Optimal. Leaf size=22 \[ \frac{\tan ^{-1}\left (\sqrt{2} \tanh (3 x+2)\right )}{3 \sqrt{2}} \]
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Rubi [A] time = 0.0155441, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {12, 2659, 206} \[ \frac{\tan ^{-1}\left (\sqrt{2} \tanh (3 x+2)\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2659
Rule 206
Rubi steps
\begin{align*} \int \frac{2}{-1+3 \cosh (4+6 x)} \, dx &=2 \int \frac{1}{-1+3 \cosh (4+6 x)} \, dx\\ &=-\left (\frac{2}{3} i \operatorname{Subst}\left (\int \frac{1}{2-4 x^2} \, dx,x,\tan \left (\frac{1}{2} (4 i+6 i x)\right )\right )\right )\\ &=\frac{\tan ^{-1}\left (\sqrt{2} \tanh (2+3 x)\right )}{3 \sqrt{2}}\\ \end{align*}
Mathematica [B] time = 0.0704974, size = 47, normalized size = 2.14 \[ \frac{\tan ^{-1}\left (\frac{\left (3+2 e^4+3 e^8\right ) \tanh (3 x)+3 \left (e^8-1\right )}{4 \sqrt{2} e^4}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 17, normalized size = 0.8 \begin{align*}{\frac{\arctan \left ( \sqrt{2}\tanh \left ( 2+3\,x \right ) \right ) \sqrt{2}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48378, size = 28, normalized size = 1.27 \begin{align*} -\frac{1}{6} \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (3 \, e^{\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 [B] time = 2.04042, size = 120, normalized size = 5.45 \begin{align*} \frac{1}{6} \, \sqrt{2} \arctan \left (\frac{3}{4} \, \sqrt{2} \cosh \left (6 \, x + 4\right ) + \frac{3}{4} \, \sqrt{2} \sinh \left (6 \, x + 4\right ) - \frac{1}{4} \, \sqrt{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.322538, size = 19, normalized size = 0.86 \begin{align*} \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \tanh{\left (3 x + 2 \right )} \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.186, size = 28, normalized size = 1.27 \begin{align*} \frac{1}{6} \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (3 \, e^{\left (6 \, x + 4\right )} - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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