Optimal. Leaf size=23 \[ \frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \sin ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {26, 201, 222}
\begin {gather*} \frac {\text {ArcSin}(x)}{2}+\frac {1}{2} \sqrt {1-x^2} x \end {gather*}
Antiderivative was successfully verified.
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Rule 26
Rule 201
Rule 222
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^4}}{\sqrt {1+x^2}} \, dx &=\int \sqrt {1-x^2} \, dx\\ &=\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(23)=46\).
time = 0.61, size = 50, normalized size = 2.17 \begin {gather*} \frac {1}{2} \left (\frac {x \sqrt {1-x^4}}{\sqrt {1+x^2}}+\tan ^{-1}\left (\frac {x \sqrt {1+x^2}}{\sqrt {1-x^4}}\right )\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. \(41\) vs.
\(2(17)=34\).
time = 0.48, size = 42, normalized size = 1.83
method | result | size |
default | \(\frac {\sqrt {-x^{4}+1}\, \left (x \sqrt {-x^{2}+1}+\arcsin \left (x \right )\right )}{2 \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}}\) | \(42\) |
risch | \(-\frac {x \left (x^{2}-1\right ) \sqrt {\frac {-x^{4}+1}{x^{2}+1}}\, \sqrt {x^{2}+1}}{2 \sqrt {-x^{2}+1}\, \sqrt {-x^{4}+1}}+\frac {\arcsin \left (x \right ) \sqrt {\frac {-x^{4}+1}{x^{2}+1}}\, \sqrt {x^{2}+1}}{2 \sqrt {-x^{4}+1}}\) | \(89\) |
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. 60 vs.
\(2 (17) = 34\).
time = 0.35, size = 60, normalized size = 2.61 \begin {gather*} \frac {\sqrt {-x^{4} + 1} \sqrt {x^{2} + 1} x - {\left (x^{2} + 1\right )} \arctan \left (\frac {\sqrt {-x^{4} + 1} \sqrt {x^{2} + 1}}{x^{3} + x}\right )}{2 \, {\left (x^{2} + 1\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 {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}{\sqrt {x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {1-x^4}}{\sqrt {x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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