Optimal. Leaf size=61 \[ \frac {x \left (1+x^2\right )}{2 \sqrt {1-x^4}}-\frac {\sqrt {1-x^2} \sqrt {1+x^2} E\left (\left .\sin ^{-1}(x)\right |-1\right )}{2 \sqrt {1-x^4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1270, 482, 435}
\begin {gather*} \frac {x \left (x^2+1\right )}{2 \sqrt {1-x^4}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} E(\text {ArcSin}(x)|-1)}{2 \sqrt {1-x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 435
Rule 482
Rule 1270
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1-x^2\right ) \sqrt {1-x^4}} \, dx &=\frac {\left (\sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {x^2}{\left (1-x^2\right )^{3/2} \sqrt {1+x^2}} \, dx}{\sqrt {1-x^4}}\\ &=\frac {x \left (1+x^2\right )}{2 \sqrt {1-x^4}}-\frac {\left (\sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx}{2 \sqrt {1-x^4}}\\ &=\frac {x \left (1+x^2\right )}{2 \sqrt {1-x^4}}-\frac {\sqrt {1-x^2} \sqrt {1+x^2} E\left (\left .\sin ^{-1}(x)\right |-1\right )}{2 \sqrt {1-x^4}}\\ \end {align*}
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Mathematica [A]
time = 10.07, size = 37, normalized size = 0.61 \begin {gather*} \frac {x+x^3-\sqrt {1-x^4} E\left (\left .\sin ^{-1}(x)\right |-1\right )}{2 \sqrt {1-x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 142 vs. \(2 (49 ) = 98\).
time = 0.16, size = 143, normalized size = 2.34
method | result | size |
risch | \(\frac {x \left (x^{2}+1\right )}{2 \sqrt {-x^{4}+1}}-\frac {\EllipticF \left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{2 \sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\) | \(88\) |
elliptic | \(-\frac {\left (-x^{2}-1\right ) x}{2 \sqrt {\left (x^{2}-1\right ) \left (-x^{2}-1\right )}}-\frac {\EllipticF \left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{2 \sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\) | \(96\) |
default | \(-\frac {\EllipticF \left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{2 \sqrt {-x^{4}+1}}-\frac {-x^{3}-x^{2}-x -1}{4 \sqrt {\left (-1+x \right ) \left (-x^{3}-x^{2}-x -1\right )}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}-\frac {-x^{3}+x^{2}-x +1}{4 \sqrt {\left (1+x \right ) \left (-x^{3}+x^{2}-x +1\right )}}\) | \(143\) |
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^{4}} - \sqrt {1 - x^{4}}}\, 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 = -0.02 \begin {gather*} -\int \frac {x^2}{\left (x^2-1\right )\,\sqrt {1-x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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