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 = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 206
Rule 3675
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*}
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Mathematica [A] time = 0.11, 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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fricas [B] time = 0.47, size = 89, normalized size = 4.05 \[ \frac {1}{12} \, \sqrt {2} \log \left (\frac {3 \, {\left (2 \, \sqrt {2} + 3\right )} \cosh \left (3 \, x + 2\right )^{2} - 4 \, {\left (3 \, \sqrt {2} + 4\right )} \cosh \left (3 \, x + 2\right ) \sinh \left (3 \, x + 2\right ) + 3 \, {\left (2 \, \sqrt {2} + 3\right )} \sinh \left (3 \, x + 2\right )^{2} - 2 \, \sqrt {2} - 3}{\cosh \left (3 \, x + 2\right )^{2} + \sinh \left (3 \, x + 2\right )^{2} - 3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.45, size = 44, normalized size = 2.00 \[ -\frac {\sqrt {2}\, \arctanh \left (\frac {\left (-2+2 \tanh \left (1+\frac {3 x}{2}\right )\right ) \sqrt {2}}{4}\right )}{6}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (1+\frac {3 x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 69, normalized size = 3.14 \[ -\frac {1}{12} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-3 \, x - 2\right )} + 1}{\sqrt {2} + e^{\left (-3 \, x - 2\right )} - 1}\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-3 \, x - 2\right )} - 1}{\sqrt {2} + e^{\left (-3 \, x - 2\right )} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 57, normalized size = 2.59 \[ \frac {\sqrt {2}\,\left (\ln \left (\sqrt {2}\,\left (3\,{\mathrm {e}}^{6\,x+4}-1\right )-4\,{\mathrm {e}}^{6\,x+4}\right )-\ln \left (-4\,{\mathrm {e}}^{6\,x+4}-\sqrt {2}\,\left (3\,{\mathrm {e}}^{6\,x+4}-1\right )\right )\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\operatorname {csch}^{2}{\left (3 x + 2 \right )}}{\coth ^{2}{\left (3 x + 2 \right )} - 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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