Waldek Hebisch schrieb:
This is the antiderivative of x/((x^2+x+1)*SQRT(x^2-x+1)). Derive 6.10
handles the postprocessing reasonably well, in 26+1 steps:
SUM(t*LN(24*t^3+6*t^2+2*t+SQRT(x^2-x+1)-x),t,SOLUTIONS(r^4+r^2/6~
+1/36,r))
" SUM(F(x),x,[x1,x2,...]) -> F(x1)+F(x2)+... "
(-SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(-SQRT(6)/12-SQRT(2)*#i/4)^3+6*~
(-SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(-SQRT(6)/12-SQRT~
(2)*#i/4))+(-SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(-SQRT(6)/12+SQRT(2)~
*#i/4)^3+6*(-SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(-SQRT~
(6)/12+SQRT(2)*#i/4))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/1~
2-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+~
2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQ~
RT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-~
x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" LN(x+y*#i) -> LN(|x+y*#i|) + #i*ATAN(y,x) "
(-SQRT(6)/12-SQRT(2)*#i/4)*(LN(ABS(SQRT(x^2-x+1)-x+SQRT(6)/2-1/2~
+#i*(SQRT(3)/2-SQRT(2)/2)))+#i*ATAN(SQRT(3)/2-SQRT(2)/2,SQRT(x^2~
-x+1)-x+SQRT(6)/2-1/2))+(-SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(-SQRT(~
6)/12+SQRT(2)*#i/4)^3+6*(-SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+~
1)-x+2*(-SQRT(6)/12+SQRT(2)*#i/4))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(~
24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQR~
T(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i~
/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4~
)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" ABS(x+#i*y) -> SQRT(x^2+y^2) "
(-SQRT(6)/12-SQRT(2)*#i/4)*(LN(SQRT((SQRT(x^2-x+1)-x+SQRT(6)/2-1~
/2)^2+(SQRT(3)/2-SQRT(2)/2)^2))+#i*ATAN(SQRT(3)/2-SQRT(2)/2,SQRT~
(x^2-x+1)-x+SQRT(6)/2-1/2))+(-SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(-S~
QRT(6)/12+SQRT(2)*#i/4)^3+6*(-SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^~
2-x+1)-x+2*(-SQRT(6)/12+SQRT(2)*#i/4))+(SQRT(6)/12-SQRT(2)*#i/4)~
*LN(24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2~
+SQRT(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2~
)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*~
#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" If -1<n<1, LN(z^n) -> n*LN(z) "
(-SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*~
x^2-SQRT(6)*x-SQRT(6)+4)/2+#i*ATAN(SQRT(3)/2-SQRT(2)/2,SQRT(x^2-~
x+1)-x+SQRT(6)/2-1/2))+(-SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(-SQRT(6~
)/12+SQRT(2)*#i/4)^3+6*(-SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1~
)-x+2*(-SQRT(6)/12+SQRT(2)*#i/4))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(2~
4*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT~
(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/~
4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)~
^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" ATAN(y,x) -> pi*SIGN(y)/2-ATAN(x/y) "
(-SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*~
x^2-SQRT(6)*x-SQRT(6)+4)/2+#i*(pi*SIGN(SQRT(3)/2-SQRT(2)/2)/2-AT~
AN((SQRT(x^2-x+1)-x+SQRT(6)/2-1/2)/(SQRT(3)/2-SQRT(2)/2))))+(-SQ~
RT(6)/12+SQRT(2)*#i/4)*LN(24*(-SQRT(6)/12+SQRT(2)*#i/4)^3+6*(-SQ~
RT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(-SQRT(6)/12+SQRT(2)*~
#i/4))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2)*#i/4)~
^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12-S~
QRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2~
)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(~
6)/12+SQRT(2)*#i/4))
" 1/(z+w) -> (z-w)/(z^2-w^2) "
(-SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*~
x^2-SQRT(6)*x-SQRT(6)+4)/2+#i*(pi/2-ATAN(4*(SQRT(x^2-x+1)-x+SQRT~
(6)/2-1/2)*(SQRT(3)/2+SQRT(2)/2))))+(-SQRT(6)/12+SQRT(2)*#i/4)*L~
N(24*(-SQRT(6)/12+SQRT(2)*#i/4)^3+6*(-SQRT(6)/12+SQRT(2)*#i/4)^2~
+SQRT(x^2-x+1)-x+2*(-SQRT(6)/12+SQRT(2)*#i/4))+(SQRT(6)/12-SQRT(~
2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)~
*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/1~
2+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+~
SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" (a+b*#i)*(c+d*#i) -> (a*c-b*d)+(a*d+b*c)*#i "
-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQRT(~
6)+4)/(2*12)-SQRT(2)*(ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*~
x+SQRT(6)-1))-pi/2)/4+#i*(SQRT(6)*(ATAN((SQRT(3)+SQRT(2))*(2*SQR~
T(x^2-x+1)-2*x+SQRT(6)-1))-pi/2)/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2~
*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/(2*4))+(-SQRT(6)/12+SQR~
T(2)*#i/4)*LN(24*(-SQRT(6)/12+SQRT(2)*#i/4)^3+6*(-SQRT(6)/12+SQR~
T(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(-SQRT(6)/12+SQRT(2)*#i/4))+(SQRT~
(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6~
)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4)~
)+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*~
(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2~
)*#i/4))
" LN(x+y*#i) -> LN(|x+y*#i|) + #i*ATAN(y,x) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(ABS(SQRT(x^2-x+1)-x+SQRT(6)/2-1/2+#i*(SQRT(2~
)/2-SQRT(3)/2)))+#i*ATAN(-SQRT(3)/2+SQRT(2)/2,SQRT(x^2-x+1)-x+SQ~
RT(6)/2-1/2))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2~
)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(~
6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12~
+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2~
*(SQRT(6)/12+SQRT(2)*#i/4))
" ABS(x+#i*y) -> SQRT(x^2+y^2) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(SQRT((SQRT(x^2-x+1)-x+SQRT(6)/2-1/2)^2+(-SQR~
T(3)/2+SQRT(2)/2)^2))+#i*ATAN(-SQRT(3)/2+SQRT(2)/2,SQRT(x^2-x+1)~
-x+SQRT(6)/2-1/2))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-S~
QRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(~
SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(~
6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1~
)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" If -1<n<1, LN(z^n) -> n*LN(z) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)~
*x-SQRT(6)+4)/2+#i*ATAN(-SQRT(3)/2+SQRT(2)/2,SQRT(x^2-x+1)-x+SQR~
T(6)/2-1/2))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2)~
*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6~
)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+~
SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*~
(SQRT(6)/12+SQRT(2)*#i/4))
" ATAN(y,x) -> pi*SIGN(y)/2-ATAN(x/y) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)~
*x-SQRT(6)+4)/2+#i*(pi*SIGN(-SQRT(3)/2+SQRT(2)/2)/2-ATAN((SQRT(x~
^2-x+1)-x+SQRT(6)/2-1/2)/(SQRT(2)/2-SQRT(3)/2))))+(SQRT(6)/12-SQ~
RT(2)*#i/4)*LN(24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT~
(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6~
)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/~
12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" 1/(z+w) -> (z-w)/(z^2-w^2) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)~
*x-SQRT(6)+4)/2+#i*(-pi/2-ATAN(4*(SQRT(x^2-x+1)-x+SQRT(6)/2-1/2)~
*(-SQRT(3)/2-SQRT(2)/2))))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT~
(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+~
1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(2~
4*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT~
(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" ATAN(-z) -> -ATAN(z) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)+(-SQRT(6)/1~
2+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)~
*x-SQRT(6)+4)/2+#i*(-pi/2+ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1~
)-2*x+SQRT(6)-1))))+(SQRT(6)/12-SQRT(2)*#i/4)*LN(24*(SQRT(6)/12-~
SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*~
(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT~
(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+~
1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" (a+b*#i)*(c+d*#i) -> (a*c-b*d)+(a*d+b*c)*#i "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/4-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)+SQRT(2))*(2*SQ~
RT(x^2-x+1)-2*x+SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQ~
RT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/8-SQRT(6)*pi/24)-SQRT(6)*LN(~
-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQRT(6)+4)/(2*12)~
+SQRT(2)*(pi/2-ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(~
6)-1)))/4+#i*(SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQ~
RT(6)*x-SQRT(6)+4)/(2*4)-SQRT(6)*(ATAN((SQRT(3)+SQRT(2))*(2*SQRT~
(x^2-x+1)-2*x+SQRT(6)-1))-pi/2)/12)+(SQRT(6)/12-SQRT(2)*#i/4)*LN~
(24*(SQRT(6)/12-SQRT(2)*#i/4)^3+6*(SQRT(6)/12-SQRT(2)*#i/4)^2+SQ~
RT(x^2-x+1)-x+2*(SQRT(6)/12-SQRT(2)*#i/4))+(SQRT(6)/12+SQRT(2)*#~
i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/~
4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" LN(x+y*#i) -> LN(|x+y*#i|) + #i*ATAN(y,x) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+(SQRT(6)/12-SQRT(2)*#i/4)*(LN(ABS(SQRT(x~
^2-x+1)-x-SQRT(6)/2-1/2-#i*(SQRT(3)/2+SQRT(2)/2)))+#i*ATAN(-SQRT~
(3)/2-SQRT(2)/2,SQRT(x^2-x+1)-x-SQRT(6)/2-1/2))+(SQRT(6)/12+SQRT~
(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2~
)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" ABS(x+#i*y) -> SQRT(x^2+y^2) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+(SQRT(6)/12-SQRT(2)*#i/4)*(LN(SQRT((SQRT~
(x^2-x+1)-x-SQRT(6)/2-1/2)^2+(-SQRT(3)/2-SQRT(2)/2)^2))+#i*ATAN(~
-SQRT(3)/2-SQRT(2)/2,SQRT(x^2-x+1)-x-SQRT(6)/2-1/2))+(SQRT(6)/12~
+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+S~
QRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" If -1<n<1, LN(z^n) -> n*LN(z) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+(SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-~
x+1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/2+#i*ATAN(-SQRT(~
3)/2-SQRT(2)/2,SQRT(x^2-x+1)-x-SQRT(6)/2-1/2))+(SQRT(6)/12+SQRT(~
2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)~
*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4))
" ATAN(y,x) -> pi*SIGN(y)/2-ATAN(x/y) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+(SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-~
x+1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/2+#i*(pi*SIGN(-S~
QRT(3)/2-SQRT(2)/2)/2+ATAN((SQRT(x^2-x+1)-x-SQRT(6)/2-1/2)/(SQRT~
(3)/2+SQRT(2)/2))))+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+~
SQRT(2)*#i/4)^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*~
(SQRT(6)/12+SQRT(2)*#i/4))
" 1/(z+w) -> (z-w)/(z^2-w^2) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+(SQRT(6)/12-SQRT(2)*#i/4)*(LN(-SQRT(x^2-~
x+1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/2+#i*(-pi/2+ATAN~
(4*(SQRT(x^2-x+1)-x-SQRT(6)/2-1/2)*(SQRT(3)/2-SQRT(2)/2))))+(SQR~
T(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)^3+6*(SQRT(~
6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+SQRT(2)*#i/4~
))
" (a+b*#i)*(c+d*#i) -> (a*c-b*d)+(a*d+b*c)*#i "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*pi/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1~
)+2*x^2+SQRT(6)*x+SQRT(6)+4)/(2*12)-SQRT(2)*(pi/2-ATAN((SQRT(3)-~
SQRT(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1)))/4-#i*(SQRT(2)*LN(-SQR~
T(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/(2*4)+SQRT~
(6)*(pi/2-ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1)~
))/12)+(SQRT(6)/12+SQRT(2)*#i/4)*LN(24*(SQRT(6)/12+SQRT(2)*#i/4)~
^3+6*(SQRT(6)/12+SQRT(2)*#i/4)^2+SQRT(x^2-x+1)-x+2*(SQRT(6)/12+S~
QRT(2)*#i/4))
" LN(x+y*#i) -> LN(|x+y*#i|) + #i*ATAN(y,x) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+(~
SQRT(6)/12+SQRT(2)*#i/4)*(LN(ABS(SQRT(x^2-x+1)-x-SQRT(6)/2-1/2+#~
i*(SQRT(3)/2+SQRT(2)/2)))+#i*ATAN(SQRT(3)/2+SQRT(2)/2,SQRT(x^2-x~
+1)-x-SQRT(6)/2-1/2))
" ABS(x+#i*y) -> SQRT(x^2+y^2) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+(~
SQRT(6)/12+SQRT(2)*#i/4)*(LN(SQRT((SQRT(x^2-x+1)-x-SQRT(6)/2-1/2~
)^2+(SQRT(3)/2+SQRT(2)/2)^2))+#i*ATAN(SQRT(3)/2+SQRT(2)/2,SQRT(x~
^2-x+1)-x-SQRT(6)/2-1/2))
" If -1<n<1, LN(z^n) -> n*LN(z) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+(~
SQRT(6)/12+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^~
2+SQRT(6)*x+SQRT(6)+4)/2+#i*ATAN(SQRT(3)/2+SQRT(2)/2,SQRT(x^2-x+~
1)-x-SQRT(6)/2-1/2))
" ATAN(y,x) -> pi*SIGN(y)/2-ATAN(x/y) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+(~
SQRT(6)/12+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^~
2+SQRT(6)*x+SQRT(6)+4)/2+#i*(pi*SIGN(SQRT(3)/2+SQRT(2)/2)/2-ATAN~
((SQRT(x^2-x+1)-x-SQRT(6)/2-1/2)/(SQRT(3)/2+SQRT(2)/2))))
" 1/(z+w) -> (z-w)/(z^2-w^2) "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+(~
SQRT(6)/12+SQRT(2)*#i/4)*(LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^~
2+SQRT(6)*x+SQRT(6)+4)/2+#i*(pi/2-ATAN(4*(SQRT(x^2-x+1)-x-SQRT(6~
)/2-1/2)*(SQRT(3)/2-SQRT(2)/2))))
" (a+b*#i)*(c+d*#i) -> (a*c-b*d)+(a*d+b*c)*#i "
-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1))~
/2-SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x-SQRT(6)+1)+2*x^2-SQRT(6)*x-SQR~
T(6)+4)/12+SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-S~
QRT(6)-1))/4+SQRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQR~
T(6)*x+SQRT(6)+4)/24+SQRT(2)*pi/8+#i*(SQRT(6)*ATAN((SQRT(3)-SQRT~
(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/12-SQRT(2)*LN(-SQRT(x^2-x+~
1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)+4)/8-SQRT(6)*pi/24)+S~
QRT(6)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1)+2*x^2+SQRT(6)*x+SQRT(6)~
+4)/(2*12)+SQRT(2)*(ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-~
SQRT(6)-1))-pi/2)/4+#i*(SQRT(2)*LN(-SQRT(x^2-x+1)*(2*x+SQRT(6)+1~
)+2*x^2+SQRT(6)*x+SQRT(6)+4)/(2*4)+SQRT(6)*(pi/2-ATAN((SQRT(3)-S~
QRT(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1)))/12)
" one final step "
SQRT(2)*ATAN((SQRT(3)-SQRT(2))*(2*SQRT(x^2-x+1)-2*x-SQRT(6)-1))/~
2-SQRT(2)*ATAN((SQRT(3)+SQRT(2))*(2*SQRT(x^2-x+1)-2*x+SQRT(6)-1)~
)/2-SQRT(6)*LN((SQRT(x^2-x+1)*(2*x-SQRT(6)+1)-2*x^2+SQRT(6)*x+SQ~
RT(6)-4)/(SQRT(x^2-x+1)*(2*x+SQRT(6)+1)-2*x^2-SQRT(6)*x-SQRT(6)-~
4))/12
However, the combined logarithmic term can be manually simplified
further to:
-SQRT(6)*LN((5-2*SQRT(6))*(SQRT(3)*SQRT(x^2-x+1)-SQRT(2)*(x-1))
/(SQRT(3)*SQRT(x^2-x+1)+SQRT(2)*(x-1)))/12
where the factor 5-2*SQRT(6) represents an additive constant that can be
omitted.
Martin.