Optimal. Leaf size=46 \[ -\text{PolyLog}\left (2,-e^{2 \text{sech}^{-1}\left (\sqrt{x}\right )}\right )+\text{sech}^{-1}\left (\sqrt{x}\right )^2-2 \text{sech}^{-1}\left (\sqrt{x}\right ) \log \left (e^{2 \text{sech}^{-1}\left (\sqrt{x}\right )}+1\right ) \]
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Rubi [A] time = 0.0952382, 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 = {6281, 5660, 3718, 2190, 2279, 2391} \[ -\text{PolyLog}\left (2,-e^{2 \text{sech}^{-1}\left (\sqrt{x}\right )}\right )+\text{sech}^{-1}\left (\sqrt{x}\right )^2-2 \text{sech}^{-1}\left (\sqrt{x}\right ) \log \left (e^{2 \text{sech}^{-1}\left (\sqrt{x}\right )}+1\right ) \]
Antiderivative was successfully verified.
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Rule 6281
Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{\text{sech}^{-1}(x)}{x} \, dx,x,\sqrt{x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{\cosh ^{-1}(x)}{x} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\left (2 \operatorname{Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\right )\\ &=\cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-4 \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\\ &=\cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )+2 \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\\ &=\cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )+\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )\\ &=\cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )-\text{Li}_2\left (-e^{2 \cosh ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )\\ \end{align*}
Mathematica [A] time = 0.0359203, size = 45, normalized size = 0.98 \[ \text{PolyLog}\left (2,-e^{-2 \text{sech}^{-1}\left (\sqrt{x}\right )}\right )-\text{sech}^{-1}\left (\sqrt{x}\right ) \left (\text{sech}^{-1}\left (\sqrt{x}\right )+2 \log \left (e^{-2 \text{sech}^{-1}\left (\sqrt{x}\right )}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.16, size = 65, normalized size = 1.4 \begin{align*} \left ({\rm arcsech} \left (\sqrt{x}\right ) \right ) ^{2}-2\,{\rm arcsech} \left (\sqrt{x}\right )\ln \left ( 1+ \left ({\frac{1}{\sqrt{x}}}+\sqrt{-1+{\frac{1}{\sqrt{x}}}}\sqrt{1+{\frac{1}{\sqrt{x}}}} \right ) ^{2} \right ) -{\it polylog} \left ( 2,- \left ({\frac{1}{\sqrt{x}}}+\sqrt{-1+{\frac{1}{\sqrt{x}}}}\sqrt{1+{\frac{1}{\sqrt{x}}}} \right ) ^{2} \right ) \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*} -\frac{1}{4} \, \log \left (x\right )^{2} + \log \left (x\right ) \log \left (\sqrt{\sqrt{x} + 1} \sqrt{-\sqrt{x} + 1} + 1\right ) - \log \left (\sqrt{x} + 1\right ) \log \left (\sqrt{x}\right ) - \log \left (\sqrt{x}\right ) \log \left (-\sqrt{x} + 1\right ) -{\rm Li}_2\left (-\sqrt{x}\right ) -{\rm Li}_2\left (\sqrt{x}\right ) + \int \frac{\log \left (x\right )}{2 \,{\left ({\left (x - 1\right )} e^{\left (\frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) + \frac{1}{2} \, \log \left (-\sqrt{x} + 1\right )\right )} + x - 1\right )}}\,{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{arsech}\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{asech}{\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{arsech}\left (\sqrt{x}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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