Optimal. Leaf size=8 \[ \frac {1}{2} \sin ^{-1}\left (x^2\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {281, 222}
\begin {gather*} \frac {\text {ArcSin}\left (x^2\right )}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 281
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1-x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \sin ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(20\) vs. \(2(8)=16\).
time = 0.06, size = 20, normalized size = 2.50 \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {1-x^4}}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 7, normalized size = 0.88
method | result | size |
default | \(\frac {\arcsin \left (x^{2}\right )}{2}\) | \(7\) |
meijerg | \(\frac {\arcsin \left (x^{2}\right )}{2}\) | \(7\) |
elliptic | \(\frac {\arcsin \left (x^{2}\right )}{2}\) | \(7\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}+1}+x^{2}\right )}{2}\) | \(30\) |
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. 16 vs.
\(2 (6) = 12\).
time = 3.83, size = 16, normalized size = 2.00 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{4} + 1}}{x^{2}}\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. 18 vs.
\(2 (6) = 12\).
time = 0.63, size = 18, normalized size = 2.25 \begin {gather*} -\arctan \left (\frac {\sqrt {-x^{4} + 1} - 1}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.41, size = 19, normalized size = 2.38 \begin {gather*} \begin {cases} - \frac {i \operatorname {acosh}{\left (x^{2} \right )}}{2} & \text {for}\: \left |{x^{4}}\right | > 1 \\\frac {\operatorname {asin}{\left (x^{2} \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, \arcsin \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 16, normalized size = 2.00 \begin {gather*} \frac {\mathrm {atan}\left (\frac {x^2}{\sqrt {1-x^4}}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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