Optimal. Leaf size=4 \[ x+\sin ^{-1}(x) \]
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Rubi [A]
time = 0.03, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2187, 8, 222}
\begin {gather*} \text {ArcSin}(x)+x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 222
Rule 2187
Rubi steps
\begin {align*} \int \frac {x^2}{-1+x^2+\sqrt {1-x^2}} \, dx &=-\int -1 \, dx+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=x+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(22\) vs. \(2(4)=8\).
time = 0.10, size = 22, normalized size = 5.50 \begin {gather*} x+2 \tan ^{-1}\left (\frac {x}{-1+\sqrt {1-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs.
\(2(4)=8\).
time = 0.08, size = 51, normalized size = 12.75
method | result | size |
trager | \(x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\) | \(29\) |
default | \(x +\frac {\ln \left (-1+x \right )}{2}-\frac {\ln \left (1+x \right )}{2}+\arctanh \left (x \right )-\frac {\sqrt {-\left (-1+x \right )^{2}+2-2 x}}{2}+\arcsin \left (x \right )+\frac {\sqrt {-\left (1+x \right )^{2}+2+2 x}}{2}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 20 vs.
\(2 (4) = 8\).
time = 0.33, size = 20, normalized size = 5.00 \begin {gather*} x - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \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 \frac {x^{2}}{x^{2} + \sqrt {1 - x^{2}} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.95, size = 4, normalized size = 1.00 \begin {gather*} x + \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 4, normalized size = 1.00 \begin {gather*} x+\mathrm {asin}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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