Integrand size = 32, antiderivative size = 170 \[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\frac {x \sqrt {1+x^4} \left (7 x^2+2 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}+x \left (4+8 x^4+2 x^8\right ) \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+x^4} \left (2 x^2+2 x^6\right )+2 \left (1+3 x^4+2 x^8\right )}-\frac {3}{2} \arctan \left (x \sqrt {x^2+\sqrt {1+x^4}}\right )+\frac {\arctan \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{2 \sqrt {2}} \]
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\[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt {x^2+\sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2}-\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{1+x^4}\right ) \, dx \\ & = 4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx-4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^4} \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ & = -\left (4 \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (i-x^2\right )}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (i+x^2\right )}\right ) \, dx\right )+4 \int \left (-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{4 \left (i-x^2\right )^2}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{4 \left (i+x^2\right )^2}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-1-x^4\right )}\right ) \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ & = -\left (2 i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i-x^2} \, dx\right )-2 i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i+x^2} \, dx-2 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-1-x^4} \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx-\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i-x^2\right )^2} \, dx-\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i+x^2\right )^2} \, dx \\ & = -\left (2 i \int \left (-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx\right )-2 i \int \left (-\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}-x\right )}-\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx-2 \int \left (-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (i-x^2\right )}-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (i+x^2\right )}\right ) \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx-\int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \left (-(-1)^{3/4}-x\right )^2}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \left (-(-1)^{3/4}+x\right )^2}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-i-x^2\right )}\right ) \, dx-\int \left (-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \left (\sqrt [4]{-1}-x\right )^2}-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \left (\sqrt [4]{-1}+x\right )^2}-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (i-x^2\right )}\right ) \, dx \\ & = \frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}-x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}-x\right )^2} \, dx+\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}+x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}+x\right )^2} \, dx-\frac {1}{2} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-i-x^2} \, dx+\frac {1}{2} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i-x^2} \, dx+i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i-x^2} \, dx+i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i+x^2} \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}-x} \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}+x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}-x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}+x} \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ & = \frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}-x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}-x\right )^2} \, dx+\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}+x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}+x\right )^2} \, dx+\frac {1}{2} i \int \left (-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx-\frac {1}{2} i \int \left (\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}-x\right )}+\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx+i \int \left (-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx+i \int \left (-\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}-x\right )}-\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}-x} \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}+x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}-x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}+x} \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ & = \frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}-x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}-x\right )^2} \, dx+\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}+x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}+x\right )^2} \, dx+\frac {1}{4} \sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}-x} \, dx+\frac {1}{4} \sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}+x} \, dx+\frac {1}{2} \sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}-x} \, dx+\frac {1}{2} \sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}+x} \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}-x} \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}+x} \, dx-\frac {1}{4} (-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}-x} \, dx-\frac {1}{4} (-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}+x} \, dx-\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}-x} \, dx-\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}+x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}-x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}+x} \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\frac {1}{4} \left (\frac {2 x \sqrt {x^2+\sqrt {1+x^4}} \left (4+8 x^4+2 x^8+7 x^2 \sqrt {1+x^4}+2 x^6 \sqrt {1+x^4}\right )}{\left (1+x^4\right ) \left (1+2 x^4+2 x^2 \sqrt {1+x^4}\right )}-6 \arctan \left (x \sqrt {x^2+\sqrt {1+x^4}}\right )+\sqrt {2} \arctan \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )\right ) \]
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\[\int \frac {\left (x^{4}-1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (x^{4}+1\right )^{2}}d x\]
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Time = 0.98 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=-\frac {2 \, \sqrt {2} {\left (x^{4} + 1\right )} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) - 3 \, {\left (x^{4} + 1\right )} \arctan \left (-\frac {4 \, {\left (3 \, x^{9} - 12 \, x^{5} - {\left (3 \, x^{7} - 5 \, x^{3}\right )} \sqrt {x^{4} + 1} + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{17 \, x^{8} - 46 \, x^{4} + 1}\right ) - 4 \, {\left (2 \, x^{5} - \sqrt {x^{4} + 1} x^{3} + 4 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{8 \, {\left (x^{4} + 1\right )}} \]
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\[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\int \frac {\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\left (x^{4} + 1\right )^{2}}\, dx \]
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\[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\int { \frac {{\left (x^{4} - 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\int { \frac {{\left (x^{4} - 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\int \frac {{\left (x^4-1\right )}^2\,\sqrt {\sqrt {x^4+1}+x^2}}{{\left (x^4+1\right )}^2} \,d x \]
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