Integrand size = 20, antiderivative size = 166 \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^4} \, dx=\frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^3}+\frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^2}+\frac {1}{4} \arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{3 \sqrt {1+x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{8 \sqrt {1+x^2+x^4}} \]
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Time = 0.37 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {1242, 1237, 1710, 1600, 1211, 1117, 1209, 1607, 1726, 1714, 1712, 209, 12, 1331} \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^4} \, dx=\frac {1}{4} \arctan \left (\frac {x}{\sqrt {x^4+x^2+1}}\right )-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{8 \sqrt {x^4+x^2+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{3 \sqrt {x^4+x^2+1}}+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^2}+\frac {\sqrt {x^4+x^2+1} x}{6 \left (x^2+1\right )^3} \]
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Rule 12
Rule 209
Rule 1117
Rule 1209
Rule 1211
Rule 1237
Rule 1242
Rule 1331
Rule 1600
Rule 1607
Rule 1710
Rule 1712
Rule 1714
Rule 1726
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\left (1+x^2\right )^4 \sqrt {1+x^2+x^4}}-\frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}}+\frac {1}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^4}}\right ) \, dx \\ & = \int \frac {1}{\left (1+x^2\right )^4 \sqrt {1+x^2+x^4}} \, dx-\int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx+\int \frac {1}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^4}} \, dx \\ & = \frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^3}-\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac {x \sqrt {1+x^2+x^4}}{2 \left (1+x^2\right )}-\frac {1}{6} \int \frac {-5+2 x^2-3 x^4}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx+\frac {1}{4} \int \frac {-3+2 x^2-x^4}{\left (1+x^2\right )^2 \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 \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^3}+\frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^2}-\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )}+\frac {1}{24} \int \frac {10-8 x^2+10 x^4}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^4}} \, dx-\frac {1}{8} \int \frac {-10 x^2-6 x^4}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\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 {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^3}+\frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^2}-\frac {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 )}{2 \sqrt {1+x^2+x^4}}-\frac {1}{48} \int \frac {8+36 x^2+28 x^4}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx-\frac {1}{8} \int \frac {x^2 \left (-10-6 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx-\int \frac {x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx \\ & = \frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^3}+\frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^2}-\frac {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 )}{2 \sqrt {1+x^2+x^4}}-\frac {1}{48} \int \frac {8+28 x^2}{\sqrt {1+x^2+x^4}} \, dx-\frac {1}{8} \int \frac {-6-10 x^2}{\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 {3}{4} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx \\ & = \frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^3}+\frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^2}+\frac {7 x \sqrt {1+x^2+x^4}}{12 \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 )}{4 \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}{4} \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {7}{12} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx-\frac {3}{4} \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx+\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx \\ & = \frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^3}+\frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^2}+\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 )}{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 )}{8 \sqrt {1+x^2+x^4}}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right ) \\ & = \frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^3}+\frac {x \sqrt {1+x^2+x^4}}{6 \left (1+x^2\right )^2}+\frac {1}{4} \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 )}{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 )}{8 \sqrt {1+x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.31 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^4} \, dx=\frac {\frac {x \left (1+x^2+x^4\right ) \left (4+5 x^2+2 x^4\right )}{\left (1+x^2\right )^3}-2 \sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \left (E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )\right )-(-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+3 (-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticPi}\left (\sqrt [3]{-1},i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )}{6 \sqrt {1+x^2+x^4}} \]
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Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.05
method | result | size |
risch | \(\frac {\sqrt {x^{4}+x^{2}+1}\, x \left (2 x^{4}+5 x^{2}+4\right )}{6 \left (x^{2}+1\right )^{3}}-\frac {\sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(341\) |
default | \(\frac {x \sqrt {x^{4}+x^{2}+1}}{6 \left (x^{2}+1\right )^{3}}+\frac {x \sqrt {x^{4}+x^{2}+1}}{6 \left (x^{2}+1\right )^{2}}+\frac {x \sqrt {x^{4}+x^{2}+1}}{3 x^{2}+3}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(438\) |
elliptic | \(\frac {x \sqrt {x^{4}+x^{2}+1}}{6 \left (x^{2}+1\right )^{3}}+\frac {x \sqrt {x^{4}+x^{2}+1}}{6 \left (x^{2}+1\right )^{2}}+\frac {x \sqrt {x^{4}+x^{2}+1}}{3 x^{2}+3}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(438\) |
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Time = 0.10 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^4} \, dx=-\frac {4 \, \sqrt {2} {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} - \sqrt {-3} {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} + 1\right )} \sqrt {\sqrt {-3} - 1} E(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - \sqrt {2} {\left (3 \, x^{6} + 9 \, x^{4} + 9 \, x^{2} - 5 \, \sqrt {-3} {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {\sqrt {-3} - 1} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 12 \, {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) - 8 \, {\left (2 \, x^{5} + 5 \, x^{3} + 4 \, x\right )} \sqrt {x^{4} + x^{2} + 1}}{48 \, {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )}} \]
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\[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^4} \, dx=\int \frac {\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}{\left (x^{2} + 1\right )^{4}}\, dx \]
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\[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^4} \, dx=\int { \frac {\sqrt {x^{4} + x^{2} + 1}}{{\left (x^{2} + 1\right )}^{4}} \,d x } \]
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\[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^4} \, dx=\int { \frac {\sqrt {x^{4} + x^{2} + 1}}{{\left (x^{2} + 1\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^4} \, dx=\int \frac {\sqrt {x^4+x^2+1}}{{\left (x^2+1\right )}^4} \,d x \]
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