rule 44
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 = 0.0709533, 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.
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Maple [A] time = 0.05, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{3+3\,\cos \left ( 3\,x \right ) }}-{\frac{\ln \left ( \cos \left ( 3\,x \right ) \right ) }{3}}+{\frac{\ln \left ( 1+\cos \left ( 3\,x \right ) \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12507, size = 36, normalized size = 1.09 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41435, size = 132, normalized size = 4. \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan{\left (3 x \right )}}{\left (\cos{\left (3 x \right )} + 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46617, size = 53, normalized size = 1.61 \begin{align*} \frac{\cos \left (3 \, x\right ) - 1}{6 \,{\left (\cos \left (3 \, x\right ) + 1\right )}} - \frac{1}{3} \, \log \left ({\left | -\frac{\cos \left (3 \, x\right ) - 1}{\cos \left (3 \, x\right ) + 1} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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