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 = {1593, 329, 212, 206, 203} \[ \tan ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 329
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x}-x^{5/2}} \, dx &=\int \frac {1}{\sqrt {x} \left (1-x^2\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {x}\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right )+\operatorname {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 = 0.00, size = 13, normalized size = 1.00 \[ \tan ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 21, normalized size = 1.62 \[ \arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 22, normalized size = 1.69 \[ \arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left ({\left | \sqrt {x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 10, normalized size = 0.77 \[ \arctanh \left (\sqrt {x}\right )+\arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.41, size = 21, normalized size = 1.62 \[ \arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.08, size = 9, normalized size = 0.69 \[ \mathrm {atan}\left (\sqrt {x}\right )+\mathrm {atanh}\left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.40, size = 26, normalized size = 2.00 \[ - \frac {\log {\left (\sqrt {x} - 1 \right )}}{2} + \frac {\log {\left (\sqrt {x} + 1 \right )}}{2} + \operatorname {atan}{\left (\sqrt {x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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