# [Maxima] problem with integrate and differentiate a function with trigonometric terms

Martin Ettl ettl.martin at gmx.de
Fri Jul 17 14:39:28 CDT 2009

```Hello,

i am using maxima to integrate a special function:

(E*sin(E)-e*sin(E)*sin(E))/(1-e*cos(E))^2;

For integration i used the following maxima command:

integrate((E*sin(E)-e*sin(E)*sin(E))/(1-e*cos(E))^2,E);

So far, this works quite well, but when i differentiate the result of the integration, i get a different result from where i was started. For differentiation i am using the following command:

(%i8) integrate((E*sin(E)-e*sin(E)*sin(E))/(1-e*cos(E))^2,E);
(%o8) (e*E*sin(2*E)^2+(-2*E*sin(E)+2*cos(E)-2*e)*sin(2*E)+e*E*cos(2*E)^2+(-2*sin(E)-2*E*cos(E)+2*e*E)*cos(2*E)+2*sin(E)-2*E*cos(E)+e*E)/(e^2*sin(2*E)^2-4*e*sin(E)*sin(2*E)+e^2*cos(2*E)^2+(2*e^2-4*e*cos(E))*cos(2*E)+4*sin(E)^2+4*cos(E)^2-4*e*cos(E)+e^2)

(%i9) diff(%, E, 1);
(%o9) (e*sin(2*E)^2-2*(-2*sin(E)-2*E*cos(E)+2*e*E)*sin(2*E)+(-4*sin(E)-2*E*cos(E))*sin(2*E)+e*cos(2*E)^2+(2*E*sin(E)-4*cos(E)+2*e)*cos(2*E)+2*(-2*E*sin(E)+2*cos(E)-2*e)*cos(2*E)+2*E*sin(E)+e)/(e^2*sin(2*E)^2-4*e*sin(E)*sin(2*E)+e^2*cos(2*E)^2+(2*e^2-4*e*cos(E))*cos(2*E)+4*sin(E)^2+4*cos(E)^2-4*e*cos(E)+e^2)-((-2*(2*e^2-4*e*cos(E))*sin(2*E)-4*e*cos(E)*sin(2*E)-4*e*sin(E)*cos(2*E)+4*e*sin(E))*(e*E*sin(2*E)^2+(-2*E*sin(E)+2*cos(E)-2*e)*sin(2*E)+e*E*cos(2*E)^2+(-2*sin(E)-2*E*cos(E)+2*e*E)*cos(2*E)+2*sin(E)-2*E*cos(E)+e*E))/((e^2*sin(2*E)^2-4*e*sin(E)*sin(2*E)+e^2*cos(2*E)^2+(2*e^2-4*e*cos(E))*cos(2*E)+4*sin(E)^2+4*cos(E)^2-4*e*cos(E)+e^2)^2

What am i doing wrong? Why do i not get the same result, from where i started to integrate. Is this a bug in maxima?