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 = 0.28, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, 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 12
Rule 203
Rule 1103
Rule 1178
Rule 1195
Rule 1210
Rule 1223
Rule 1228
Rule 1317
Rule 1698
Rule 1712
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*}
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Mathematica [C] time = 0.38, size = 168, normalized size = 1.51 \[ \frac {-2 x \left (x^2+1\right ) \left (x^2+2\right )-\sqrt [3]{-1} \left (x^2+1\right ) \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} \left (\left (5 \sqrt [3]{-1}-1\right ) F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-12 \sqrt [3]{-1} \Pi \left (\sqrt [3]{-1};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )\right )+3 x \left (x^4+x^2+1\right )}{6 \left (x^2+1\right ) \sqrt {x^4+x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + x^{2} + 1}}{x^{12} + 4 \, x^{10} + 8 \, x^{8} + 10 \, x^{6} + 8 \, x^{4} + 4 \, x^{2} + 1}, x\right ) \]
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 \[ \int \frac {1}{{\left (x^{4} + x^{2} + 1\right )}^{\frac {3}{2}} {\left (x^{2} + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 419, normalized size = 3.77 \[ \frac {\sqrt {x^{4}+x^{2}+1}\, x}{2 x^{2}+2}-\frac {2 \sqrt {\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}+1}\, \sqrt {\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}+1}\, \EllipticE \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {5 \sqrt {\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}+1}\, \sqrt {\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}+1}\, \EllipticF \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {2 \sqrt {\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}+1}\, \sqrt {\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}+1}\, \EllipticF \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {2 \sqrt {\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}+1}\, \sqrt {\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}+1}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {2 \left (\frac {1}{6} x^{3}+\frac {1}{3} x \right )}{\sqrt {x^{4}+x^{2}+1}} \]
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 \[ \int \frac {1}{{\left (x^{4} + x^{2} + 1\right )}^{\frac {3}{2}} {\left (x^{2} + 1\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{{\left (x^2+1\right )}^2\,{\left (x^4+x^2+1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {3}{2}} \left (x^{2} + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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