Optimal. Leaf size=25 \[ -\frac {1}{\sqrt {x}}+\tanh ^{-1}\left (\sqrt {x}\right )-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x} \]
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Rubi [A] time = 0.01, antiderivative size = 25, 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 = {6098, 51, 63, 206} \[ -\frac {1}{\sqrt {x}}+\tanh ^{-1}\left (\sqrt {x}\right )-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 6098
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx &=-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \int \frac {1}{(1-x) x^{3/2}} \, dx\\ &=-\frac {1}{\sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \int \frac {1}{(1-x) \sqrt {x}} \, dx\\ &=-\frac {1}{\sqrt {x}}-\frac {\coth ^{-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 {\coth ^{-1}\left (\sqrt {x}\right )}{x}+\tanh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 1.80 \[ -\frac {1}{\sqrt {x}}-\frac {1}{2} \log \left (1-\sqrt {x}\right )+\frac {1}{2} \log \left (\sqrt {x}+1\right )-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 30, normalized size = 1.20 \[ \frac {{\left (x - 1\right )} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) - 2 \, \sqrt {x}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (\sqrt {x}\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 1.28 \[ -\frac {\mathrm {arccoth}\left (\sqrt {x}\right )}{x}-\frac {1}{\sqrt {x}}-\frac {\ln \left (-1+\sqrt {x}\right )}{2}+\frac {\ln \left (1+\sqrt {x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 31, normalized size = 1.24 \[ -\frac {\operatorname {arcoth}\left (\sqrt {x}\right )}{x} - \frac {1}{\sqrt {x}} + \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 18, normalized size = 0.72 \[ \mathrm {atanh}\left (\sqrt {x}\right )-\frac {\mathrm {acoth}\left (\sqrt {x}\right )+\sqrt {x}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.12, size = 92, normalized size = 3.68 \[ \frac {x^{\frac {5}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 x^{\frac {3}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {\sqrt {x} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {x^{2}}{x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {x}{x^{\frac {5}{2}} - x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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