Integrand size = 18, antiderivative size = 145 \[ \int \left (1+x^2\right ) \sqrt {1+x^2+x^4} \, dx=\frac {3 x \sqrt {1+x^2+x^4}}{5 \left (1+x^2\right )}+\frac {1}{5} x \left (2+x^2\right ) \sqrt {1+x^2+x^4}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{5 \sqrt {1+x^2+x^4}}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{5 \sqrt {1+x^2+x^4}} \]
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Time = 0.03 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1190, 1211, 1117, 1209} \[ \int \left (1+x^2\right ) \sqrt {1+x^2+x^4} \, dx=\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{5 \sqrt {x^4+x^2+1}}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{5 \sqrt {x^4+x^2+1}}+\frac {1}{5} \left (x^2+2\right ) \sqrt {x^4+x^2+1} x+\frac {3 \sqrt {x^4+x^2+1} x}{5 \left (x^2+1\right )} \]
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Rule 1117
Rule 1190
Rule 1209
Rule 1211
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x \left (2+x^2\right ) \sqrt {1+x^2+x^4}+\frac {1}{15} \int \frac {9+9 x^2}{\sqrt {1+x^2+x^4}} \, dx \\ & = \frac {1}{5} x \left (2+x^2\right ) \sqrt {1+x^2+x^4}-\frac {3}{5} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx+\frac {6}{5} \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx \\ & = \frac {3 x \sqrt {1+x^2+x^4}}{5 \left (1+x^2\right )}+\frac {1}{5} x \left (2+x^2\right ) \sqrt {1+x^2+x^4}-\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 )}{5 \sqrt {1+x^2+x^4}}+\frac {3 \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 )}{5 \sqrt {1+x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 4.31 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.16 \[ \int \left (1+x^2\right ) \sqrt {1+x^2+x^4} \, dx=\frac {2 x+3 x^3+3 x^5+x^7+3 \sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+\frac {3}{2} \sqrt {2+\left (1-i \sqrt {3}\right ) x^2} \sqrt {2+\left (1+i \sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \left (x+i \sqrt {3} x\right )\right ),\frac {1}{2} i \left (i+\sqrt {3}\right )\right )}{5 \sqrt {1+x^2+x^4}} \]
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Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.54
method | result | size |
risch | \(\frac {x \left (x^{2}+2\right ) \sqrt {x^{4}+x^{2}+1}}{5}+\frac {6 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{5 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {12 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{5 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}\) | \(223\) |
default | \(\frac {2 x \sqrt {x^{4}+x^{2}+1}}{5}+\frac {6 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{5 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {12 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{5 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {x^{3} \sqrt {x^{4}+x^{2}+1}}{5}\) | \(233\) |
elliptic | \(\frac {2 x \sqrt {x^{4}+x^{2}+1}}{5}+\frac {6 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{5 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {12 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{5 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {x^{3} \sqrt {x^{4}+x^{2}+1}}{5}\) | \(233\) |
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Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.76 \[ \int \left (1+x^2\right ) \sqrt {1+x^2+x^4} \, dx=\frac {3 \, \sqrt {2} {\left (\sqrt {-3} x - x\right )} \sqrt {\sqrt {-3} - 1} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-3} - 1}}{2 \, x}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) + 6 \, \sqrt {2} x \sqrt {\sqrt {-3} - 1} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-3} - 1}}{2 \, x}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) + 4 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {x^{4} + x^{2} + 1}}{20 \, x} \]
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\[ \int \left (1+x^2\right ) \sqrt {1+x^2+x^4} \, dx=\int \sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )\, dx \]
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\[ \int \left (1+x^2\right ) \sqrt {1+x^2+x^4} \, dx=\int { \sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )} \,d x } \]
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\[ \int \left (1+x^2\right ) \sqrt {1+x^2+x^4} \, dx=\int { \sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )} \,d x } \]
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Timed out. \[ \int \left (1+x^2\right ) \sqrt {1+x^2+x^4} \, dx=\int \left (x^2+1\right )\,\sqrt {x^4+x^2+1} \,d x \]
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