### Related

##### risch-sin-sqrt

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

Calculate

##### risch-sqrt

risch(sqrt(x+sqrt(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-sqrt

xgtr1:d - (d/r_1)*(sq...

risch(xgtr1, t);

Calculate

##### risch-sqrt

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

risch(xgtr1, u);

Calculate

##### risch

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

Calculate

##### risch-sqrt

risch ((x^2 + 2*x + 1...

Calculate

##### risch

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

Calculate

##### risch-sqrt

risch ((x^2 + 2*x + 1...

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
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) subst(y_1^2 + x^2 = r_1^2, subst([y_0^2 + x^2 = r_0^2, t_11 = (y_0 + y_1)/v, t_1 = (y_0 - y_1)/v], expand(subst(t = t_11, risch(xltr1, t)) - subst(t = t_1, risch(xltr1, t)) + 2*(subst(t = t_1, risch(xltr0, t)) - subst(t = 0, risch(xltr0, t))))));
2                          2    2
d_3 (y_1 + y_0) log((y_1 + y_0)  - 2 y_0 (y_1 + y_0) + y_0  + x )
(%o3) - -----------------------------------------------------------------
2 log(r_1) v - 2 log(r_2) v
2                          2    2
d_2 (y_1 + y_0) log((y_1 + y_0)  - 2 y_0 (y_1 + y_0) + y_0  + x )
+ -----------------------------------------------------------------
2 log(r_1) v - 2 log(r_2) v
2                          2    2
d_3 y_0 log((y_1 + y_0)  - 2 y_0 (y_1 + y_0) + y_0  + x )
+ ---------------------------------------------------------
2 log(r_1) v - 2 log(r_2) v
2                          2    2
d_2 y_0 log((y_1 + y_0)  - 2 y_0 (y_1 + y_0) + y_0  + x )
- ---------------------------------------------------------
2 log(r_1) v - 2 log(r_2) v
2      y_1 + y_0   y_0            2      y_1 + y_0   y_0
2 d_3 x  atan(--------- - ---)    2 d_2 x  atan(--------- - ---)
x        x                        x        x
- ------------------------------- + -------------------------------
2 log(r_1) v x - 2 log(r_2) v x   2 log(r_1) v x - 2 log(r_2) v x
2 d_2 log(r_2) (y_1 + y_0)    2 d_3 log(r_1) (y_1 + y_0)
- --------------------------- + ---------------------------
2 log(r_1) v - 2 log(r_2) v   2 log(r_1) v - 2 log(r_2) v
2 d_3 (y_1 + y_0)             2 d_2 (y_1 + y_0)
+ --------------------------- - ---------------------------
2 log(r_1) v - 2 log(r_2) v   2 log(r_1) v - 2 log(r_2) v
2      2    2
d_3 log(- 2 y_0 (y_0 - y_1) + (y_0 - y_1)  + y_0  + x ) (y_0 - y_1)
+ -------------------------------------------------------------------
2 log(r_1) v - 2 log(r_2) v
2      2    2
d_2 log(- 2 y_0 (y_0 - y_1) + (y_0 - y_1)  + y_0  + x ) (y_0 - y_1)
- -------------------------------------------------------------------
2 log(r_1) v - 2 log(r_2) v
2      2    2
2 d_1 log(- 2 y_0 (y_0 - y_1) + (y_0 - y_1)  + y_0  + x ) (y_0 - y_1)
- ---------------------------------------------------------------------
2 log(r_0) v - 2 log(r_1) v
2      2    2
2 d_0 log(- 2 y_0 (y_0 - y_1) + (y_0 - y_1)  + y_0  + x ) (y_0 - y_1)
+ ---------------------------------------------------------------------
2 log(r_0) v - 2 log(r_1) v
2 d_2 log(r_2) (y_0 - y_1)    2 d_3 log(r_1) (y_0 - y_1)
+ --------------------------- - ---------------------------
2 log(r_1) v - 2 log(r_2) v   2 log(r_1) v - 2 log(r_2) v
2 d_3 (y_0 - y_1)             2 d_2 (y_0 - y_1)
- --------------------------- + ---------------------------
2 log(r_1) v - 2 log(r_2) v   2 log(r_1) v - 2 log(r_2) v
4 d_0 log(r_1) (y_0 - y_1)    4 d_1 log(r_0) (y_0 - y_1)
- --------------------------- + ---------------------------
2 log(r_0) v - 2 log(r_1) v   2 log(r_0) v - 2 log(r_1) v
4 d_1 (y_0 - y_1)             4 d_0 (y_0 - y_1)
+ --------------------------- - ---------------------------
2 log(r_0) v - 2 log(r_1) v   2 log(r_0) v - 2 log(r_1) v
2      2    2
d_3 y_0 log(- 2 y_0 (y_0 - y_1) + (y_0 - y_1)  + y_0  + x )
- -----------------------------------------------------------
2 log(r_1) v - 2 log(r_2) v
2      2    2
d_2 y_0 log(- 2 y_0 (y_0 - y_1) + (y_0 - y_1)  + y_0  + x )
+ -----------------------------------------------------------
2 log(r_1) v - 2 log(r_2) v
2      2    2
2 d_1 y_0 log(- 2 y_0 (y_0 - y_1) + (y_0 - y_1)  + y_0  + x )
+ -------------------------------------------------------------
2 log(r_0) v - 2 log(r_1) v
2      2    2
2 d_0 y_0 log(- 2 y_0 (y_0 - y_1) + (y_0 - y_1)  + y_0  + x )
- -------------------------------------------------------------
2 log(r_0) v - 2 log(r_1) v
2      y_0 - y_1   y_0            2      y_0 - y_1   y_0
2 d_3 x  atan(--------- - ---)    2 d_2 x  atan(--------- - ---)
x        x                        x        x
+ ------------------------------- - -------------------------------
2 log(r_1) v x - 2 log(r_2) v x   2 log(r_1) v x - 2 log(r_2) v x
2      y_0 - y_1   y_0            2      y_0 - y_1   y_0
4 d_1 x  atan(--------- - ---)    4 d_0 x  atan(--------- - ---)
x        x                        x        x
- ------------------------------- + -------------------------------
2 log(r_0) v x - 2 log(r_1) v x   2 log(r_0) v x - 2 log(r_1) v x
2      y_0                        2      y_0
4 d_1 x  atan(---)                4 d_0 x  atan(---)
x                                 x
- ------------------------------- + -------------------------------
2 log(r_0) v x - 2 log(r_1) v x   2 log(r_0) v x - 2 log(r_1) v x
4 d_1 log(r_0) y_0            4 d_0 log(r_0) y_0
- --------------------------- + ---------------------------
2 log(r_0) v - 2 log(r_1) v   2 log(r_0) v - 2 log(r_1) v
(%i4) 

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