Optimal. Leaf size=58 \[ \sqrt{\sqrt{\frac{1}{x}}+1} x-\frac{3 \sqrt{\sqrt{\frac{1}{x}}+1}}{2 \sqrt{\frac{1}{x}}}+\frac{3}{2} \tanh ^{-1}\left (\sqrt{\sqrt{\frac{1}{x}}+1}\right ) \]
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Rubi [A] time = 0.0158496, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {255, 190, 51, 63, 207} \[ \sqrt{\sqrt{\frac{1}{x}}+1} x-\frac{3 \sqrt{\sqrt{\frac{1}{x}}+1}}{2 \sqrt{\frac{1}{x}}}+\frac{3}{2} \tanh ^{-1}\left (\sqrt{\sqrt{\frac{1}{x}}+1}\right ) \]
Antiderivative was successfully verified.
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Rule 255
Rule 190
Rule 51
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+\sqrt{\frac{1}{x}}}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{1}{\sqrt{x}}}} \, dx,\sqrt{x},\frac{1}{\sqrt{\frac{1}{x}}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1+x}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{1}{\sqrt{\frac{1}{x}}}\right )\\ &=\sqrt{1+\sqrt{\frac{1}{x}}} x+\operatorname{Subst}\left (\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{1}{\sqrt{\frac{1}{x}}}\right )\\ &=-\frac{3 \sqrt{1+\sqrt{\frac{1}{x}}}}{2 \sqrt{\frac{1}{x}}}+\sqrt{1+\sqrt{\frac{1}{x}}} x-\operatorname{Subst}\left (\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{1}{\sqrt{\frac{1}{x}}}\right )\\ &=-\frac{3 \sqrt{1+\sqrt{\frac{1}{x}}}}{2 \sqrt{\frac{1}{x}}}+\sqrt{1+\sqrt{\frac{1}{x}}} x-\operatorname{Subst}\left (\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+\frac{1}{\sqrt{x}}}\right ),\sqrt{x},\frac{1}{\sqrt{\frac{1}{x}}}\right )\\ &=-\frac{3 \sqrt{1+\sqrt{\frac{1}{x}}}}{2 \sqrt{\frac{1}{x}}}+\sqrt{1+\sqrt{\frac{1}{x}}} x+\frac{3}{2} \tanh ^{-1}\left (\sqrt{1+\sqrt{\frac{1}{x}}}\right )\\ \end{align*}
Mathematica [A] time = 0.130816, size = 70, normalized size = 1.21 \[ \frac{1}{4} \left (2 \left (2-3 \sqrt{\frac{1}{x}}\right ) \sqrt{\sqrt{\frac{1}{x}}+1} x-3 \log \left (1-\frac{1}{\sqrt{\sqrt{\frac{1}{x}}+1}}\right )+3 \log \left (\frac{1}{\sqrt{\sqrt{\frac{1}{x}}+1}}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 92, normalized size = 1.6 \begin{align*} -{\frac{1}{4}\sqrt{1+\sqrt{{x}^{-1}}}\sqrt{x} \left ( 6\,\sqrt{\sqrt{{x}^{-1}}x+x}\sqrt{{x}^{-1}}\sqrt{x}-4\,\sqrt{\sqrt{{x}^{-1}}x+x}\sqrt{x}-3\,\ln \left ( 1/2\,\sqrt{{x}^{-1}}\sqrt{x}+\sqrt{x}+\sqrt{\sqrt{{x}^{-1}}x+x} \right ) \right ){\frac{1}{\sqrt{x \left ( 1+\sqrt{{x}^{-1}} \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957679, size = 84, normalized size = 1.45 \begin{align*} -\frac{3 \,{\left (\frac{1}{\sqrt{x}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{\frac{1}{\sqrt{x}} + 1}}{2 \,{\left ({\left (\frac{1}{\sqrt{x}} + 1\right )}^{2} - \frac{2}{\sqrt{x}} - 1\right )}} + \frac{3}{4} \, \log \left (\sqrt{\frac{1}{\sqrt{x}} + 1} + 1\right ) - \frac{3}{4} \, \log \left (\sqrt{\frac{1}{\sqrt{x}} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37454, size = 161, normalized size = 2.78 \begin{align*} \frac{1}{2} \,{\left (2 \, x - 3 \, \sqrt{x}\right )} \sqrt{\frac{x + \sqrt{x}}{x}} + \frac{3}{4} \, \log \left (\sqrt{\frac{x + \sqrt{x}}{x}} + 1\right ) - \frac{3}{4} \, \log \left (\sqrt{\frac{x + \sqrt{x}}{x}} - 1\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{1}{\sqrt{\sqrt{\frac{1}{x}} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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