Optimal. Leaf size=98 \[ \frac{1-x}{2 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{3/2}}+\frac{\sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{2 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x} \]
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Rubi [A] time = 0.0233795, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6345, 12, 51, 63, 206} \[ \frac{1-x}{2 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{3/2}}+\frac{\sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{2 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 6345
Rule 12
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x^2} \, dx &=-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x}-\frac{\sqrt{1-x} \int \frac{1}{2 \sqrt{1-x} x^2} \, dx}{\sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x}-\frac{\sqrt{1-x} \int \frac{1}{\sqrt{1-x} x^2} \, dx}{2 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{2 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x}-\frac{\sqrt{1-x} \int \frac{1}{\sqrt{1-x} x} \, dx}{4 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{2 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x}+\frac{\sqrt{1-x} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x}\right )}{2 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{2 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x}+\frac{\sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{2 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0771281, size = 111, normalized size = 1.13 \[ \frac{\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}} \left (\sqrt{x}+1\right )+x \log \left (\sqrt{x} \sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}}+\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}}+1\right )-\frac{1}{2} x \log (x)-2 \text{sech}^{-1}\left (\sqrt{x}\right )}{2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.134, size = 64, normalized size = 0.7 \begin{align*} -{\frac{1}{x}{\rm arcsech} \left (\sqrt{x}\right )}+{\frac{1}{2}\sqrt{-{ \left ( -1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{{ \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}} \left ({\it Artanh} \left ({\frac{1}{\sqrt{1-x}}} \right ) x+\sqrt{1-x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1-x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980277, size = 88, normalized size = 0.9 \begin{align*} -\frac{\sqrt{x} \sqrt{\frac{1}{x} - 1}}{2 \,{\left (x{\left (\frac{1}{x} - 1\right )} - 1\right )}} - \frac{\operatorname{arsech}\left (\sqrt{x}\right )}{x} + \frac{1}{4} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} - 1} + 1\right ) - \frac{1}{4} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} - 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87779, size = 111, normalized size = 1.13 \begin{align*} \frac{{\left (x - 2\right )} \log \left (\frac{x \sqrt{-\frac{x - 1}{x}} + \sqrt{x}}{x}\right ) + \sqrt{x} \sqrt{-\frac{x - 1}{x}}}{2 \, x} \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^{2}}\, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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