Optimal. Leaf size=20 \[ \frac {1}{2 (1-\cosh (x))}-\frac {1}{2} \tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.07, antiderivative size = 24, normalized size of antiderivative = 1.20, number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4397, 2706, 2606, 30, 2611, 3770} \[ -\frac {1}{2} \text {csch}^2(x)-\frac {1}{2} \tanh ^{-1}(\cosh (x))-\frac {1}{2} \coth (x) \text {csch}(x) \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2611
Rule 2706
Rule 3770
Rule 4397
Rubi steps
\begin {align*} \int \frac {1}{\sinh (x)-\tanh (x)} \, dx &=-\left (i \int \frac {\coth (x)}{i-i \cosh (x)} \, dx\right )\\ &=\int \coth ^2(x) \text {csch}(x) \, dx+\int \coth (x) \text {csch}^2(x) \, dx\\ &=-\frac {1}{2} \coth (x) \text {csch}(x)+\frac {1}{2} \int \text {csch}(x) \, dx+\operatorname {Subst}(\int x \, dx,x,-i \text {csch}(x))\\ &=-\frac {1}{2} \tanh ^{-1}(\cosh (x))-\frac {1}{2} \coth (x) \text {csch}(x)-\frac {\text {csch}^2(x)}{2}\\ \end {align*}
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Mathematica [B] time = 0.06, size = 50, normalized size = 2.50 \[ -\frac {1}{4} \text {csch}^2\left (\frac {x}{2}\right ) \left (\log \left (\sinh \left (\frac {x}{2}\right )\right )-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\cosh (x) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )+1\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 96, normalized size = 4.80 \[ -\frac {{\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 2 \, \cosh \relax (x) + 2 \, \sinh \relax (x)}{2 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 43, normalized size = 2.15 \[ -\frac {e^{\left (-x\right )} + e^{x} + 2}{4 \, {\left (e^{\left (-x\right )} + e^{x} - 2\right )}} - \frac {1}{4} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac {1}{4} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 17, normalized size = 0.85 \[ -\frac {1}{4 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 40, normalized size = 2.00 \[ \frac {e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} - 1} - \frac {1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 41, normalized size = 2.05 \[ \frac {\ln \left (1-{\mathrm {e}}^x\right )}{2}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{2}-\frac {1}{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1}-\frac {1}{{\mathrm {e}}^x-1} \]
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 \[ \int \frac {1}{\sinh {\relax (x )} - \tanh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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