Optimal. Leaf size=137 \[ \frac {4 \sqrt {x^4+x^2+1} x}{3 \left (x^2+1\right )}+\frac {1}{3} \sqrt {x^4+x^2+1} x+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {x^4+x^2+1}}-\frac {4 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{3 \sqrt {x^4+x^2+1}} \]
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Rubi [A] time = 0.04, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1206, 1197, 1103, 1195} \[ \frac {4 \sqrt {x^4+x^2+1} x}{3 \left (x^2+1\right )}+\frac {1}{3} \sqrt {x^4+x^2+1} x+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {x^4+x^2+1}}-\frac {4 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{3 \sqrt {x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1195
Rule 1197
Rule 1206
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right )^2}{\sqrt {1+x^2+x^4}} \, dx &=\frac {1}{3} x \sqrt {1+x^2+x^4}+\frac {1}{3} \int \frac {2+4 x^2}{\sqrt {1+x^2+x^4}} \, dx\\ &=\frac {1}{3} x \sqrt {1+x^2+x^4}-\frac {4}{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 {1}{3} x \sqrt {1+x^2+x^4}+\frac {4 x \sqrt {1+x^2+x^4}}{3 \left (1+x^2\right )}-\frac {4 \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*}
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Mathematica [C] time = 0.14, size = 143, normalized size = 1.04 \[ \frac {x^5+x^3+2 \sqrt [3]{-1} \left (\sqrt [3]{-1}-2\right ) \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+4 \sqrt [3]{-1} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+x}{3 \sqrt {x^4+x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4} + 2 \, x^{2} + 1}{\sqrt {x^{4} + x^{2} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{2} + 1\right )}^{2}}{\sqrt {x^{4} + x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 218, normalized size = 1.59 \[ \frac {\sqrt {x^{4}+x^{2}+1}\, x}{3}+\frac {4 \sqrt {-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {16 \sqrt {-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )+\EllipticF \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{2} + 1\right )}^{2}}{\sqrt {x^{4} + x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (x^2+1\right )}^2}{\sqrt {x^4+x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x^{2} + 1\right )^{2}}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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