Optimal. Leaf size=520 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^6-6 \text {$\#$1}^4+2 \text {$\#$1}^2+1\& ,\frac {-3 \text {$\#$1}^6 \log \left (\sqrt {x^4+1}+x^2-1\right )+3 \text {$\#$1}^6 \log \left (\text {$\#$1} \sqrt {x^4+1}+\text {$\#$1} x^2-\text {$\#$1}+\sqrt {2} \sqrt {\sqrt {x^4+1}+x^2} x\right )-\text {$\#$1}^4 \log \left (\sqrt {x^4+1}+x^2-1\right )+\text {$\#$1}^4 \log \left (\text {$\#$1} \sqrt {x^4+1}+\text {$\#$1} x^2-\text {$\#$1}+\sqrt {2} \sqrt {\sqrt {x^4+1}+x^2} x\right )+7 \text {$\#$1}^2 \log \left (\sqrt {x^4+1}+x^2-1\right )-7 \text {$\#$1}^2 \log \left (\text {$\#$1} \sqrt {x^4+1}+\text {$\#$1} x^2-\text {$\#$1}+\sqrt {2} \sqrt {\sqrt {x^4+1}+x^2} x\right )-\log \left (\text {$\#$1} \sqrt {x^4+1}+\text {$\#$1} x^2-\text {$\#$1}+\sqrt {2} \sqrt {\sqrt {x^4+1}+x^2} x\right )+\log \left (\sqrt {x^4+1}+x^2-1\right )}{2 \text {$\#$1}^7-3 \text {$\#$1}^5-6 \text {$\#$1}^3+\text {$\#$1}}\& \right ]}{5 \sqrt {2}}+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\frac {-2 x \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2} \left (2 x^4+3 x^2+2\right )-2 x \left (2 x^6+3 x^4+3 x^2+1\right ) \sqrt {\sqrt {x^4+1}+x^2}}{10 \sqrt {x^4+1} \left (x^4+x^2-1\right ) x^2+5 \left (x^4+x^2-1\right ) \left (2 x^4+1\right )} \]
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Rubi [C] time = 11.28, antiderivative size = 2475, normalized size of antiderivative = 4.76, number of steps used = 112, number of rules used = 8, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {6742, 2132, 206, 2133, 731, 725, 6725, 6728}
result too large to display
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 206
Rule 725
Rule 731
Rule 2132
Rule 2133
Rule 6725
Rule 6728
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx &=\int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )}\right ) \, dx\\ &=4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx+4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx+\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx\\ &=4 \int \left (-\frac {2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {5} \left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}}-\frac {2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {5} \left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+4 \int \left (\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \left (-1+\sqrt {5}-2 x^2\right )^2 \sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \sqrt {5} \left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \left (1+\sqrt {5}+2 x^2\right )^2 \sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \sqrt {5} \left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {16}{5} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}-2 x^2\right )^2 \sqrt {1+x^4}} \, dx+\frac {16}{5} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right )^2 \sqrt {1+x^4}} \, dx+\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5}}+\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5}}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5}}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5}}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {16}{5} \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-1+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-1+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}\right ) \left (2 \left (-1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+\frac {16}{5} \int \left (-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}\right ) \left (-2 \left (1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+\frac {16 \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{5 \sqrt {5}}+\frac {16 \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{5 \sqrt {5}}-\frac {8 \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{\sqrt {5}}-\frac {8 \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{\sqrt {5}}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )}+\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (2 \left (-1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )}-\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-2 \left (1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )}+\frac {(8 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {(8 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(4 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(4 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+i x^2}} \, dx}{1-\sqrt {5}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+i x^2}} \, dx}{1-\sqrt {5}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1-i x^2}} \, dx}{1-\sqrt {5}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1-i x^2}} \, dx}{1-\sqrt {5}}+\frac {16 \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{4 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{4 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{5 \left (-1+\sqrt {5}\right )}+\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {(2-2 i) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {(2-2 i) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {(2+2 i) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {(2+2 i) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1-i x^2}} \, dx}{1+\sqrt {5}}-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1-i x^2}} \, dx}{1+\sqrt {5}}-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+i x^2}} \, dx}{1+\sqrt {5}}-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+i x^2}} \, dx}{1+\sqrt {5}}-\frac {16 \int \left (-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{5 \left (1+\sqrt {5}\right )}--\frac {(2-2 i) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}--\frac {(2-2 i) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(2+2 i) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(2+2 i) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}\\ &=\frac {2 i \sqrt {1-i x^2}}{\left ((10+5 i)-5 \sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((-2-i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}-\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+-\frac {\left ((2-2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right ) \sqrt {1-i x^2}} \, dx}{(5-10 i)-5 \sqrt {5}}+-\frac {\left ((2-2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right ) \sqrt {1-i x^2}} \, dx}{(5-10 i)-5 \sqrt {5}}+-\frac {\left ((2+2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right ) \sqrt {1+i x^2}} \, dx}{(5+10 i)-5 \sqrt {5}}+-\frac {\left ((2+2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \int \frac {1}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right ) \sqrt {1+i x^2}} \, dx}{(5+10 i)-5 \sqrt {5}}+\frac {4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )^{3/2}}+\frac {4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )^{3/2}}--\frac {(2+2 i) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}--\frac {(2+2 i) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}--\frac {(2-2 i) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}--\frac {(2-2 i) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {(4 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )^{3/2}}+\frac {(4 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )^{3/2}}--\frac {(2+2 i) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}--\frac {(2+2 i) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}+-\frac {\left (\frac {4}{5}+\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\left (\frac {4}{5}-\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(2-2 i) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(2-2 i) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {5 \left (1+\sqrt {5}\right )}}--\frac {\left ((2+2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right ) \sqrt {1-i x^2}} \, dx}{(5-10 i)+5 \sqrt {5}}--\frac {\left ((2+2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right ) \sqrt {1-i x^2}} \, dx}{(5-10 i)+5 \sqrt {5}}-\frac {\left ((2-2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right ) \sqrt {1+i x^2}} \, dx}{(5+10 i)+5 \sqrt {5}}-\frac {\left ((2-2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \int \frac {1}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right ) \sqrt {1+i x^2}} \, dx}{(5+10 i)+5 \sqrt {5}}\\ &=\frac {2 i \sqrt {1-i x^2}}{\left ((10+5 i)-5 \sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((-2-i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}-\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}-\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}+\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}-\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}+\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {\left ((2-2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4-2 i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {1-i x^2}}\right )}{(5-10 i)-5 \sqrt {5}}+\frac {\left ((2-2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4-2 i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {1-i x^2}}\right )}{(5-10 i)-5 \sqrt {5}}+\frac {\left ((2+2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4+2 i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {1+i x^2}}\right )}{(5+10 i)-5 \sqrt {5}}+\frac {\left ((2+2 i) \sqrt {\frac {2}{-1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4+2 i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {1+i x^2}}\right )}{(5+10 i)-5 \sqrt {5}}+\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\left (-1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\left (-1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\left (-1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\left (-1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\left (1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\left (1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\left (1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\left (1+\sqrt {5}\right )^{3/2}}-\frac {\left ((2+2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4+2 i \left (1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {1-i x^2}}\right )}{(5-10 i)+5 \sqrt {5}}-\frac {\left ((2+2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4+2 i \left (1+\sqrt {5}\right )-x^2} \, dx,x,\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {1-i x^2}}\right )}{(5-10 i)+5 \sqrt {5}}--\frac {\left ((2-2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4-2 i \left (1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {1+i x^2}}\right )}{(5+10 i)+5 \sqrt {5}}--\frac {\left ((2-2 i) \sqrt {\frac {2}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{4-2 i \left (1+\sqrt {5}\right )-x^2} \, dx,x,\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {1+i x^2}}\right )}{(5+10 i)+5 \sqrt {5}}\\ &=\frac {2 i \sqrt {1-i x^2}}{\left ((10+5 i)-5 \sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((-2-i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}-\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}-\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}+\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}+\frac {2 i \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)+\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(3+i)-(1+i) \sqrt {5}} \left ((-1-2 i)+\sqrt {5}\right )}-\frac {2 i \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)+\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(3+i)-(1+i) \sqrt {5}} \left ((-1-2 i)+\sqrt {5}\right )}-\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}+\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}+\frac {2 \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2-\sqrt {2 \left (1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)-\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \left ((1+2 i)+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}-\frac {2 \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2+\sqrt {2 \left (1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)-\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \left ((1+2 i)+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {1}{25}-\frac {3 i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)-2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3-\sqrt {5}}-\frac {\left (\frac {1}{25}-\frac {3 i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)-2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3-\sqrt {5}}+\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{25}+\frac {i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((2+i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)+2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3+\sqrt {5}}+\frac {\left (\frac {3}{25}+\frac {i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((2+i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)+2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3+\sqrt {5}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+-\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\left (-1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\left (-1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\left (-1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\left (-1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\left (1+\sqrt {5}\right )^{3/2}}+-\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\left (1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\left (1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\left (1+\sqrt {5}\right )^{3/2}}\\ &=\frac {2 i \sqrt {1-i x^2}}{\left ((10+5 i)-5 \sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((-2-i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}-\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {1+i x^2}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )}+\frac {2 i \sqrt {1+i x^2}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )}-\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}+\frac {2 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \left (-1+\sqrt {5}\right )^{3/2} \sqrt {\frac {1}{2} \left ((-1-2 i)+\sqrt {5}\right )}}+\frac {6 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}-\frac {2 i \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{5 \left (-1+\sqrt {5}\right )^{3/2} \sqrt {\frac {1}{2} \left ((-1-2 i)+\sqrt {5}\right )}}+\frac {2 i \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)+\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(3+i)-(1+i) \sqrt {5}} \left ((-1-2 i)+\sqrt {5}\right )}-\frac {2 i \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)+\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(3+i)-(1+i) \sqrt {5}} \left ((-1-2 i)+\sqrt {5}\right )}+\frac {4 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-2-4 i)-2 \sqrt {5}} \left (1+\sqrt {5}\right )^{3/2}}-\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}-\frac {4 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-2-4 i)-2 \sqrt {5}} \left (1+\sqrt {5}\right )^{3/2}}+\frac {6 \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}+\frac {2 \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2-\sqrt {2 \left (1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)-\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \left ((1+2 i)+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}-\frac {2 \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2+\sqrt {2 \left (1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)-\sqrt {5}} \sqrt {1+i x^2}}\right )}{5 \left ((1+2 i)+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(2+i)-i \sqrt {5}} \left (-1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}-\frac {\left (\frac {2}{5}-\frac {2 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(2+i)-i \sqrt {5}} \left (-1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {1}{25}-\frac {3 i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)-2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3-\sqrt {5}}-\frac {\left (\frac {1}{25}-\frac {3 i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)-2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3-\sqrt {5}}-\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(2+i)+i \sqrt {5}} \left (1+\sqrt {5}\right )^{3/2}}+\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}+\frac {\left (\frac {2}{5}+\frac {2 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(2+i)+i \sqrt {5}} \left (1+\sqrt {5}\right )^{3/2}}-\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{25}+\frac {i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((2+i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)+2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3+\sqrt {5}}+\frac {\left (\frac {3}{25}+\frac {i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((2+i)+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)+2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3+\sqrt {5}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [F] time = 2.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.17, size = 828, normalized size = 1.59 \begin {gather*} \frac {-2 x \sqrt {1+x^4} \left (2+3 x^2+2 x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}-2 x \left (1+3 x^2+3 x^4+2 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}}{10 x^2 \sqrt {1+x^4} \left (-1+x^2+x^4\right )+5 \left (-1+x^2+x^4\right ) \left (1+2 x^4\right )}+\sqrt {2} \tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+2 \sqrt {2} \text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 \log \left (1+x^2+\sqrt {1+x^4}\right )+8 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right )+3 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2-3 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]+\frac {\text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {163 \log \left (1+x^2+\sqrt {1+x^4}\right )-163 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right )-59 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2+59 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2+13 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4-13 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{5 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {{\left (x^{4} + x^{2} + 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + x^{2} - 1\right )}^{2} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}+x^{2}+1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}\, \left (x^{4}+x^{2}-1\right )^{2}}\, dx\]
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 \begin {gather*} \int \frac {{\left (x^{4} + x^{2} + 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + x^{2} - 1\right )}^{2} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x^4+1}+x^2}\,{\left (x^4+x^2+1\right )}^2}{\sqrt {x^4+1}\,{\left (x^4+x^2-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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