Optimal. Leaf size=144 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{4}\right )}{\sqrt{x^4+x^2+1}}+\frac{2 \sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}-\frac{\left (1-x^2\right ) x}{3 \sqrt{x^4+x^2+1}}-\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]
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Rubi [A] time = 0.0445113, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1205, 1197, 1103, 1195} \[ \frac{2 \sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}-\frac{\left (1-x^2\right ) x}{3 \sqrt{x^4+x^2+1}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{\sqrt{x^4+x^2+1}}-\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1205
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \frac{\left (1+x^2\right )^3}{\left (1+x^2+x^4\right )^{3/2}} \, dx &=-\frac{x \left (1-x^2\right )}{3 \sqrt{1+x^2+x^4}}+\frac{1}{3} \int \frac{4+2 x^2}{\sqrt{1+x^2+x^4}} \, dx\\ &=-\frac{x \left (1-x^2\right )}{3 \sqrt{1+x^2+x^4}}-\frac{2}{3} \int \frac{1-x^2}{\sqrt{1+x^2+x^4}} \, dx+2 \int \frac{1}{\sqrt{1+x^2+x^4}} \, dx\\ &=-\frac{x \left (1-x^2\right )}{3 \sqrt{1+x^2+x^4}}+\frac{2 x \sqrt{1+x^2+x^4}}{3 \left (1+x^2\right )}-\frac{2 \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 )}{\sqrt{1+x^2+x^4}}\\ \end{align*}
Mathematica [F] time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [C] time = 0.033, size = 268, normalized size = 1.9 \begin{align*} -4\,{\frac{-x/6+1/6\,{x}^{3}}{\sqrt{{x}^{4}+{x}^{2}+1}}}+{\frac{8}{3\,\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}{\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{8}{3\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}-6\,{\frac{1/6\,{x}^{3}+x/3}{\sqrt{{x}^{4}+{x}^{2}+1}}}-6\,{\frac{-1/3\,{x}^{3}-x/6}{\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{{\left (x^{2} + 1\right )}^{3}}{{\left (x^{4} + x^{2} + 1\right )}^{\frac{3}{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{{\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} \sqrt{x^{4} + x^{2} + 1}}{x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, 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{\left (x^{2} + 1\right )^{3}}{\left (\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{3}{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{{\left (x^{2} + 1\right )}^{3}}{{\left (x^{4} + x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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