Optimal. Leaf size=61 \[ \frac{\log \left (\sqrt{2} \cos (3 x+2)-\sin (3 x+2)\right )}{6 \sqrt{2}}-\frac{\log \left (\sin (3 x+2)+\sqrt{2} \cos (3 x+2)\right )}{6 \sqrt{2}} \]
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Rubi [A] time = 0.046606, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3675, 206} \[ \frac{\log \left (\sqrt{2} \cos (3 x+2)-\sin (3 x+2)\right )}{6 \sqrt{2}}-\frac{\log \left (\sin (3 x+2)+\sqrt{2} \cos (3 x+2)\right )}{6 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 206
Rubi steps
\begin{align*} \int \frac{\csc ^2(2+3 x)}{1-2 \cot ^2(2+3 x)} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\cot (2+3 x)\right )\right )\\ &=\frac{\log \left (\sqrt{2} \cos (2+3 x)-\sin (2+3 x)\right )}{6 \sqrt{2}}-\frac{\log \left (\sqrt{2} \cos (2+3 x)+\sin (2+3 x)\right )}{6 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0349107, size = 22, normalized size = 0.36 \[ -\frac{\tanh ^{-1}\left (\frac{\tan (3 x+2)}{\sqrt{2}}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 18, normalized size = 0.3 \begin{align*} -{\frac{\sqrt{2}}{6}{\it Artanh} \left ({\frac{\tan \left ( 2+3\,x \right ) \sqrt{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66734, size = 43, normalized size = 0.7 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \tan \left (3 \, x + 2\right )}{\sqrt{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 = 1.79598, size = 230, normalized size = 3.77 \begin{align*} \frac{1}{24} \, \sqrt{2} \log \left (-\frac{7 \, \cos \left (3 \, x + 2\right )^{4} - 10 \, \cos \left (3 \, x + 2\right )^{2} + 4 \,{\left (\sqrt{2} \cos \left (3 \, x + 2\right )^{3} + \sqrt{2} \cos \left (3 \, x + 2\right )\right )} \sin \left (3 \, x + 2\right ) - 1}{9 \, \cos \left (3 \, x + 2\right )^{4} - 6 \, \cos \left (3 \, x + 2\right )^{2} + 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{\csc ^{2}{\left (3 x + 2 \right )}}{2 \cot ^{2}{\left (3 x + 2 \right )} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.51047, size = 53, normalized size = 0.87 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \tan \left (3 \, x + 2\right ) \right |}}{{\left | 2 \, \sqrt{2} + 2 \, \tan \left (3 \, x + 2\right ) \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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