Optimal. Leaf size=19 \[ -\frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \left (2-\tanh \left (\frac{x}{2}\right )\right )\right ) \]
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Rubi [A] time = 0.0502114, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3166, 3124, 618, 204} \[ -\frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \left (2-\tanh \left (\frac{x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Rule 3166
Rule 3124
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\text{csch}(x)}{2+2 \coth (x)+3 \text{csch}(x)} \, dx &=i \int \frac{1}{3 i+2 i \cosh (x)+2 i \sinh (x)} \, dx\\ &=2 i \operatorname{Subst}\left (\int \frac{1}{5 i+4 i x-i x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=-\left (4 i \operatorname{Subst}\left (\int \frac{1}{-36-x^2} \, dx,x,4 i-2 i \tanh \left (\frac{x}{2}\right )\right )\right )\\ &=-\frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \left (2-\tanh \left (\frac{x}{2}\right )\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0685395, size = 28, normalized size = 1.47 \[ \frac{x}{6}-\frac{1}{3} \log \left (5 \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 20, normalized size = 1.1 \begin{align*}{\frac{1}{3}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{3}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -5 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09324, size = 15, normalized size = 0.79 \begin{align*} -\frac{1}{3} \, \log \left (3 \, e^{\left (-x\right )} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31758, size = 59, normalized size = 3.11 \begin{align*} \frac{1}{3} \, x - \frac{1}{3} \, \log \left (2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right ) + 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}{\left (x \right )}}{2 \coth{\left (x \right )} + 3 \operatorname{csch}{\left (x \right )} + 2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15271, size = 18, normalized size = 0.95 \begin{align*} \frac{1}{3} \, x - \frac{1}{3} \, \log \left (2 \, e^{x} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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