Optimal. Leaf size=69 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right )+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {x^4+x^2+1}} \]
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Rubi [A] time = 0.06, antiderivative size = 69, 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 = {1210, 1103, 1698, 203} \[ \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right )+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 1103
Rule 1210
Rule 1698
Rubi steps
\begin {align*} \int \frac {1}{\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 {\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} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\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}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 72, normalized size = 1.04 \[ \frac {(-1)^{2/3} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} \Pi \left (\sqrt [3]{-1};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )}{\sqrt {x^4+x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + x^{2} + 1}}{x^{6} + 2 \, x^{4} + 2 \, x^{2} + 1}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 104, normalized size = 1.51 \[ \frac {\sqrt {\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}+1}\, \sqrt {\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}+1}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}} \]
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 \[ \int \frac {1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (x^2+1\right )\,\sqrt {x^4+x^2+1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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