Optimal. Leaf size=23 \[ \frac {1}{\sqrt {x}}+\tan ^{-1}\left (\sqrt {x}\right )-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{x} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5034, 51, 63, 203} \[ \frac {1}{\sqrt {x}}+\tan ^{-1}\left (\sqrt {x}\right )-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 203
Rule 5034
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx &=-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{x}-\frac {1}{2} \int \frac {1}{x^{3/2} (1+x)} \, dx\\ &=\frac {1}{\sqrt {x}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=\frac {1}{\sqrt {x}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{x}+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {1}{\sqrt {x}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{x}+\tan ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 29, normalized size = 1.26 \[ \frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-x\right )}{\sqrt {x}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 19, normalized size = 0.83 \[ -\frac {{\left (x + 1\right )} \operatorname {arccot}\left (\sqrt {x}\right ) - \sqrt {x}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 19, normalized size = 0.83 \[ -\frac {\arctan \left (\frac {1}{\sqrt {x}}\right )}{x} + \frac {1}{\sqrt {x}} - \arctan \left (\frac {1}{\sqrt {x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 18, normalized size = 0.78 \[ -\frac {\mathrm {arccot}\left (\sqrt {x}\right )}{x}+\arctan \left (\sqrt {x}\right )+\frac {1}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 17, normalized size = 0.74 \[ -\frac {\operatorname {arccot}\left (\sqrt {x}\right )}{x} + \frac {1}{\sqrt {x}} + \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 17, normalized size = 0.74 \[ \mathrm {atan}\left (\sqrt {x}\right )-\frac {\mathrm {acot}\left (\sqrt {x}\right )}{x}+\frac {1}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.94, size = 92, normalized size = 4.00 \[ - \frac {x^{\frac {5}{2}} \operatorname {acot}{\left (\sqrt {x} \right )}}{x^{\frac {5}{2}} + x^{\frac {3}{2}}} - \frac {2 x^{\frac {3}{2}} \operatorname {acot}{\left (\sqrt {x} \right )}}{x^{\frac {5}{2}} + x^{\frac {3}{2}}} - \frac {\sqrt {x} \operatorname {acot}{\left (\sqrt {x} \right )}}{x^{\frac {5}{2}} + x^{\frac {3}{2}}} + \frac {x^{2}}{x^{\frac {5}{2}} + x^{\frac {3}{2}}} + \frac {x}{x^{\frac {5}{2}} + x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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