Optimal. Leaf size=57 \[ \frac {x \left (x^2+1\right )}{2 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} E\left (\left .\sin ^{-1}(x)\right |-1\right )}{2 \sqrt {x^4-1}} \]
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Rubi [A] time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1256, 471, 426, 424} \[ \frac {x \left (x^2+1\right )}{2 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} E\left (\left .\sin ^{-1}(x)\right |-1\right )}{2 \sqrt {x^4-1}} \]
Antiderivative was successfully verified.
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Rule 424
Rule 426
Rule 471
Rule 1256
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 (\left (-1-x^2\right ) \sqrt {1-x^2}\right ) \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx}{2 \sqrt {1+x^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 = 0.07, size = 35, normalized size = 0.61 \[ \frac {-\sqrt {1-x^4} E\left (\left .\sin ^{-1}(x)\right |-1\right )+x^3+x}{2 \sqrt {x^4-1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x^{4} - 1} x^{2}}{x^{6} - x^{4} - x^{2} + 1}, x\right ) \]
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 \[ \int -\frac {x^{2}}{\sqrt {x^{4} - 1} {\left (x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 134, normalized size = 2.35 \[ \frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{2 \sqrt {x^{4}-1}}+\frac {x^{3}-x^{2}+x -1}{4 \sqrt {\left (x +1\right ) \left (x^{3}-x^{2}+x -1\right )}}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (-\EllipticE \left (i x , i\right )+\EllipticF \left (i x , i\right )\right )}{2 \sqrt {x^{4}-1}}+\frac {x^{3}+x^{2}+x +1}{4 \sqrt {\left (x -1\right ) \left (x^{3}+x^{2}+x +1\right )}} \]
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 \[ -\int \frac {x^{2}}{\sqrt {x^{4} - 1} {\left (x^{2} - 1\right )}}\,{d x} \]
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 \[ -\int \frac {x^2}{\left (x^2-1\right )\,\sqrt {x^4-1}} \,d x \]
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 \[ - \int \frac {x^{2}}{x^{2} \sqrt {x^{4} - 1} - \sqrt {x^{4} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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