Optimal. Leaf size=38 \[ -\frac{\sqrt{x-1}}{2 x}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x-1}\right )-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{x} \]
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Rubi [A] time = 0.0154438, antiderivative size = 38, 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 = {5271, 12, 51, 63, 203} \[ -\frac{\sqrt{x-1}}{2 x}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x-1}\right )-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 5271
Rule 12
Rule 51
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{\csc ^{-1}\left (\sqrt{x}\right )}{x^2} \, dx &=-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{x}-\int \frac{1}{2 \sqrt{-1+x} x^2} \, dx\\ &=-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{x}-\frac{1}{2} \int \frac{1}{\sqrt{-1+x} x^2} \, dx\\ &=-\frac{\sqrt{-1+x}}{2 x}-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{x}-\frac{1}{4} \int \frac{1}{\sqrt{-1+x} x} \, dx\\ &=-\frac{\sqrt{-1+x}}{2 x}-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{x}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+x}\right )\\ &=-\frac{\sqrt{-1+x}}{2 x}-\frac{\csc ^{-1}\left (\sqrt{x}\right )}{x}-\frac{1}{2} \tan ^{-1}\left (\sqrt{-1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0241616, size = 32, normalized size = 0.84 \[ -\frac{\sqrt{x-1}-x \sin ^{-1}\left (\frac{1}{\sqrt{x}}\right )+2 \csc ^{-1}\left (\sqrt{x}\right )}{2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.114, size = 46, normalized size = 1.2 \begin{align*} -{\frac{1}{x}{\rm arccsc} \left (\sqrt{x}\right )}+{\frac{1}{2}\sqrt{x-1} \left ( \arctan \left ({\frac{1}{\sqrt{x-1}}} \right ) x-\sqrt{x-1} \right ){\frac{1}{\sqrt{{\frac{x-1}{x}}}}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50806, size = 69, normalized size = 1.82 \begin{align*} \frac{\sqrt{x} \sqrt{-\frac{1}{x} + 1}}{2 \,{\left (x{\left (\frac{1}{x} - 1\right )} - 1\right )}} - \frac{\operatorname{arccsc}\left (\sqrt{x}\right )}{x} - \frac{1}{2} \, \arctan \left (\sqrt{x} \sqrt{-\frac{1}{x} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.76804, size = 63, normalized size = 1.66 \begin{align*} \frac{{\left (x - 2\right )} \operatorname{arccsc}\left (\sqrt{x}\right ) - \sqrt{x - 1}}{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{acsc}{\left (\sqrt{x} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09947, size = 43, normalized size = 1.13 \begin{align*} -{\left (\frac{1}{x} - 1\right )} \arcsin \left (\frac{1}{\sqrt{x}}\right ) - \frac{\sqrt{-\frac{1}{x} + 1}}{2 \, \sqrt{x}} - \frac{1}{2} \, \arcsin \left (\frac{1}{\sqrt{x}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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