\[ \int \frac {\left (1-x^4\right ) \sqrt [4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}}{1+x^4} \, dx \]
Optimal antiderivative \[ \operatorname {Unintegrable} \]
command
Int[((1 - x^4)*(1 - x - 4*x^2 + 4*x^3 + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9)^(1/4))/(1 + x^4),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\left (1-x^4\right ) \sqrt [4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}}{1+x^4} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {\sqrt [4]{-4-(2+2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{-1-\sqrt [4]{-1}}}\right )}{(x-1)^{5/4} (x+1)}-\frac {\sqrt [4]{-4+(2+2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{\sqrt [4]{-1}-1}}\right )}{(x-1)^{5/4} (x+1)}-\frac {\sqrt [4]{-4+(2-2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{-1-(-1)^{3/4}}}\right )}{(x-1)^{5/4} (x+1)}+\frac {\sqrt [4]{-4-(2-2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{(-1)^{3/4}-1}}\right )}{(x-1)^{5/4} (x+1)}+\frac {\sqrt [4]{-4-(2+2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{-1-\sqrt [4]{-1}}}\right )}{(x-1)^{5/4} (x+1)}-\frac {\sqrt [4]{-4+(2+2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{\sqrt [4]{-1}-1}}\right )}{(x-1)^{5/4} (x+1)}-\frac {\sqrt [4]{-4+(2-2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{-1-(-1)^{3/4}}}\right )}{(x-1)^{5/4} (x+1)}+\frac {\sqrt [4]{-4-(2-2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{(-1)^{3/4}-1}}\right )}{(x-1)^{5/4} (x+1)}-\frac {4 \sqrt [4]{(1-x)^5 (x+1)^4} (1-x)^2}{13 (x+1)}+\frac {8 \sqrt [4]{(1-x)^5 (x+1)^4} (1-x)}{9 (x+1)} \]