Optimal. Leaf size=60 \[ \frac{\log \left (\cos (3 x+2)-\sqrt{2} \sin (3 x+2)\right )}{6 \sqrt{2}}-\frac{\log \left (\sqrt{2} \sin (3 x+2)+\cos (3 x+2)\right )}{6 \sqrt{2}} \]
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Rubi [A] time = 0.0304304, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {207} \[ \frac{\log \left (\cos (3 x+2)-\sqrt{2} \sin (3 x+2)\right )}{6 \sqrt{2}}-\frac{\log \left (\sqrt{2} \sin (3 x+2)+\cos (3 x+2)\right )}{6 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 207
Rubi steps
\begin{align*} \int \frac{1}{-\cos ^2(2+3 x)+2 \sin ^2(2+3 x)} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+2 x^2} \, dx,x,\tan (2+3 x)\right )\\ &=\frac{\log \left (\cos (2+3 x)-\sqrt{2} \sin (2+3 x)\right )}{6 \sqrt{2}}-\frac{\log \left (\cos (2+3 x)+\sqrt{2} \sin (2+3 x)\right )}{6 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0319567, size = 22, normalized size = 0.37 \[ -\frac{\tanh ^{-1}\left (\sqrt{2} \tan (3 x+2)\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 17, normalized size = 0.3 \begin{align*} -{\frac{\sqrt{2}{\it Artanh} \left ( \tan \left ( 2+3\,x \right ) \sqrt{2} \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60005, size = 46, normalized size = 0.77 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - 2 \, \tan \left (3 \, x + 2\right )}{\sqrt{2} + 2 \, \tan \left (3 \, x + 2\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40656, size = 232, normalized size = 3.87 \begin{align*} \frac{1}{24} \, \sqrt{2} \log \left (-\frac{7 \, \cos \left (3 \, x + 2\right )^{4} - 4 \, \cos \left (3 \, x + 2\right )^{2} - 4 \,{\left (\sqrt{2} \cos \left (3 \, x + 2\right )^{3} - 2 \, \sqrt{2} \cos \left (3 \, x + 2\right )\right )} \sin \left (3 \, x + 2\right ) - 4}{9 \, \cos \left (3 \, x + 2\right )^{4} - 12 \, \cos \left (3 \, x + 2\right )^{2} + 4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22547, size = 53, normalized size = 0.88 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 4 \, \tan \left (3 \, x + 2\right ) \right |}}{{\left | 2 \, \sqrt{2} + 4 \, \tan \left (3 \, x + 2\right ) \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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