Integrand size = 27, antiderivative size = 186 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=-4 \sqrt [4]{2} \text {RootSum}\left [1-4 \text {$\#$1}^2-122 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (-\sqrt {2}+2^{3/4} x-x^2\right ) \text {$\#$1}+\log \left (\sqrt {2+x^4}+\sqrt {2} \text {$\#$1}-2^{3/4} x \text {$\#$1}+x^2 \text {$\#$1}\right ) \text {$\#$1}-\log \left (-\sqrt {2}+2^{3/4} x-x^2\right ) \text {$\#$1}^3+\log \left (\sqrt {2+x^4}+\sqrt {2} \text {$\#$1}-2^{3/4} x \text {$\#$1}+x^2 \text {$\#$1}\right ) \text {$\#$1}^3}{-1-61 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \]
[Out]
Result contains complex when optimal does not.
Time = 1.88 (sec) , antiderivative size = 1539, normalized size of antiderivative = 8.27, number of steps used = 20, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6860, 415, 226, 418, 1231, 1721} \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=-\frac {\sqrt {1-i \sqrt {7}} \left (5 i+\sqrt {7}\right ) \arctan \left (\frac {\sqrt {1-i \sqrt {7}} x}{\sqrt [4]{2 \left (-3-i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2 \left (3 i-\sqrt {7}\right ) \left (2 \left (-3-i \sqrt {7}\right )\right )^{3/4}}-\frac {\left (1-3 i \sqrt {7}\right ) \arctan \left (\frac {\sqrt {-1-i \sqrt {7}} x}{\sqrt [4]{2 \left (-3+i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2^{3/4} \sqrt {-1-i \sqrt {7}} \left (-3+i \sqrt {7}\right )^{7/4}}-\frac {\left (1+3 i \sqrt {7}\right ) \arctan \left (\frac {\sqrt {-1+i \sqrt {7}} x}{\sqrt [4]{2 \left (-3-i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2^{3/4} \left (-3-i \sqrt {7}\right )^{7/4} \sqrt {-1+i \sqrt {7}}}-\frac {\left (5 i-\sqrt {7}\right ) \sqrt {1+i \sqrt {7}} \arctan \left (\frac {\sqrt {1+i \sqrt {7}} x}{\sqrt [4]{2 \left (-3+i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2 \left (2 \left (-3+i \sqrt {7}\right )\right )^{3/4} \left (3 i+\sqrt {7}\right )}+\frac {\left (1+i \sqrt {7}\right )^2 \left (1+\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7-i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1+i \sqrt {7}\right )^2 \left (1-\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7-i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1-i \sqrt {7}\right )^2 \left (1+\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7+i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1-i \sqrt {7}\right )^2 \left (1-\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7+i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1+i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {x^4+2}}+\frac {\left (1-i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left (3 i-\sqrt {7}\right ) \left (2+\sqrt {-3-i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (2-\sqrt {-3-i \sqrt {7}}\right )^2}{8 \sqrt {-3-i \sqrt {7}}},2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i-5 \sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (3 i-\sqrt {7}\right ) \left (2-\sqrt {-3-i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (2+\sqrt {-3-i \sqrt {7}}\right )^2}{8 \sqrt {-3-i \sqrt {7}}},2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i-5 \sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (3 i+\sqrt {7}\right ) \left (2+\sqrt {-3+i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (2-\sqrt {-3+i \sqrt {7}}\right )^2}{8 \sqrt {-3+i \sqrt {7}}},2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i+5 \sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (3 i+\sqrt {7}\right ) \left (2-\sqrt {-3+i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (2+\sqrt {-3+i \sqrt {7}}\right )^2}{8 \sqrt {-3+i \sqrt {7}}},2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i+5 \sqrt {7}\right ) \sqrt {x^4+2}} \]
[In]
[Out]
Rule 226
Rule 415
Rule 418
Rule 1231
Rule 1721
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1+i \sqrt {7}\right ) \sqrt {2+x^4}}{3-i \sqrt {7}+2 x^4}+\frac {\left (1-i \sqrt {7}\right ) \sqrt {2+x^4}}{3+i \sqrt {7}+2 x^4}\right ) \, dx \\ & = \left (1-i \sqrt {7}\right ) \int \frac {\sqrt {2+x^4}}{3+i \sqrt {7}+2 x^4} \, dx+\left (1+i \sqrt {7}\right ) \int \frac {\sqrt {2+x^4}}{3-i \sqrt {7}+2 x^4} \, dx \\ & = \left (-3-i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4} \left (3+i \sqrt {7}+2 x^4\right )} \, dx+\frac {1}{2} \left (1-i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx+\left (-3+i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4} \left (3-i \sqrt {7}+2 x^4\right )} \, dx+\frac {1}{2} \left (1+i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx \\ & = \frac {\left (1-i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}+\frac {\left (1+i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {1}{2} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (-3-i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (-3-i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (-3+i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (-3+i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx \\ & = \frac {\left (1-i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}+\frac {\left (1+i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {\left (1-\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{2 \left (1+\frac {4 i}{3 i-\sqrt {7}}\right )}-\frac {\left (1+\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{2 \left (1+\frac {4 i}{3 i-\sqrt {7}}\right )}-\frac {\left (2-\sqrt {-3-i \sqrt {7}}\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (-3-i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx}{7+i \sqrt {7}}-\frac {\left (2+\sqrt {-3-i \sqrt {7}}\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (-3-i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx}{7+i \sqrt {7}}-\frac {\left (2-\sqrt {-3+i \sqrt {7}}\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (-3+i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx}{7-i \sqrt {7}}-\frac {\left (2+\sqrt {-3+i \sqrt {7}}\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (-3+i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx}{7-i \sqrt {7}}-\frac {\left (1-\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{2 \left (1+\frac {4 i}{3 i+\sqrt {7}}\right )}-\frac {\left (1+\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{2 \left (1+\frac {4 i}{3 i+\sqrt {7}}\right )} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.18 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=-\frac {1}{2} \arctan \left (\frac {x}{\sqrt {2+x^4}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {2+x^4}}\right ) \]
[In]
[Out]
Time = 5.33 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.22
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+2}}{x}\right )}{2}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+2}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(41\) |
trager | \(-\frac {\ln \left (-\frac {x^{4}+2 x \sqrt {x^{4}+2}+x^{2}+2}{x^{4}-x^{2}+2}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{4}+2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{2}+2}\right )}{4}\) | \(98\) |
default | \(\frac {\operatorname {arctanh}\left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {1+2 \sqrt {2}}+2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}-\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {-1+2 \sqrt {2}}+2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}+\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {-1+2 \sqrt {2}}-2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}-\frac {\operatorname {arctanh}\left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {1+2 \sqrt {2}}-2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}\) | \(142\) |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {1+2 \sqrt {2}}+2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}-\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {-1+2 \sqrt {2}}+2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}+\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {-1+2 \sqrt {2}}-2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}-\frac {\operatorname {arctanh}\left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {1+2 \sqrt {2}}-2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}\) | \(142\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.32 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=-\frac {1}{4} \, \arctan \left (\frac {2 \, \sqrt {x^{4} + 2} x}{x^{4} - x^{2} + 2}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} + x^{2} - 2 \, \sqrt {x^{4} + 2} x + 2}{x^{4} - x^{2} + 2}\right ) \]
[In]
[Out]
Timed out. \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=\text {Timed out} \]
[In]
[Out]
Not integrable
Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=\int { \frac {\sqrt {x^{4} + 2} {\left (x^{4} - 2\right )}}{x^{8} + 3 \, x^{4} + 4} \,d x } \]
[In]
[Out]
Not integrable
Time = 1.65 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=\int { \frac {\sqrt {x^{4} + 2} {\left (x^{4} - 2\right )}}{x^{8} + 3 \, x^{4} + 4} \,d x } \]
[In]
[Out]
Not integrable
Time = 6.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=\int \frac {\left (x^4-2\right )\,\sqrt {x^4+2}}{x^8+3\,x^4+4} \,d x \]
[In]
[Out]