Integrand size = 42, antiderivative size = 349 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=-\frac {\text {RootSum}\left [1+2 \text {$\#$1}^2-6 \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (-1+x^2+\sqrt {1+x^4}\right )-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right )-\log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2-3 \log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4+3 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-\log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{\text {$\#$1}-6 \text {$\#$1}^3-3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{2 \sqrt {2}} \]
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Result contains complex when optimal does not.
Time = 1.43 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.95, number of steps used = 26, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {6860, 6857, 2158, 739, 212} \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=-\frac {i \left (1+\sqrt {5}\right ) \arctan \left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {\sqrt {5}-1} x\right )}{\sqrt {2 \left (\sqrt {5}+(-1-2 i)\right )} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left (\sqrt {5}+(-2+i)\right )}}+\frac {i \left (1+\sqrt {5}\right ) \arctan \left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {\sqrt {5}-1} x\right )}{\sqrt {2 \left (\sqrt {5}+(-1-2 i)\right )} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left (\sqrt {5}+(-2+i)\right )}}-\frac {\left (1-\sqrt {5}\right ) \arctan \left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {-2 \sqrt {5}+(-2-4 i)} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left (\sqrt {5}+(2-i)\right )}}+\frac {\left (1-\sqrt {5}\right ) \arctan \left (\frac {(1+i) \left (\sqrt {1+\sqrt {5}} x+\sqrt {2}\right )}{\sqrt {-2 \sqrt {5}+(-2-4 i)} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left (\sqrt {5}+(2-i)\right )}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (1+\sqrt {5}\right ) \text {arctanh}\left (\frac {\sqrt {2}-i \sqrt {\sqrt {5}-1} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(-2-i)\right )}}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (1+\sqrt {5}\right ) \text {arctanh}\left (\frac {\sqrt {2}+i \sqrt {\sqrt {5}-1} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(-2-i)\right )}}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (1-\sqrt {5}\right ) \text {arctanh}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(2+i)\right )}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (1-\sqrt {5}\right ) \text {arctanh}\left (\frac {\sqrt {1+\sqrt {5}} x+\sqrt {2}}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(2+i)\right )}} \]
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Rule 212
Rule 739
Rule 2158
Rule 6857
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1+\frac {1}{\sqrt {5}}\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}}+\frac {\left (1-\frac {1}{\sqrt {5}}\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}}\right ) \, dx \\ & = \frac {1}{5} \left (5-\sqrt {5}\right ) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{5} \left (5+\sqrt {5}\right ) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx \\ & = \frac {1}{5} \left (5-\sqrt {5}\right ) \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+\frac {1}{5} \left (5+\sqrt {5}\right ) \int \left (\frac {\sqrt {-1+\sqrt {5}} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1-\sqrt {5}\right ) \left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {\sqrt {-1+\sqrt {5}} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1-\sqrt {5}\right ) \left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx \\ & = \left (i \sqrt {\frac {1}{10} \left (-2+\sqrt {5}\right )}\right ) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx+\left (i \sqrt {\frac {1}{10} \left (-2+\sqrt {5}\right )}\right ) \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 {\frac {1}{10} \left (2+\sqrt {5}\right )} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx-\sqrt {\frac {1}{10} \left (2+\sqrt {5}\right )} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx \\ & = \left (\left (-\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (-2+\sqrt {5}\right )}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx+\left (\left (-\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (-2+\sqrt {5}\right )}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx+\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (-2+\sqrt {5}\right )}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx+\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (-2+\sqrt {5}\right )}\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx-\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (2+\sqrt {5}\right )}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx-\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (2+\sqrt {5}\right )}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx-\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (2+\sqrt {5}\right )}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx-\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (2+\sqrt {5}\right )}\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx \\ & = \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (-2+\sqrt {5}\right )}\right ) \text {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 (\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (-2+\sqrt {5}\right )}\right ) \text {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 (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (-2+\sqrt {5}\right )}\right ) \text {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 (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (-2+\sqrt {5}\right )}\right ) \text {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 (\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (2+\sqrt {5}\right )}\right ) \text {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 (\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (2+\sqrt {5}\right )}\right ) \text {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 (\left (-\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (2+\sqrt {5}\right )}\right ) \text {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 (\left (-\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {1}{10} \left (2+\sqrt {5}\right )}\right ) \text {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 ) \\ & = -\frac {i \arctan \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 )}{2 \sqrt {5 \left ((7+4 i)-(3+2 i) \sqrt {5}\right )}}+\frac {i \arctan \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 )}{2 \sqrt {5 \left ((7+4 i)-(3+2 i) \sqrt {5}\right )}}+\frac {\arctan \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 )}{2 \sqrt {5 \left ((-7-4 i)-(3+2 i) \sqrt {5}\right )}}-\frac {\arctan \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 )}{2 \sqrt {5 \left ((-7-4 i)-(3+2 i) \sqrt {5}\right )}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \text {arctanh}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {10 \left ((-4-7 i)+(2+3 i) \sqrt {5}\right )}}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \text {arctanh}\left (\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {10 \left ((-4-7 i)+(2+3 i) \sqrt {5}\right )}}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \text {arctanh}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {10 \left ((4+7 i)+(2+3 i) \sqrt {5}\right )}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \text {arctanh}\left (\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {10 \left ((4+7 i)+(2+3 i) \sqrt {5}\right )}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.03 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\frac {\text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (1+x^2+\sqrt {1+x^4}\right )-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right )+3 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2-3 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{2 \sqrt {2}} \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.10
\[\int \frac {\left (x^{2}+1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}\, \left (x^{4}+x^{2}-1\right )}d x\]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 16.15 (sec) , antiderivative size = 10165, normalized size of antiderivative = 29.13 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\text {Too large to display} \]
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Not integrable
Time = 24.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int \frac {\left (x^{2} + 1\right ) \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} \left (x^{4} + x^{2} - 1\right )}\, dx \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{2} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
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Not integrable
Time = 0.39 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{2} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int \frac {\left (x^2+1\right )\,\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}\,\left (x^4+x^2-1\right )} \,d x \]
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