Optimal. Leaf size=46 \[ -\text{PolyLog}\left (2,e^{2 \text{csch}^{-1}\left (\sqrt{x}\right )}\right )+\text{csch}^{-1}\left (\sqrt{x}\right )^2-2 \text{csch}^{-1}\left (\sqrt{x}\right ) \log \left (1-e^{2 \text{csch}^{-1}\left (\sqrt{x}\right )}\right ) \]
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Rubi [A] time = 0.0965395, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6282, 5659, 3716, 2190, 2279, 2391} \[ -\text{PolyLog}\left (2,e^{2 \text{csch}^{-1}\left (\sqrt{x}\right )}\right )+\text{csch}^{-1}\left (\sqrt{x}\right )^2-2 \text{csch}^{-1}\left (\sqrt{x}\right ) \log \left (1-e^{2 \text{csch}^{-1}\left (\sqrt{x}\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 6282
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\text{csch}^{-1}\left (\sqrt{x}\right )}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{\text{csch}^{-1}(x)}{x} \, dx,x,\sqrt{x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{\sinh ^{-1}(x)}{x} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\left (2 \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\right )\\ &=\sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2+4 \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\\ &=\sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )+2 \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\\ &=\sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )+\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )\\ &=\sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )-\text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )\\ \end{align*}
Mathematica [A] time = 0.0353073, size = 45, normalized size = 0.98 \[ \text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}\left (\sqrt{x}\right )}\right )-\text{csch}^{-1}\left (\sqrt{x}\right ) \left (\text{csch}^{-1}\left (\sqrt{x}\right )+2 \log \left (1-e^{-2 \text{csch}^{-1}\left (\sqrt{x}\right )}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.17, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\rm arccsch} \left (\sqrt{x}\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 \frac{\operatorname{arcsch}\left (\sqrt{x}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcsch}\left (\sqrt{x}\right )}{x}, x\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{acsch}{\left (\sqrt{x} \right )}}{x}\, dx \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 \frac{\operatorname{arcsch}\left (\sqrt{x}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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