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risch

risch(e^x/(1-e^2*x),x);

Calculate

risch-sqrt

xgtr1:d-(d*sqrt((y_0-...

risch(xgtr1, u);

Calculate

risch

risch(e^x/(1-e^2*x),x);

Calculate

risch

risch(e^x/(1-e^2*x),x);

Calculate

risch

risch(e^x/(1-e^2*x),x);

Calculate

risch-sqrt

risch(sqrt(x+sqrt(x))...

Calculate

risch-sqrt

risch(sqrt(x+sqrt(x))...

Calculate

risch

risch((x^2-x^3)^(-1/3...

Calculate

risch

risch((x^2-x^3)^(-1/3...

Calculate

risch-sin-sqrt

risch(sqrt(sin(x)),x);

Calculate

risch

Run Example
(%i1)xltr0:d_1 - (d_1 - d_0)*log(sqrt(x^2 + (y_0 - v*t)^2)/r_1)/(log(r_0/r_1));
                                                      2    2
                                      sqrt((y_0 - t v)  + x )
                      (d_1 - d_0) log(-----------------------)
                                                r_1
(%o1)           d_1 - ----------------------------------------
                                          r_0
                                      log(---)
                                          r_1
(%i2) xltr1:d_3 - (d_3 - d_2)*log(sqrt(x^2 + (y_0 - v*t)^2)/r_2)/(log(r_1/r_2));
                                                      2    2
                                      sqrt((y_0 - t v)  + x )
                      (d_3 - d_2) log(-----------------------)
                                                r_2
(%o2)           d_3 - ----------------------------------------
                                          r_1
                                      log(---)
                                          r_2
(%i3) tex(expand(subst(y_0^2 + x^2 = r_0^2, subst(y_1^2 + x^2 = r_1^2, ratsimp(subst([t_11 = (y_0 + y_1)/v, t_1 = (y_0 - y_1)/v], ratsimp(subst(t = t_11, risch(xltr1, t))) - ratsimp(subst(t = t_1, risch(xltr1, t))) + 2*(ratsimp(subst(t = t_1, risch(xltr0, t))) - ratsimp(subst(t = 0, risch(xltr0, t))))))))));
$$-{{2\,{\it d\__1}\,\log {\it r\__2}\,x\,\arctan \left({{{\it y\__1}
 }\over{x}}\right)}\over{\log {\it r\__1}\,\log {\it r\__2}\,v-\log 
 {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}+{{2\,{\it d\__0}\,\log 
 {\it r\__2}\,x\,\arctan \left({{{\it y\__1}}\over{x}}\right)}\over{
 \log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}+{{2\,{\it d\__3}\,\log {\it r\__1}\,x\,\arctan 
 \left({{{\it y\__1}}\over{x}}\right)}\over{\log {\it r\__1}\,\log 
 {\it r\__2}\,v-\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2
 {\it r\__1}\,v+\log {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,
 {\it d\__2}\,\log {\it r\__1}\,x\,\arctan \left({{{\it y\__1}}\over{
 x}}\right)}\over{\log {\it r\__1}\,\log {\it r\__2}\,v-\log 
 {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}+{{2\,{\it d\__1}\,\log 
 {\it r\__1}\,x\,\arctan \left({{{\it y\__1}}\over{x}}\right)}\over{
 \log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}-{{2\,{\it d\__0}\,\log {\it r\__1}\,x\,\arctan 
 \left({{{\it y\__1}}\over{x}}\right)}\over{\log {\it r\__1}\,\log 
 {\it r\__2}\,v-\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2
 {\it r\__1}\,v+\log {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,
 {\it d\__3}\,\log {\it r\__0}\,x\,\arctan \left({{{\it y\__1}}\over{
 x}}\right)}\over{\log {\it r\__1}\,\log {\it r\__2}\,v-\log 
 {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}+{{2\,{\it d\__2}\,\log 
 {\it r\__0}\,x\,\arctan \left({{{\it y\__1}}\over{x}}\right)}\over{
 \log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}+{{2\,{\it d\__2}\,\log {\it r\__1}\,\log 
 {\it r\__2}\,{\it y\__1}}\over{\log {\it r\__1}\,\log {\it r\__2}\,v
 -\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,{\it d\__1}\,\log 
 {\it r\__1}\,\log {\it r\__2}\,{\it y\__1}}\over{\log {\it r\__1}\,
 \log {\it r\__2}\,v-\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2
 {\it r\__1}\,v+\log {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,
 {\it d\__2}\,\log {\it r\__0}\,\log {\it r\__2}\,{\it y\__1}}\over{
 \log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}+{{2\,{\it d\__1}\,\log {\it r\__0}\,\log 
 {\it r\__2}\,{\it y\__1}}\over{\log {\it r\__1}\,\log {\it r\__2}\,v
 -\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}+{{2\,{\it d\__1}\,\log 
 {\it r\__2}\,{\it y\__1}}\over{\log {\it r\__1}\,\log {\it r\__2}\,v
 -\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,{\it d\__0}\,\log 
 {\it r\__2}\,{\it y\__1}}\over{\log {\it r\__1}\,\log {\it r\__2}\,v
 -\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,{\it d\__2}\,\log ^2
 {\it r\__1}\,{\it y\__1}}\over{\log {\it r\__1}\,\log {\it r\__2}\,v
 -\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}+{{2\,{\it d\__1}\,\log ^2
 {\it r\__1}\,{\it y\__1}}\over{\log {\it r\__1}\,\log {\it r\__2}\,v
 -\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}+{{2\,{\it d\__2}\,\log 
 {\it r\__0}\,\log {\it r\__1}\,{\it y\__1}}\over{\log {\it r\__1}\,
 \log {\it r\__2}\,v-\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2
 {\it r\__1}\,v+\log {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,
 {\it d\__1}\,\log {\it r\__0}\,\log {\it r\__1}\,{\it y\__1}}\over{
 \log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}-{{2\,{\it d\__3}\,\log {\it r\__1}\,{\it y\__1}
 }\over{\log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}+{{2\,{\it d\__2}\,\log {\it r\__1}\,{\it y\__1}
 }\over{\log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}-{{2\,{\it d\__1}\,\log {\it r\__1}\,{\it y\__1}
 }\over{\log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}+{{2\,{\it d\__0}\,\log {\it r\__1}\,{\it y\__1}
 }\over{\log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}+{{2\,{\it d\__3}\,\log {\it r\__0}\,{\it y\__1}
 }\over{\log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}-{{2\,{\it d\__2}\,\log {\it r\__0}\,{\it y\__1}
 }\over{\log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}+{{2\,{\it d\__1}\,\log {\it r\__2}\,x\,\arctan 
 \left({{{\it y\__0}}\over{x}}\right)}\over{\log {\it r\__1}\,\log 
 {\it r\__2}\,v-\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2
 {\it r\__1}\,v+\log {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,
 {\it d\__0}\,\log {\it r\__2}\,x\,\arctan \left({{{\it y\__0}}\over{
 x}}\right)}\over{\log {\it r\__1}\,\log {\it r\__2}\,v-\log 
 {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,{\it d\__1}\,\log 
 {\it r\__1}\,x\,\arctan \left({{{\it y\__0}}\over{x}}\right)}\over{
 \log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}+{{2\,{\it d\__0}\,\log {\it r\__1}\,x\,\arctan 
 \left({{{\it y\__0}}\over{x}}\right)}\over{\log {\it r\__1}\,\log 
 {\it r\__2}\,v-\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2
 {\it r\__1}\,v+\log {\it r\__0}\,\log {\it r\__1}\,v}}+{{2\,
 {\it d\__0}\,\log {\it r\__1}\,\log {\it r\__2}\,{\it y\__0}}\over{
 \log {\it r\__1}\,\log {\it r\__2}\,v-\log {\it r\__0}\,\log 
 {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log {\it r\__0}\,\log 
 {\it r\__1}\,v}}-{{2\,{\it d\__0}\,\log {\it r\__0}\,\log 
 {\it r\__2}\,{\it y\__0}}\over{\log {\it r\__1}\,\log {\it r\__2}\,v
 -\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,{\it d\__1}\,\log 
 {\it r\__2}\,{\it y\__0}}\over{\log {\it r\__1}\,\log {\it r\__2}\,v
 -\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}+{{2\,{\it d\__0}\,\log 
 {\it r\__2}\,{\it y\__0}}\over{\log {\it r\__1}\,\log {\it r\__2}\,v
 -\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,{\it d\__0}\,\log ^2
 {\it r\__1}\,{\it y\__0}}\over{\log {\it r\__1}\,\log {\it r\__2}\,v
 -\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2{\it r\__1}\,v+\log 
 {\it r\__0}\,\log {\it r\__1}\,v}}+{{2\,{\it d\__0}\,\log 
 {\it r\__0}\,\log {\it r\__1}\,{\it y\__0}}\over{\log {\it r\__1}\,
 \log {\it r\__2}\,v-\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2
 {\it r\__1}\,v+\log {\it r\__0}\,\log {\it r\__1}\,v}}+{{2\,
 {\it d\__1}\,\log {\it r\__1}\,{\it y\__0}}\over{\log {\it r\__1}\,
 \log {\it r\__2}\,v-\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2
 {\it r\__1}\,v+\log {\it r\__0}\,\log {\it r\__1}\,v}}-{{2\,
 {\it d\__0}\,\log {\it r\__1}\,{\it y\__0}}\over{\log {\it r\__1}\,
 \log {\it r\__2}\,v-\log {\it r\__0}\,\log {\it r\__2}\,v-\log ^2
 {\it r\__1}\,v+\log {\it r\__0}\,\log {\it r\__1}\,v}}$$
(%o3)                                false
(%i4) 
Run Example
xgtr1:d - (d/(log(r_2/r_1)))*log(sqrt(x^2 + (y_0 - v*t)^2)/r_1);
                                                2    2
                                sqrt((y_0 - t v)  + x )
                          d log(-----------------------)
                                          r_1
(%o1)                 d - ------------------------------
                                         r_2
                                     log(---)
                                         r_1
(%i2) risch(xgtr1, t);
                              2                2    2  2
(%o2) ((d t v - d y_0) log(y_0  - 2 t v y_0 + x  + t  v )
 + d t (- 2 log(r_2) v - 2 v))/(2 log(r_1) v - 2 log(r_2) v)
                 2
            2 t v  - 2 v y_0
   d x atan(----------------)
                 2 v x
 + --------------------------
    (log(r_1) - log(r_2)) v
(%i3) 
Run Example
xgtr1:d - (d/(log(r_0/r_1)))*log(sqrt(x^2 + (y_0 - v*t)^2)/r_1);
                                                2    2
                                sqrt((y_0 - t v)  + x )
                          d log(-----------------------)
                                          r_1
(%o1)                 d - ------------------------------
                                         r_0
                                     log(---)
                                         r_1
(%i2) ratsimp(subst(y_0^2 + x^2 = r_0^2, ratsimp(subst(t = t_2, risch(xgtr1, t)) - subst(t = t_1, risch(xgtr1, t)))));
                                  2                  2      2  2
(%o2) - ((d y_0 - d t_2 v) log(y_0  - 2 t_2 v y_0 + x  + t_2  v )
                            2                  2      2  2
 + (d t_1 v - d y_0) log(y_0  - 2 t_1 v y_0 + x  + t_1  v )
              y_0 - t_2 v               y_0 - t_1 v
 + 2 d x atan(-----------) - 2 d x atan(-----------)
                   x                         x
 + ((2 d log(r_0) + 2 d) t_2 + (- 2 d log(r_0) - 2 d) t_1) v)
/((2 log(r_1) - 2 log(r_0)) v)
(%i3) 

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