Optimal. Leaf size=20 \[ \frac{2 x}{\sqrt{\text{csch}(x)}}-\frac{4 \text{sech}(x)}{\text{csch}^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.166912, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6742, 5445, 3771, 2639, 2626} \[ \frac{2 x}{\sqrt{\text{csch}(x)}}-\frac{4 \text{sech}(x)}{\text{csch}^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 5445
Rule 3771
Rule 2639
Rule 2626
Rubi steps
\begin{align*} \int \sqrt{\text{csch}(x)} (x \cosh (x)-4 \text{sech}(x) \tanh (x)) \, dx &=\int \left (x \cosh (x) \sqrt{\text{csch}(x)}-\frac{4 \text{sech}^2(x)}{\sqrt{\text{csch}(x)}}\right ) \, dx\\ &=-\left (4 \int \frac{\text{sech}^2(x)}{\sqrt{\text{csch}(x)}} \, dx\right )+\int x \cosh (x) \sqrt{\text{csch}(x)} \, dx\\ &=\frac{2 x}{\sqrt{\text{csch}(x)}}-\frac{4 \text{sech}(x)}{\text{csch}^{\frac{3}{2}}(x)}\\ \end{align*}
Mathematica [A] time = 1.10116, size = 17, normalized size = 0.85 \[ \frac{2 (x \text{csch}(x)-2 \text{sech}(x))}{\text{csch}^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.158, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{\rm csch} \left (x\right )} \left ( x\cosh \left ( x \right ) -4\,{\rm sech} \left (x\right )\tanh \left ( x \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x \cosh \left (x\right ) - 4 \, \operatorname{sech}\left (x\right ) \tanh \left (x\right )\right )} \sqrt{\operatorname{csch}\left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x \cosh \left (x\right ) - 4 \, \operatorname{sech}\left (x\right ) \tanh \left (x\right )\right )} \sqrt{\operatorname{csch}\left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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