Integrand size = 20, antiderivative size = 142 \[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac {1}{4} \arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{4 \sqrt {1+x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}} \]
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Time = 0.17 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1237, 1710, 1607, 1726, 1209, 1714, 1117, 1712, 209} \[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\frac {1}{4} \arctan \left (\frac {x}{\sqrt {x^4+x^2+1}}\right )-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{2 \sqrt {x^4+x^2+1}}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{4 \sqrt {x^4+x^2+1}}+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2} \]
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Rule 209
Rule 1117
Rule 1209
Rule 1237
Rule 1607
Rule 1710
Rule 1712
Rule 1714
Rule 1726
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}-\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 {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac {3 x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )}+\frac {1}{8} \int \frac {-10 x^2-6 x^4}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx \\ & = \frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac {3 x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )}+\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 \\ & = \frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac {3 x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )}+\frac {1}{8} \int \frac {-6-10 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}}{4 \left (1+x^2\right )^2}+\frac {3 \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 {1}{4} \int \frac {1-x^2}{\left (1+x^2\right ) \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}}{4 \left (1+x^2\right )^2}+\frac {3 \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 )}{2 \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}}{4 \left (1+x^2\right )^2}+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {3 \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 )}{2 \sqrt {1+x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.26 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\frac {\frac {x \left (4+3 x^2\right ) \left (1+x^2+x^4\right )}{\left (1+x^2\right )^2}-3 \sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \left (E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )\right )-2 (-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+2 (-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticPi}\left (\sqrt [3]{-1},i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )}{4 \sqrt {1+x^2+x^4}} \]
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Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.37
method | result | size |
risch | \(\frac {\sqrt {x^{4}+x^{2}+1}\, x \left (3 x^{2}+4\right )}{4 \left (x^{2}+1\right )^{2}}-\frac {\sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {3 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(336\) |
default | \(\frac {x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )^{2}}+\frac {3 x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {3 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {3 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(418\) |
elliptic | \(\frac {x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )^{2}}+\frac {3 x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {3 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {3 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(418\) |
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Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=-\frac {3 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} - \sqrt {-3} {\left (x^{4} + 2 \, x^{2} + 1\right )} + 1\right )} \sqrt {\sqrt {-3} - 1} E(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 2 \, \sqrt {2} {\left (2 \, x^{4} + 4 \, x^{2} - \sqrt {-3} {\left (x^{4} + 2 \, x^{2} + 1\right )} + 2\right )} \sqrt {\sqrt {-3} - 1} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) - 4 \, \sqrt {x^{4} + x^{2} + 1} {\left (3 \, x^{3} + 4 \, x\right )}}{16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
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\[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\int \frac {1}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )^{3}}\, dx \]
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\[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\int \frac {1}{{\left (x^2+1\right )}^3\,\sqrt {x^4+x^2+1}} \,d x \]
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