Chapter 3
Listing of integrals solved by CAS which has no known antiderivatives

3.1 Test file Number [5]

3.1.1 Maxima

Integral number [145] \[ \int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx \]

[C]   time = 0.222839 (sec), size = 240 ,normalized size = 21.82 \[ -\frac {{\left (x e^{\left (\frac {4 \, k \cos \left (2 \, x\right ) \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} + \frac {4 \, k \sin \left (2 \, x\right ) \sin \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )} + x e^{\left (\frac {4 \, k \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )}\right )} e^{\left (-\frac {2 \, k \cos \left (2 \, x\right ) \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} - \frac {2 \, k \sin \left (2 \, x\right ) \sin \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} - \frac {2 \, k \cos \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )} \sin \left (\frac {2 \, {\left (k \cos \left (x\right ) \sin \left (2 \, x\right ) - k \cos \left (2 \, x\right ) \sin \left (x\right ) + k \sin \left (x\right )\right )}}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )}{2 \, k} \]

[In]

integrate(x*cos(x)*cos(k/sin(x))/sin(x)^2,x, algorithm=""maxima"")

output

-1/2*(x*e^(4*k*cos(2*x)*cos(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1) + 4* 
k*sin(2*x)*sin(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)) + x*e^(4*k*cos(x 
)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)))*e^(-2*k*cos(2*x)*cos(x)/(cos(2* 
x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1) - 2*k*sin(2*x)*sin(x)/(cos(2*x)^2 + sin(2* 
x)^2 - 2*cos(2*x) + 1) - 2*k*cos(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1) 
)*sin(2*(k*cos(x)*sin(2*x) - k*cos(2*x)*sin(x) + k*sin(x))/(cos(2*x)^2 + sin(2 
*x)^2 - 2*cos(2*x) + 1))/k