\[ \int \frac {\cos (c+d x)}{a-b \sin ^4(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {b^{\frac {1}{4}} \sin \left (d x +c \right )}{a^{\frac {1}{4}}}\right )}{2 a^{\frac {3}{4}} b^{\frac {1}{4}} d}+\frac {\arctanh \left (\frac {b^{\frac {1}{4}} \sin \left (d x +c \right )}{a^{\frac {1}{4}}}\right )}{2 a^{\frac {3}{4}} b^{\frac {1}{4}} d} \]
command
integrate(cos(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{2} \, \left (\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \arctan \left (a^{2} b d^{3} \left (\frac {1}{a^{3} b d^{4}}\right )^{\frac {3}{4}} \sin \left (d x + c\right ) + \sqrt {a^{2} d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} - \cos \left (d x + c\right )^{2} + 1} a^{2} b d^{3} \left (\frac {1}{a^{3} b d^{4}}\right )^{\frac {3}{4}}\right ) - \frac {1}{2} \, \left (\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \arctan \left (-a^{2} b d^{3} \left (\frac {1}{a^{3} b d^{4}}\right )^{\frac {3}{4}} \sin \left (d x + c\right ) + \sqrt {a^{2} d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} - \cos \left (d x + c\right )^{2} + 1} a^{2} b d^{3} \left (\frac {1}{a^{3} b d^{4}}\right )^{\frac {3}{4}}\right ) + \frac {1}{8} \, \left (\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, a^{2} d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} + \frac {1}{2} \, a d \left (\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) - \frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) - \frac {1}{8} \, \left (\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, a^{2} d^{2} \sqrt {\frac {1}{a^{3} b d^{4}}} - \frac {1}{2} \, a d \left (\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) - \frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]