Optimal. Leaf size=74 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{2}\right )}{4 \sqrt{-x^4-1}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{-x^4-1}}\right )}{2 \sqrt{2}} \]
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Rubi [A] time = 0.0635063, antiderivative size = 74, 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} \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{-x^4-1}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{-x^4-1}}\right )}{2 \sqrt{2}} \]
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 &=\frac{1}{2} \int \frac{1}{\sqrt{-1-x^4}} \, dx-\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 = 0.092931, size = 60, normalized size = 0.81 \[ \frac{\sqrt [4]{-1} \sqrt{x^4+1} \left (\Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt [4]{-1} x\right ),-1\right )\right )}{\sqrt{-x^4-1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.033, size = 168, normalized size = 2.3 \begin{align*}{\frac{{\it EllipticF} \left ( \left ({\frac{\sqrt{2}}{2}}-{\frac{i}{2}}\sqrt{2} \right ) x,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}}}}-{{\frac{i}{2}}\sqrt{-i}\sqrt{1+i{x}^{2}}\sqrt{1-i{x}^{2}}{\it EllipticPi} \left ( \sqrt{-i}x,-i,{\frac{\sqrt [4]{-1}}{\sqrt{-i}}} \right ){\frac{1}{\sqrt{-{x}^{4}-1}}}}-{\frac{1}{2\,\sqrt{-i}}\sqrt{1+i{x}^{2}}\sqrt{1-i{x}^{2}}{\it EllipticPi} \left ( \sqrt{-i}x,-i,{\frac{\sqrt [4]{-1}}{\sqrt{-i}}} \right ){\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*} -\frac{1}{8} \, \sqrt{2} \log \left (\frac{\sqrt{2} x + \sqrt{-x^{4} - 1}}{x^{2} + 1}\right ) + \frac{1}{8} \, \sqrt{2} \log \left (-\frac{\sqrt{2} x - \sqrt{-x^{4} - 1}}{x^{2} + 1}\right ) +{\rm integral}\left (-\frac{\sqrt{-x^{4} - 1}}{2 \,{\left (x^{4} + 1\right )}}, 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}}{\left (x^{2} + 1\right ) \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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