Optimal. Leaf size=23 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1699, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1699
Rubi steps
\begin {align*} \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 36, normalized size = 1.57 \[ \sqrt [4]{-1} \left (\operatorname {EllipticF}\left (i \sinh ^{-1}\left (\sqrt [4]{-1} x\right ),-1\right )-2 \Pi \left (i;\left .\sin ^{-1}\left ((-1)^{3/4} x\right )\right |-1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 42, normalized size = 1.83 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} + 2 \, \sqrt {2} \sqrt {x^{4} + 1} x + 2 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) \]
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 \[ \int -\frac {x^{2} + 1}{\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 112, normalized size = 4.87 \[ -\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {2 \left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{\sqrt {x^{4}+1}} \]
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 \[ -\int \frac {x^{2} + 1}{\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int -\frac {x^2+1}{\left (x^2-1\right )\,\sqrt {x^4+1}} \,d x \]
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 \[ - \int \frac {x^{2}}{x^{2} \sqrt {x^{4} + 1} - \sqrt {x^{4} + 1}}\, dx - \int \frac {1}{x^{2} \sqrt {x^{4} + 1} - \sqrt {x^{4} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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