Optimal. Leaf size=22 \[ \sqrt {x}+x \coth ^{-1}\left (\sqrt {x}\right )-\tanh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6022, 52, 65,
212} \begin {gather*} \sqrt {x}-\tanh ^{-1}\left (\sqrt {x}\right )+x \coth ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 6022
Rubi steps
\begin {align*} \int \coth ^{-1}\left (\sqrt {x}\right ) \, dx &=x \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {\sqrt {x}}{1-x} \, dx\\ &=\sqrt {x}+x \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {1}{(1-x) \sqrt {x}} \, dx\\ &=\sqrt {x}+x \coth ^{-1}\left (\sqrt {x}\right )-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x}+x \coth ^{-1}\left (\sqrt {x}\right )-\tanh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} \sqrt {x}+x \coth ^{-1}\left (\sqrt {x}\right )-\tanh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 27, normalized size = 1.23
method | result | size |
derivativedivides | \(x \,\mathrm {arccoth}\left (\sqrt {x}\right )+\sqrt {x}+\frac {\ln \left (\sqrt {x}-1\right )}{2}-\frac {\ln \left (\sqrt {x}+1\right )}{2}\) | \(27\) |
default | \(x \,\mathrm {arccoth}\left (\sqrt {x}\right )+\sqrt {x}+\frac {\ln \left (\sqrt {x}-1\right )}{2}-\frac {\ln \left (\sqrt {x}+1\right )}{2}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 26, normalized size = 1.18 \begin {gather*} x \operatorname {arcoth}\left (\sqrt {x}\right ) + \sqrt {x} - \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 24, normalized size = 1.09 \begin {gather*} \frac {1}{2} \, {\left (x - 1\right )} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {acoth}{\left (\sqrt {x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (16) = 32\).
time = 0.39, size = 65, normalized size = 2.95 \begin {gather*} \frac {2}{\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1} + \frac {2 \, {\left (\sqrt {x} + 1\right )} \log \left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{{\left (\sqrt {x} - 1\right )} {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.24, size = 16, normalized size = 0.73 \begin {gather*} x\,\mathrm {acoth}\left (\sqrt {x}\right )-\mathrm {acoth}\left (\sqrt {x}\right )+\sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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