Optimal. Leaf size=22 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x^2}{\sqrt {a^4-x^4}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {281, 223, 209}
\begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {x^2}{\sqrt {a^4-x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 281
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a^4-x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {a^4-x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x^2}{\sqrt {a^4-x^4}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {x^2}{\sqrt {a^4-x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 22, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a^4-x^4}}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.35, size = 33, normalized size = 1.50 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\left (-\frac {I}{2}\right ) \text {ArcCosh}\left [\frac {x^2}{a^2}\right ],\text {Abs}\left [\frac {x^4}{a^4}\right ]>1\right \}\right \},\frac {\text {ArcSin}\left [\frac {x^2}{a^2}\right ]}{2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.05, size = 19, normalized size = 0.86
method | result | size |
default | \(\frac {\arctan \left (\frac {x^{2}}{\sqrt {a^{4}-x^{4}}}\right )}{2}\) | \(19\) |
elliptic | \(\frac {\arctan \left (\frac {x^{2}}{\sqrt {a^{4}-x^{4}}}\right )}{2}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 18, normalized size = 0.82 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {\sqrt {a^{4} - x^{4}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 25, normalized size = 1.14 \begin {gather*} -\arctan \left (-\frac {a^{2} - \sqrt {a^{4} - x^{4}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.46, size = 29, normalized size = 1.32 \begin {gather*} \begin {cases} - \frac {i \operatorname {acosh}{\left (\frac {x^{2}}{a^{2}} \right )}}{2} & \text {for}\: \left |{\frac {x^{4}}{a^{4}}}\right | > 1 \\\frac {\operatorname {asin}{\left (\frac {x^{2}}{a^{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 = 0.00, size = 12, normalized size = 0.55 \begin {gather*} \frac {\arcsin \left (\frac {x^{2}}{a^{2}}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 18, normalized size = 0.82 \begin {gather*} \frac {\mathrm {atan}\left (\frac {x^2}{\sqrt {a^4-x^4}}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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