Optimal. Leaf size=111 \[ \frac{\sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}-\frac{\left (x^2+2\right ) x}{3 \sqrt{x^4+x^2+1}}+\tan ^{-1}\left (\frac{x}{\sqrt{x^4+x^2+1}}\right )+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{6 \sqrt{x^4+x^2+1}} \]
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Rubi [A] time = 0.278156, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {1228, 1178, 1195, 1223, 1712, 12, 1317, 1103, 1698, 203, 1210} \[ \frac{\sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}-\frac{\left (x^2+2\right ) x}{3 \sqrt{x^4+x^2+1}}+\tan ^{-1}\left (\frac{x}{\sqrt{x^4+x^2+1}}\right )+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{6 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1228
Rule 1178
Rule 1195
Rule 1223
Rule 1712
Rule 12
Rule 1317
Rule 1103
Rule 1698
Rule 203
Rule 1210
Rubi steps
\begin{align*} \int \frac{1}{\left (1+x^2\right )^2 \left (1+x^2+x^4\right )^{3/2}} \, dx &=\int \left (\frac{-1-x^2}{\left (1+x^2+x^4\right )^{3/2}}+\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-x^2}{\left (1+x^2+x^4\right )^{3/2}} \, 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 \left (2+x^2\right )}{3 \sqrt{1+x^2+x^4}}+\frac{x \sqrt{1+x^2+x^4}}{2 \left (1+x^2\right )}+\frac{1}{3} \int \frac{-1+x^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 \left (2+x^2\right )}{3 \sqrt{1+x^2+x^4}}+\frac{5 x \sqrt{1+x^2+x^4}}{6 \left (1+x^2\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 )}{3 \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}{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 \left (2+x^2\right )}{3 \sqrt{1+x^2+x^4}}+\frac{x \sqrt{1+x^2+x^4}}{3 \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 )}{6 \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}}-\int \frac{x^2}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx\\ &=-\frac{x \left (2+x^2\right )}{3 \sqrt{1+x^2+x^4}}+\frac{x \sqrt{1+x^2+x^4}}{3 \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 )}{6 \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}{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{x \left (2+x^2\right )}{3 \sqrt{1+x^2+x^4}}+\frac{x \sqrt{1+x^2+x^4}}{3 \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 )}{6 \sqrt{1+x^2+x^4}}+\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 \left (2+x^2\right )}{3 \sqrt{1+x^2+x^4}}+\frac{x \sqrt{1+x^2+x^4}}{3 \left (1+x^2\right )}+\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 )}{6 \sqrt{1+x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.379126, size = 168, normalized size = 1.51 \[ \frac{-\sqrt [3]{-1} \left (x^2+1\right ) \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left (\left (5 \sqrt [3]{-1}-1\right ) \text{EllipticF}\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+12 \sqrt [3]{-1} \Pi \left (\sqrt [3]{-1};-i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )\right )-2 x \left (x^2+1\right ) \left (x^2+2\right )+3 x \left (x^4+x^2+1\right )}{6 \left (x^2+1\right ) \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.025, size = 419, normalized size = 3.8 \begin{align*}{\frac{x}{2\,{x}^{2}+2}\sqrt{{x}^{4}+{x}^{2}+1}}-2\,{\frac{1/6\,{x}^{3}+x/3}{\sqrt{{x}^{4}+{x}^{2}+1}}}-{\frac{5}{3\,\sqrt{-2+2\,i\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 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{2}{3\,\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{2}{3\,\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}}}}+2\,{\frac{\sqrt{1+1/2\,{x}^{2}-i/2{x}^{2}\sqrt{3}}\sqrt{1+1/2\,{x}^{2}+i/2{x}^{2}\sqrt{3}}}{\sqrt{-1/2+i/2\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1}}{\it EllipticPi} \left ( \sqrt{-1/2+i/2\sqrt{3}}x,- \left ( -1/2+i/2\sqrt{3} \right ) ^{-1},{\frac{\sqrt{-1/2-i/2\sqrt{3}}}{\sqrt{-1/2+i/2\sqrt{3}}}} \right ) } \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{1}{{\left (x^{4} + x^{2} + 1\right )}^{\frac{3}{2}}{\left (x^{2} + 1\right )}^{2}}\,{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^{12} + 4 \, x^{10} + 8 \, x^{8} + 10 \, x^{6} + 8 \, x^{4} + 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{1}{\left (\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{3}{2}} \left (x^{2} + 1\right )^{2}}\, 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{1}{{\left (x^{4} + x^{2} + 1\right )}^{\frac{3}{2}}{\left (x^{2} + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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