Optimal. Leaf size=93 \[ \frac{\sqrt{x^4+x^2+1} x}{4 \left (x^2+1\right )^2}+\frac{1}{4} \tan ^{-1}\left (\frac{x}{\sqrt{x^4+x^2+1}}\right )+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{4 \sqrt{x^4+x^2+1}} \]
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Rubi [A] time = 0.508155, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {1228, 1223, 1696, 1593, 1712, 1195, 1700, 1103, 1698, 203, 12, 1317, 1210} \[ \frac{\sqrt{x^4+x^2+1} x}{4 \left (x^2+1\right )^2}+\frac{1}{4} \tan ^{-1}\left (\frac{x}{\sqrt{x^4+x^2+1}}\right )+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{4 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1228
Rule 1223
Rule 1696
Rule 1593
Rule 1712
Rule 1195
Rule 1700
Rule 1103
Rule 1698
Rule 203
Rule 12
Rule 1317
Rule 1210
Rubi steps
\begin{align*} \int \frac{\sqrt{1+x^2+x^4}}{\left (1+x^2\right )^3} \, dx &=\int \left (\frac{1}{\left (1+x^2\right )^3 \sqrt{1+x^2+x^4}}-\frac{1}{\left (1+x^2\right )^2 \sqrt{1+x^2+x^4}}+\frac{1}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}}\right ) \, dx\\ &=\int \frac{1}{\left (1+x^2\right )^3 \sqrt{1+x^2+x^4}} \, dx-\int \frac{1}{\left (1+x^2\right )^2 \sqrt{1+x^2+x^4}} \, dx+\int \frac{1}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx\\ &=\frac{x \sqrt{1+x^2+x^4}}{4 \left (1+x^2\right )^2}-\frac{x \sqrt{1+x^2+x^4}}{2 \left (1+x^2\right )}-\frac{1}{4} \int \frac{-3+2 x^2-x^4}{\left (1+x^2\right )^2 \sqrt{1+x^2+x^4}} \, dx+\frac{1}{2} \int \frac{1}{\sqrt{1+x^2+x^4}} \, dx+\frac{1}{2} \int \frac{1-x^2}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx+\frac{1}{2} \int \frac{-1+2 x^2+x^4}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx\\ &=\frac{x \sqrt{1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac{x \sqrt{1+x^2+x^4}}{4 \left (1+x^2\right )}+\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{4 \sqrt{1+x^2+x^4}}+\frac{1}{8} \int \frac{-10 x^2-6 x^4}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx-\frac{1}{2} \int \frac{1-x^2}{\sqrt{1+x^2+x^4}} \, dx+\frac{1}{2} \int \frac{2 x^2}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{x}{\sqrt{1+x^2+x^4}}\right )\\ &=\frac{x \sqrt{1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac{3 x \sqrt{1+x^2+x^4}}{4 \left (1+x^2\right )}+\frac{1}{2} \tan ^{-1}\left (\frac{x}{\sqrt{1+x^2+x^4}}\right )-\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{2 \sqrt{1+x^2+x^4}}+\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{4 \sqrt{1+x^2+x^4}}+\frac{1}{8} \int \frac{x^2 \left (-10-6 x^2\right )}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx+\int \frac{x^2}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx\\ &=\frac{x \sqrt{1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac{3 x \sqrt{1+x^2+x^4}}{4 \left (1+x^2\right )}+\frac{1}{2} \tan ^{-1}\left (\frac{x}{\sqrt{1+x^2+x^4}}\right )-\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{2 \sqrt{1+x^2+x^4}}+\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{4 \sqrt{1+x^2+x^4}}+\frac{1}{8} \int \frac{-6-10 x^2}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx+\frac{1}{2} \int \frac{1}{\sqrt{1+x^2+x^4}} \, dx-\frac{1}{2} \int \frac{1-x^2}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx+\frac{3}{4} \int \frac{1-x^2}{\sqrt{1+x^2+x^4}} \, dx\\ &=\frac{x \sqrt{1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac{1}{2} \tan ^{-1}\left (\frac{x}{\sqrt{1+x^2+x^4}}\right )+\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{4 \sqrt{1+x^2+x^4}}+\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{2 \sqrt{1+x^2+x^4}}+\frac{1}{4} \int \frac{1-x^2}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{x}{\sqrt{1+x^2+x^4}}\right )-\int \frac{1}{\sqrt{1+x^2+x^4}} \, dx\\ &=\frac{x \sqrt{1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{4 \sqrt{1+x^2+x^4}}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{x}{\sqrt{1+x^2+x^4}}\right )\\ &=\frac{x \sqrt{1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac{1}{4} \tan ^{-1}\left (\frac{x}{\sqrt{1+x^2+x^4}}\right )+\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{4 \sqrt{1+x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.301769, size = 176, normalized size = 1.89 \[ \frac{\sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left (\text{EllipticF}\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )-E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )\right )+\frac{x \left (x^2+2\right ) \left (x^4+x^2+1\right )}{\left (x^2+1\right )^2}-2 (-1)^{2/3} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \Pi \left (\sqrt [3]{-1};-i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )}{4 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 333, normalized size = 3.6 \begin{align*}{\frac{x}{4\, \left ({x}^{2}+1 \right ) ^{2}}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{x}{4\,{x}^{2}+4}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{1}{\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1+{\frac{{x}^{2}}{2}}-{\frac{i}{2}}{x}^{2}\sqrt{3}}\sqrt{1+{\frac{{x}^{2}}{2}}+{\frac{i}{2}}{x}^{2}\sqrt{3}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}-{\frac{1}{\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1+{\frac{{x}^{2}}{2}}-{\frac{i}{2}}{x}^{2}\sqrt{3}}\sqrt{1+{\frac{{x}^{2}}{2}}+{\frac{i}{2}}{x}^{2}\sqrt{3}}{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}+{\frac{1}{2\,\sqrt{-1/2+i/2\sqrt{3}}}\sqrt{1+{\frac{{x}^{2}}{2}}-{\frac{i}{2}}{x}^{2}\sqrt{3}}\sqrt{1+{\frac{{x}^{2}}{2}}+{\frac{i}{2}}{x}^{2}\sqrt{3}}{\it EllipticPi} \left ( \sqrt{-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}x,- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) ^{-1},{\frac{\sqrt{-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3}}}{\sqrt{-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+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{\sqrt{x^{4} + x^{2} + 1}}{{\left (x^{2} + 1\right )}^{3}}\,{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} + x^{2} + 1}}{x^{6} + 3 \, x^{4} + 3 \, 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{\sqrt{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}{\left (x^{2} + 1\right )^{3}}\, 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{\sqrt{x^{4} + x^{2} + 1}}{{\left (x^{2} + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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