Optimal. Leaf size=70 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{2}\right )}{4 \sqrt{x^4+1}} \]
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Rubi [A] time = 0.0607269, antiderivative size = 70, normalized size of antiderivative = 1., 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 = {1318, 220, 1699, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Rule 1318
Rule 220
Rule 1699
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2}{\left (1-x^2\right ) \sqrt{1+x^4}} \, dx &=-\left (\frac{1}{2} \int \frac{1}{\sqrt{1+x^4}} \, dx\right )+\frac{1}{2} \int \frac{1+x^2}{\left (1-x^2\right ) \sqrt{1+x^4}} \, dx\\ &=-\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{1+x^4}}+\frac{1}{2} \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 )}{2 \sqrt{2}}-\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{1+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0891999, size = 36, normalized size = 0.51 \[ \sqrt [4]{-1} \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt [4]{-1} x\right ),-1\right )-\Pi \left (i;\left .\sin ^{-1}\left ((-1)^{3/4} x\right )\right |-1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 112, normalized size = 1.6 \begin{align*} -{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{ \left ( -1 \right ) ^{{\frac{3}{4}}}{\it EllipticPi} \left ( \sqrt [4]{-1}x,-i,\sqrt{-i}- \left ( -1 \right ) ^{{\frac{3}{4}}} \right ) \sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2}}{\sqrt{x^{4} + 1}{\left (x^{2} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x^{4} + 1} x^{2}}{x^{6} - x^{4} + x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{2}}{x^{2} \sqrt{x^{4} + 1} - \sqrt{x^{4} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{2}}{\sqrt{x^{4} + 1}{\left (x^{2} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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