Optimal. Leaf size=56 \[ i \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (\sqrt{x}\right )}\right )+i \sec ^{-1}\left (\sqrt{x}\right )^2-2 \sec ^{-1}\left (\sqrt{x}\right ) \log \left (1+e^{2 i \sec ^{-1}\left (\sqrt{x}\right )}\right ) \]
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Rubi [A] time = 0.0788201, antiderivative size = 56, 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 = {5218, 4626, 3719, 2190, 2279, 2391} \[ i \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (\sqrt{x}\right )}\right )+i \sec ^{-1}\left (\sqrt{x}\right )^2-2 \sec ^{-1}\left (\sqrt{x}\right ) \log \left (1+e^{2 i \sec ^{-1}\left (\sqrt{x}\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 5218
Rule 4626
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sec ^{-1}\left (\sqrt{x}\right )}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{\sec ^{-1}(x)}{x} \, dx,x,\sqrt{x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{\cos ^{-1}(x)}{x} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=2 \operatorname{Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\\ &=i \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-4 i \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\\ &=i \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )+2 \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )\right )\\ &=i \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )-i \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )\\ &=i \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )^2-2 \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )+i \text{Li}_2\left (-e^{2 i \cos ^{-1}\left (\frac{1}{\sqrt{x}}\right )}\right )\\ \end{align*}
Mathematica [A] time = 0.0262626, size = 54, normalized size = 0.96 \[ i \left (\text{PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (\sqrt{x}\right )}\right )+\sec ^{-1}\left (\sqrt{x}\right ) \left (\sec ^{-1}\left (\sqrt{x}\right )+2 i \log \left (1+e^{2 i \sec ^{-1}\left (\sqrt{x}\right )}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.2, size = 63, normalized size = 1.1 \begin{align*} i \left ({\rm arcsec} \left (\sqrt{x}\right ) \right ) ^{2}-2\,{\rm arcsec} \left (\sqrt{x}\right )\ln \left ( 1+ \left ({\frac{1}{\sqrt{x}}}+i\sqrt{1-{x}^{-1}} \right ) ^{2} \right ) +i{\it polylog} \left ( 2,- \left ({\frac{1}{\sqrt{x}}}+i\sqrt{1-{x}^{-1}} \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*} \int \frac{\operatorname{arcsec}\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{arcsec}\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{asec}{\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{arcsec}\left (\sqrt{x}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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