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 = 0.0457299, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {12, 3166, 2659, 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 12
Rule 3166
Rule 2659
Rule 206
Rubi steps
\begin{align*} \int -\frac{2 \csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx &=-\left (2 \int \frac{\csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx\right )\\ &=-\left (2 \int \frac{1}{1+3 \cos (4+6 x)} \, dx\right )\\ &=-\left (\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{4-2 x^2} \, dx,x,\tan \left (\frac{1}{2} (4+6 x)\right )\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 = 0.0348746, 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 = 0.079, 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 = 1.49522, size = 72, normalized size = 1.18 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{\sin \left (6 \, x + 4\right )}{\cos \left (6 \, x + 4\right ) + 1}}{\sqrt{2} + \frac{\sin \left (6 \, x + 4\right )}{\cos \left (6 \, x + 4\right ) + 1}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75495, size = 207, normalized size = 3.39 \begin{align*} \frac{1}{24} \, \sqrt{2} \log \left (-\frac{7 \, \cos \left (6 \, x + 4\right )^{2} + 4 \,{\left (\sqrt{2} \cos \left (6 \, x + 4\right ) + 3 \, \sqrt{2}\right )} \sin \left (6 \, x + 4\right ) - 6 \, \cos \left (6 \, x + 4\right ) - 17}{9 \, \cos \left (6 \, x + 4\right )^{2} + 6 \, \cos \left (6 \, x + 4\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*} - 2 \int \frac{\csc{\left (6 x + 4 \right )}}{3 \cot{\left (6 x + 4 \right )} + \csc{\left (6 x + 4 \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31549, 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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