Optimal. Leaf size=99 \[ -\frac {x \left (1-x^2\right )}{2 \sqrt {1-x^4}}-\frac {\sqrt {1-x^2} \sqrt {1+x^2} E\left (\left .\sin ^{-1}(x)\right |-1\right )}{2 \sqrt {1-x^4}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} F\left (\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {1-x^4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1270, 482, 434,
435, 254, 227} \begin {gather*} \frac {\sqrt {x^2+1} \sqrt {1-x^2} F(\text {ArcSin}(x)|-1)}{\sqrt {1-x^4}}-\frac {\sqrt {x^2+1} \sqrt {1-x^2} E(\text {ArcSin}(x)|-1)}{2 \sqrt {1-x^4}}-\frac {x \left (1-x^2\right )}{2 \sqrt {1-x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 254
Rule 434
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}{\sqrt {1-x^2} \left (1+x^2\right )^{3/2}} \, 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 {\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 {\left (\sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2}} \, dx}{\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}}+\frac {\left (\sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^4}} \, dx}{\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}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} F\left (\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {1-x^4}}\\ \end {align*}
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Mathematica [A]
time = 10.10, size = 46, normalized size = 0.46 \begin {gather*} \frac {1}{2} \left (-\frac {x}{\sqrt {1-x^4}}+\frac {x^3}{\sqrt {1-x^4}}-E\left (\left .\sin ^{-1}(x)\right |-1\right )+2 F\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 96, normalized size = 0.97
method | result | size |
risch | \(\frac {x \left (x^{2}-1\right )}{2 \sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}+\frac {\EllipticF \left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{2 \sqrt {-x^{4}+1}}\) | \(88\) |
default | \(\frac {\EllipticF \left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{2 \sqrt {-x^{4}+1}}-\frac {\left (-x^{2}+1\right ) x}{2 \sqrt {\left (-x^{2}+1\right ) \left (x^{2}+1\right )}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\) | \(96\) |
elliptic | \(\frac {\EllipticF \left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{2 \sqrt {-x^{4}+1}}-\frac {\left (-x^{2}+1\right ) x}{2 \sqrt {\left (-x^{2}+1\right ) \left (x^{2}+1\right )}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\) | \(96\) |
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}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, 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 = -0.01 \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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