\[ \sin ^2(x) y''(x)-y(x) \left (a^2 \cos ^2(x)+\frac {b^2}{(2 a-3)^2}+3 a+b \cos (x)+2\right )=0 \] ✓ Mathematica : cpu = 4.52568 (sec), leaf count = 4128
\[\left \{\left \{y(x)\to \frac {c_1 (\cos (x)+1)^{\frac {1}{2} \left (-\frac {8 a^2}{-16 a^2+48 a-36}+\frac {24 a}{-16 a^2+48 a-36}+a-\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{2 \left (-16 a^2+48 a-36\right )}+\frac {1}{8} \left (-8 a+\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}+\sqrt {\left (8 a-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}\right )^2-16 \left (4 a^2-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right ) a}{-16 a^2+48 a-36}+2 b+\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{-16 a^2+48 a-36}-1\right )}\right )+\frac {\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}}{4 \left (-16 a^2+48 a-36\right )}+\frac {1}{8} \sqrt {\left (8 a-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}\right )^2-16 \left (4 a^2-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right ) a}{-16 a^2+48 a-36}+2 b+\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{-16 a^2+48 a-36}-1\right )}-\frac {18}{-16 a^2+48 a-36}+1\right )} \, _2F_1\left (-\frac {8 a^2}{-16 a^2+48 a-36}+\frac {24 a}{-16 a^2+48 a-36}+a+\frac {\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}}{4 \left (-16 a^2+48 a-36\right )}+\frac {1}{8} \sqrt {\left (8 a-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}\right )^2-16 \left (4 a^2-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right ) a}{-16 a^2+48 a-36}+2 b+\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{-16 a^2+48 a-36}-1\right )}-\frac {18}{-16 a^2+48 a-36},\frac {1}{8} \left (-8 a+\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}+\sqrt {\left (8 a-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}\right )^2-16 \left (4 a^2-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right ) a}{-16 a^2+48 a-36}+2 b+\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{-16 a^2+48 a-36}-1\right )}\right );\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{2 \left (-16 a^2+48 a-36\right )};\frac {1}{2} (1-\cos (x))\right ) (\cos (x)-1)^{\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{4 \left (-16 a^2+48 a-36\right )}}}{\sqrt [4]{\cos ^2(x)-1}}+\frac {\left (-\frac {1}{2}\right )^{1-\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{2 \left (-16 a^2+48 a-36\right )}} c_2 (1-\cos (x))^{1-\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{2 \left (-16 a^2+48 a-36\right )}} (\cos (x)+1)^{\frac {1}{2} \left (-\frac {8 a^2}{-16 a^2+48 a-36}+\frac {24 a}{-16 a^2+48 a-36}+a-\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{2 \left (-16 a^2+48 a-36\right )}+\frac {1}{8} \left (-8 a+\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}+\sqrt {\left (8 a-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}\right )^2-16 \left (4 a^2-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right ) a}{-16 a^2+48 a-36}+2 b+\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{-16 a^2+48 a-36}-1\right )}\right )+\frac {\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}}{4 \left (-16 a^2+48 a-36\right )}+\frac {1}{8} \sqrt {\left (8 a-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}\right )^2-16 \left (4 a^2-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right ) a}{-16 a^2+48 a-36}+2 b+\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{-16 a^2+48 a-36}-1\right )}-\frac {18}{-16 a^2+48 a-36}+1\right )} \, _2F_1\left (-\frac {8 a^2}{-16 a^2+48 a-36}+\frac {24 a}{-16 a^2+48 a-36}+a-\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{2 \left (-16 a^2+48 a-36\right )}+\frac {\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}}{4 \left (-16 a^2+48 a-36\right )}+\frac {1}{8} \sqrt {\left (8 a-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}\right )^2-16 \left (4 a^2-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right ) a}{-16 a^2+48 a-36}+2 b+\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{-16 a^2+48 a-36}-1\right )}-\frac {18}{-16 a^2+48 a-36}+1,-\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{2 \left (-16 a^2+48 a-36\right )}+\frac {1}{8} \left (-8 a+\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}+\sqrt {\left (8 a-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right )}{-16 a^2+48 a-36}\right )^2-16 \left (4 a^2-\frac {2 \left (-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72\right ) a}{-16 a^2+48 a-36}+2 b+\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{-16 a^2+48 a-36}-1\right )}\right )+1;2-\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{2 \left (-16 a^2+48 a-36\right )};\frac {1}{2} (1-\cos (x))\right ) (\cos (x)-1)^{\frac {-32 a^2+96 a+\sqrt {\left (32 a^2-96 a+72\right )^2-4 \left (-16 a^2+48 a-36\right ) \left (16 a^4+16 b a^2-88 a^2-48 b a+48 a+4 b^2+36 b+45\right )}-72}{4 \left (-16 a^2+48 a-36\right )}}}{\sqrt [4]{\cos ^2(x)-1}}\right \}\right \}\] ✓ Maple : cpu = 2.636 (sec), leaf count = 549
\[ \left \{ y \left ( x \right ) ={ \left ( {\frac {\cos \left ( x \right ) }{2}}-{\frac {1}{2}} \right ) ^{{\frac {1}{8\,a-12} \left ( 4\,a-6+\sqrt {4\,{b}^{2}+16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81} \right ) }} \left ( {\mbox {$_2$F$_1$}({\frac {1}{8\,a-12} \left ( 8\,{a}^{2}-\sqrt {4\,{b}^{2}-16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81}+\sqrt {4\,{b}^{2}+16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81}-8\,a-6 \right ) },{\frac {1}{8\,a-12} \left ( -8\,{a}^{2}-\sqrt {4\,{b}^{2}-16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81}+\sqrt {4\,{b}^{2}+16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81}+16\,a-6 \right ) };\,{\frac {1}{4\,a-6} \left ( 4\,a-6-\sqrt {4\,{b}^{2}-16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81} \right ) };\,{\frac {\cos \left ( x \right ) }{2}}+{\frac {1}{2}})} \left ( 2\,\cos \left ( x \right ) +2 \right ) ^{{\frac {1}{8\,a-12} \left ( 4\,a-6-\sqrt {4\,{b}^{2}-16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81} \right ) }}{\it \_C1}+{\mbox {$_2$F$_1$}({\frac {1}{8\,a-12} \left ( 8\,{a}^{2}+\sqrt {4\,{b}^{2}-16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81}+\sqrt {4\,{b}^{2}+16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81}-8\,a-6 \right ) },{\frac {1}{8\,a-12} \left ( -8\,{a}^{2}+\sqrt {4\,{b}^{2}-16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81}+\sqrt {4\,{b}^{2}+16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81}+16\,a-6 \right ) };\,{\frac {1}{4\,a-6} \left ( 4\,a-6+\sqrt {4\,{b}^{2}-16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81} \right ) };\,{\frac {\cos \left ( x \right ) }{2}}+{\frac {1}{2}})} \left ( 2\,\cos \left ( x \right ) +2 \right ) ^{{\frac {1}{8\,a-12} \left ( 4\,a-6+\sqrt {4\,{b}^{2}-16\, \left ( a-3/2 \right ) ^{2}b+16\,{a}^{4}-72\,{a}^{2}+81} \right ) }}{\it \_C2} \right ) {\frac {1}{\sqrt {\sin \left ( x \right ) }}}} \right \} \]