Optimal. Leaf size=19 \[ \frac{1}{3} x \tan (3 x)+\frac{1}{9} \log (\cos (3 x)) \]
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Rubi [A] time = 0.0185204, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4184, 3475} \[ \frac{1}{3} x \tan (3 x)+\frac{1}{9} \log (\cos (3 x)) \]
Antiderivative was successfully verified.
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Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int x \sec ^2(3 x) \, dx &=\frac{1}{3} x \tan (3 x)-\frac{1}{3} \int \tan (3 x) \, dx\\ &=\frac{1}{9} \log (\cos (3 x))+\frac{1}{3} x \tan (3 x)\\ \end{align*}
Mathematica [A] time = 0.0084289, size = 19, normalized size = 1. \[ \frac{1}{3} x \tan (3 x)+\frac{1}{9} \log (\cos (3 x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 16, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( \cos \left ( 3\,x \right ) \right ) }{9}}+{\frac{x\tan \left ( 3\,x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.46553, size = 100, normalized size = 5.26 \begin{align*} \frac{{\left (\cos \left (6 \, x\right )^{2} + \sin \left (6 \, x\right )^{2} + 2 \, \cos \left (6 \, x\right ) + 1\right )} \log \left (\cos \left (6 \, x\right )^{2} + \sin \left (6 \, x\right )^{2} + 2 \, \cos \left (6 \, x\right ) + 1\right ) + 12 \, x \sin \left (6 \, x\right )}{18 \,{\left (\cos \left (6 \, x\right )^{2} + \sin \left (6 \, x\right )^{2} + 2 \, \cos \left (6 \, x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00315, size = 74, normalized size = 3.89 \begin{align*} \frac{\cos \left (3 \, x\right ) \log \left (-\cos \left (3 \, x\right )\right ) + 3 \, x \sin \left (3 \, x\right )}{9 \, \cos \left (3 \, x\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 x \sec ^{2}{\left (3 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14042, size = 139, normalized size = 7.32 \begin{align*} \frac{\log \left (\frac{4 \,{\left (\tan \left (\frac{3}{2} \, x\right )^{4} - 2 \, \tan \left (\frac{3}{2} \, x\right )^{2} + 1\right )}}{\tan \left (\frac{3}{2} \, x\right )^{4} + 2 \, \tan \left (\frac{3}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac{3}{2} \, x\right )^{2} - 12 \, x \tan \left (\frac{3}{2} \, x\right ) - \log \left (\frac{4 \,{\left (\tan \left (\frac{3}{2} \, x\right )^{4} - 2 \, \tan \left (\frac{3}{2} \, x\right )^{2} + 1\right )}}{\tan \left (\frac{3}{2} \, x\right )^{4} + 2 \, \tan \left (\frac{3}{2} \, x\right )^{2} + 1}\right )}{18 \,{\left (\tan \left (\frac{3}{2} \, x\right )^{2} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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