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 = 0.05, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 204
Rule 618
Rule 3124
Rule 3166
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.41, size = 17, normalized size = 0.89 \[ \frac {1}{3} \, x - \frac {1}{3} \, \log \left (2 \, \cosh \relax (x) + 2 \, \sinh \relax (x) + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 13, normalized size = 0.68 \[ \frac {1}{3} \, x - \frac {1}{3} \, \log \left (2 \, e^{x} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 20, normalized size = 1.05 \[ \frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{3}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-5\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 11, normalized size = 0.58 \[ -\frac {1}{3} \, \log \left (3 \, e^{\left (-x\right )} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 11, normalized size = 0.58 \[ \frac {x}{3}-\frac {\ln \left ({\mathrm {e}}^x+\frac {3}{2}\right )}{3} \]
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 \[ \int \frac {\operatorname {csch}{\relax (x )}}{2 \coth {\relax (x )} + 3 \operatorname {csch}{\relax (x )} + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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