Optimal. Leaf size=22 \[ \frac {\tan ^{-1}\left (\frac {\tanh (3 x+2)}{\sqrt {2}}\right )}{3 \sqrt {2}} \]
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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, 203} \[ \frac {\tan ^{-1}\left (\frac {\tanh (3 x+2)}{\sqrt {2}}\right )}{3 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3675
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 {\tan ^{-1}\left (\frac {\tanh (2+3 x)}{\sqrt {2}}\right )}{3 \sqrt {2}}\\ \end {align*}
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Mathematica [B] time = 0.11, 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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fricas [B] time = 0.42, size = 47, normalized size = 2.14 \[ -\frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {2 \, \sqrt {2} \cosh \left (3 \, x + 2\right ) + \sqrt {2} \sinh \left (3 \, x + 2\right )}{2 \, {\left (\cosh \left (3 \, x + 2\right ) - \sinh \left (3 \, x + 2\right )\right )}}\right ) \]
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 = 0.54, size = 132, normalized size = 6.00 \[ \frac {\sqrt {3}\, \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{\sqrt {6}-\sqrt {2}}\right )}{3 \sqrt {6}-3 \sqrt {2}}-\frac {\arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{\sqrt {6}-\sqrt {2}}\right )}{3 \left (\sqrt {6}-\sqrt {2}\right )}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{\sqrt {6}+\sqrt {2}}\right )}{3 \left (\sqrt {6}+\sqrt {2}\right )}-\frac {\arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{\sqrt {6}+\sqrt {2}}\right )}{3 \left (\sqrt {6}+\sqrt {2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 21, normalized size = 0.95 \[ -\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, e^{\left (-6 \, x - 4\right )} + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 21, normalized size = 0.95 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (3\,{\mathrm {e}}^{6\,x+4}+1\right )}{4}\right )}{6} \]
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 \[ \int \frac {\operatorname {csch}^{2}{\left (3 x + 2 \right )}}{2 \coth ^{2}{\left (3 x + 2 \right )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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