Optimal. Leaf size=20 \[ \frac{1}{2 (1-\cosh (x))}-\frac{1}{2} \tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.0732545, antiderivative size = 24, normalized size of antiderivative = 1.2, 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 4397
Rule 2706
Rule 2606
Rule 30
Rule 2611
Rule 3770
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*}
Mathematica [B] time = 0.0431123, size = 50, normalized size = 2.5 \[ -\frac{1}{4} \text{csch}^2\left (\frac{x}{2}\right ) \left (\log \left (\sinh \left (\frac{x}{2}\right )\right )-\log \left (\cosh \left (\frac{x}{2}\right )\right )+\cosh (x) \left (\log \left (\cosh \left (\frac{x}{2}\right )\right )-\log \left (\sinh \left (\frac{x}{2}\right )\right )\right )+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 17, normalized size = 0.9 \begin{align*} -{\frac{1}{4} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.22083, size = 54, normalized size = 2.7 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.69748, size = 386, normalized size = 19.3 \begin{align*} -\frac{{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) -{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )}{2 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\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{1}{\sinh{\left (x \right )} - \tanh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11335, size = 58, normalized size = 2.9 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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