Optimal. Leaf size=22 \[ -\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (3 x+2)\right )}{3 \sqrt{2}} \]
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Rubi [A] time = 0.0386269, 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} \[ -\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (3 x+2)\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(2+3 x)}{2-\coth ^2(2+3 x)} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\coth (2+3 x)\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (2+3 x)\right )}{3 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.100644, 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 = 0.052, size = 44, normalized size = 2. \begin{align*} -{\frac{\sqrt{2}}{6}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( 1+3/2\,x \right ) -2 \right ) } \right ) }-{\frac{\sqrt{2}}{6}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( 1+3/2\,x \right ) +2 \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.60329, size = 93, normalized size = 4.23 \begin{align*} -\frac{1}{12} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-3 \, x - 2\right )} + 1}{\sqrt{2} + e^{\left (-3 \, x - 2\right )} - 1}\right ) + \frac{1}{12} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-3 \, x - 2\right )} - 1}{\sqrt{2} + e^{\left (-3 \, x - 2\right )} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11168, 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 )}}{\coth ^{2}{\left (3 x + 2 \right )} - 2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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