<s>[INST] A closed form for the antiderivative of $\frac 1{\sin^5 x +\cos^5 x} $ Does there exist a closed form expression for  $$\int \dfrac {dx}{\sin^5 x +\cos^5 x}? $$ 
 [/INST] Computer algebra systems can do this.  Maybe this is not needed, since the question (answered by Steven) was only whether closed form exists (not to exhibit it).  But just for fun I gave it to Maple...
$$
1/5\,i\sqrt {\sqrt {5}+2}\ln  \left(  \left( \tan \left( 1/2\,x
 \right)  \right) ^{2}+ \left( 1/2\,i \left( \sqrt {5}+2 \right) ^{3/2
}+1/2+1/2\,\sqrt {5}-5/2\,i\sqrt {\sqrt {5}+2} \right) \tan \left( 1/2
\,x \right) -1/2\,i \left( \sqrt {5}+2 \right) ^{3/2}+1/2-1/2\,\sqrt {
5}+5/2\,i\sqrt {\sqrt {5}+2} \right) 
\\-1/5\,i\sqrt {\sqrt {5}+2}\ln 
 \left(  \left( \tan \left( 1/2\,x \right)  \right) ^{2}+ \left( -1/2
\,i \left( \sqrt {5}+2 \right) ^{3/2}+1/2+1/2\,\sqrt {5}+5/2\,i\sqrt {
\sqrt {5}+2} \right) \tan \left( 1/2\,x \right) +1/2\,i \left( \sqrt {
5}+2 \right) ^{3/2}+1/2-1/2\,\sqrt {5}-5/2\,i\sqrt {\sqrt {5}+2}
 \right) 
\\+1/5\,\sqrt {-2+\sqrt {5}}\ln  \left(  \left( \tan \left( 1/2
\,x \right)  \right) ^{2}+ \left( -1/2\, \left( -2+\sqrt {5} \right) ^
{3/2}+1/2-1/2\,\sqrt {5}-5/2\,\sqrt {-2+\sqrt {5}} \right) \tan
 \left( 1/2\,x \right) +1/2\, \left( -2+\sqrt {5} \right) ^{3/2}+1/2+1
/2\,\sqrt {5}+5/2\,\sqrt {-2+\sqrt {5}} \right) 
\\-1/5\,\sqrt {-2+\sqrt 
{5}}\ln  \left(  \left( \tan \left( 1/2\,x \right)  \right) ^{2}+
 \left( 1/2\, \left( -2+\sqrt {5} \right) ^{3/2}+1/2-1/2\,\sqrt {5}+5/
2\,\sqrt {-2+\sqrt {5}} \right) \tan \left( 1/2\,x \right) -1/2\,
 \left( -2+\sqrt {5} \right) ^{3/2}+1/2+1/2\,\sqrt {5}-5/2\,\sqrt {-2+
\sqrt {5}} \right) 
\\+4/5\,\sqrt {2}{\rm{ arctanh}} \left( 1/4\, \left( 2
\,\tan \left( 1/2\,x \right) -2 \right) \sqrt {2} \right) 
$$
In this case, we only had to solve two successive quadratic equations.
</s>