Optimal. Leaf size=46 \[ -\text {Li}_2\left (-e^{2 \text {sech}^{-1}\left (\sqrt {x}\right )}\right )+\text {sech}^{-1}\left (\sqrt {x}\right )^2-2 \text {sech}^{-1}\left (\sqrt {x}\right ) \log \left (e^{2 \text {sech}^{-1}\left (\sqrt {x}\right )}+1\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, 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 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 6281
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*}
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Mathematica [A] time = 0.04, size = 45, normalized size = 0.98 \[ \text {Li}_2\left (-e^{-2 \text {sech}^{-1}\left (\sqrt {x}\right )}\right )-\text {sech}^{-1}\left (\sqrt {x}\right ) \left (\text {sech}^{-1}\left (\sqrt {x}\right )+2 \log \left (e^{-2 \text {sech}^{-1}\left (\sqrt {x}\right )}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsech}\left (\sqrt {x}\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsech}\left (\sqrt {x}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 65, normalized size = 1.41 \[ \mathrm {arcsech}\left (\sqrt {x}\right )^{2}-2 \,\mathrm {arcsech}\left (\sqrt {x}\right ) \ln \left (1+\left (\frac {1}{\sqrt {x}}+\sqrt {-1+\frac {1}{\sqrt {x}}}\, \sqrt {1+\frac {1}{\sqrt {x}}}\right )^{2}\right )-\polylog \left (2, -\left (\frac {1}{\sqrt {x}}+\sqrt {-1+\frac {1}{\sqrt {x}}}\, \sqrt {1+\frac {1}{\sqrt {x}}}\right )^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{4} \, \log \relax (x)^{2} + \log \relax (x) \log \left (\sqrt {\sqrt {x} + 1} \sqrt {-\sqrt {x} + 1} + 1\right ) - \log \left (\sqrt {x} + 1\right ) \log \left (\sqrt {x}\right ) - \log \left (\sqrt {x}\right ) \log \left (-\sqrt {x} + 1\right ) - {\rm Li}_2\left (-\sqrt {x}\right ) - {\rm Li}_2\left (\sqrt {x}\right ) + \int \frac {\log \relax (x)}{2 \, {\left ({\left (x - 1\right )} e^{\left (\frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) + \frac {1}{2} \, \log \left (-\sqrt {x} + 1\right )\right )} + x - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {asech}{\left (\sqrt {x} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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