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 = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 12
Rule 51
Rule 63
Rule 203
Rule 5271
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.86, size = 21, normalized size = 0.55 \[ \frac {{\left (x - 2\right )} \operatorname {arccsc}\left (\sqrt {x}\right ) - \sqrt {x - 1}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 32, normalized size = 0.84 \[ -{\left (\frac {1}{x} - 1\right )} \arcsin \left (\frac {1}{\sqrt {x}}\right ) - \frac {\sqrt {-\frac {1}{x} + 1}}{2 \, \sqrt {x}} - \frac {1}{2} \, \arcsin \left (\frac {1}{\sqrt {x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 46, normalized size = 1.21 \[ -\frac {\mathrm {arccsc}\left (\sqrt {x}\right )}{x}+\frac {\sqrt {x -1}\, \left (\arctan \left (\frac {1}{\sqrt {x -1}}\right ) x -\sqrt {x -1}\right )}{2 \sqrt {\frac {x -1}{x}}\, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 51, normalized size = 1.34 \[ \frac {\sqrt {x} \sqrt {-\frac {1}{x} + 1}}{2 \, {\left (x {\left (\frac {1}{x} - 1\right )} - 1\right )}} - \frac {\operatorname {arccsc}\left (\sqrt {x}\right )}{x} - \frac {1}{2} \, \arctan \left (\sqrt {x} \sqrt {-\frac {1}{x} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 28, normalized size = 0.74 \[ -\frac {\sqrt {1-\frac {1}{x}}}{2\,\sqrt {x}}-\frac {\mathrm {asin}\left (\frac {1}{\sqrt {x}}\right )\,\left (\frac {2}{x}-1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 17.55, size = 78, normalized size = 2.05 \[ - \frac {\begin {cases} i \operatorname {acosh}{\left (\frac {1}{\sqrt {x}} \right )} + \frac {i \sqrt {-1 + \frac {1}{x}}}{\sqrt {x}} & \text {for}\: \frac {1}{\left |{x}\right |} > 1 \\- \operatorname {asin}{\left (\frac {1}{\sqrt {x}} \right )} + \frac {1}{\sqrt {x} \sqrt {1 - \frac {1}{x}}} - \frac {1}{x^{\frac {3}{2}} \sqrt {1 - \frac {1}{x}}} & \text {otherwise} \end {cases}}{2} - \frac {\operatorname {acsc}{\left (\sqrt {x} \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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