Optimal. Leaf size=42 \[ \frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sin (6 x+4)}{\cos (6 x+4)+2 \sqrt{2}+3}\right )}{3 \sqrt{2}} \]
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Rubi [A] time = 0.0370735, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {12, 3166, 2657} \[ \frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sin (6 x+4)}{\cos (6 x+4)+2 \sqrt{2}+3}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3166
Rule 2657
Rubi steps
\begin{align*} \int \frac{2 \csc (4+6 x)}{\cot (4+6 x)+3 \csc (4+6 x)} \, dx &=2 \int \frac{\csc (4+6 x)}{\cot (4+6 x)+3 \csc (4+6 x)} \, dx\\ &=2 \int \frac{1}{3+\cos (4+6 x)} \, dx\\ &=\frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sin (4+6 x)}{3+2 \sqrt{2}+\cos (4+6 x)}\right )}{3 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0220547, size = 22, normalized size = 0.52 \[ \frac{\tan ^{-1}\left (\frac{\tan (3 x+2)}{\sqrt{2}}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 18, normalized size = 0.4 \begin{align*}{\frac{\sqrt{2}}{6}\arctan \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 = 1.74898, size = 36, normalized size = 0.86 \begin{align*} \frac{1}{6} \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sin \left (6 \, x + 4\right )}{2 \,{\left (\cos \left (6 \, x + 4\right ) + 1\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84853, size = 101, normalized size = 2.4 \begin{align*} -\frac{1}{12} \, \sqrt{2} \arctan \left (\frac{3 \, \sqrt{2} \cos \left (6 \, x + 4\right ) + \sqrt{2}}{4 \, \sin \left (6 \, x + 4\right )}\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 )}}{\cot{\left (6 x + 4 \right )} + 3 \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 = 1.1875, size = 77, normalized size = 1.83 \begin{align*} \frac{1}{6} \, \sqrt{2}{\left (3 \, x + \arctan \left (-\frac{\sqrt{2} \sin \left (6 \, x + 4\right ) - \sin \left (6 \, x + 4\right )}{\sqrt{2} \cos \left (6 \, x + 4\right ) + \sqrt{2} - \cos \left (6 \, x + 4\right ) + 1}\right ) + 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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