Optimal. Leaf size=22 \[ -\frac{\tanh ^{-1}\left (\frac{\tanh (3 x+2)}{\sqrt{2}}\right )}{3 \sqrt{2}} \]
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Rubi [A] time = 0.0394867, antiderivative size = 22, 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} \[ -\frac{\tanh ^{-1}\left (\frac{\tanh (3 x+2)}{\sqrt{2}}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(2+3 x)}{1-2 \coth ^2(2+3 x)} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\coth (2+3 x)\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\tanh (2+3 x)}{\sqrt{2}}\right )}{3 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.104643, size = 42, normalized size = 1.91 \[ \frac{\tanh ^{-1}\left (\frac{\left (1-6 e^4+e^8\right ) \tanh (3 x)+e^8-1}{4 \sqrt{2} e^4}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 102, normalized size = 4.6 \begin{align*} -{\frac{\sqrt{2}}{24}\ln \left ({ \left ( \left ( \tanh \left ( 1+{\frac{3\,x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ( 1+{\frac{3\,x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ( 1+{\frac{3\,x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ( 1+{\frac{3\,x}{2}} \right ) +1 \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{24}\ln \left ({ \left ( \left ( \tanh \left ( 1+{\frac{3\,x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ( 1+{\frac{3\,x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ( 1+{\frac{3\,x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ( 1+{\frac{3\,x}{2}} \right ) +1 \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.67463, size = 51, normalized size = 2.32 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (-6 \, x - 4\right )} - 3}{2 \, \sqrt{2} + e^{\left (-6 \, x - 4\right )} + 3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.09315, size = 263, normalized size = 11.95 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (\frac{3 \,{\left (2 \, \sqrt{2} + 3\right )} \cosh \left (3 \, x + 2\right )^{2} - 4 \,{\left (3 \, \sqrt{2} + 4\right )} \cosh \left (3 \, x + 2\right ) \sinh \left (3 \, x + 2\right ) + 3 \,{\left (2 \, \sqrt{2} + 3\right )} \sinh \left (3 \, x + 2\right )^{2} + 2 \, \sqrt{2} + 3}{\cosh \left (3 \, x + 2\right )^{2} + \sinh \left (3 \, x + 2\right )^{2} + 3}\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{\operatorname{csch}^{2}{\left (3 x + 2 \right )}}{2 \coth ^{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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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