Optimal. Leaf size=13 \[ \tan ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1598, 335, 218,
212, 209} \begin {gather*} \text {ArcTan}\left (\sqrt {x}\right )+\tanh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 1598
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{x-x^3} \, dx &=\int \frac {1}{\sqrt {x} \left (1-x^2\right )} \, dx\\ &=2 \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {x}\right )\\ &=\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=\tan ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 13, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 10, normalized size = 0.77
method | result | size |
derivativedivides | \(\arctan \left (\sqrt {x}\right )+\arctanh \left (\sqrt {x}\right )\) | \(10\) |
default | \(\arctan \left (\sqrt {x}\right )+\arctanh \left (\sqrt {x}\right )\) | \(10\) |
meijerg | \(-\frac {\sqrt {x}\, \left (\ln \left (1-\left (x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{2}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (x^{2}\right )^{\frac {1}{4}}\right )\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}\) | \(40\) |
trager | \(\frac {\ln \left (\frac {2 \sqrt {x}+1+x}{-1+x}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {x \RootOf \left (\textit {\_Z}^{2}+1\right )-\RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {x}}{1+x}\right )}{2}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs.
\(2 (9) = 18\).
time = 0.49, size = 21, normalized size = 1.62 \begin {gather*} \arctan \left (\sqrt {x}\right ) + \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs.
\(2 (9) = 18\).
time = 0.35, size = 21, normalized size = 1.62 \begin {gather*} \arctan \left (\sqrt {x}\right ) + \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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (12) = 24\).
time = 0.17, size = 26, normalized size = 2.00 \begin {gather*} - \frac {\log {\left (\sqrt {x} - 1 \right )}}{2} + \frac {\log {\left (\sqrt {x} + 1 \right )}}{2} + \operatorname {atan}{\left (\sqrt {x} \right )} \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. 22 vs.
\(2 (9) = 18\).
time = 5.45, size = 22, normalized size = 1.69 \begin {gather*} \arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left ({\left | \sqrt {x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 9, normalized size = 0.69 \begin {gather*} \mathrm {atan}\left (\sqrt {x}\right )+\mathrm {atanh}\left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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