# Calculating this difficult integral (equation of a circle)

1. Jul 22, 2009

I have tried numerous methods to calculate this integral and can't seem to figure it out.
I have tried to use maple and mathematica but i am not very strong using these programs. I was wondering if someone would be able to help me solve this integral. This equation stemmed from the equation of a circle... and i got the following function:

$$h = h0 + R - \sqrt{R+x}\sqrt{R-x}$$

i am trying to integrate the following:
$$\int\frac{1}{h(x)^2}dx$$

and

$$\int\frac{1}{h(x)^3}dx$$

MT

2. Jul 22, 2009

### g_edgar

Do you really need symbolic answers? For the first one, Maple says:
$$-1/2\, \left( 2\,i{R}^{3}{\it h0}\,{\rm arctanh} \left( {\frac {{R}^{2 }+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) +2\,i{R}^{3}{\it h0}\,{\rm arctanh} \left( {\frac {-{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) -4\,{R}^{ 3}{\it h0}\,\arctan \left( {\frac {x}{\sqrt {{\it h0}}\sqrt {{\it h0}+ 2\,R}}} \right) +i{R}^{2}{{\it h0}}^{2}{\rm arctanh} \left( {\frac {{R }^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) +i{R}^{2}{{\it h0}}^{2}{ \rm arctanh} \left( {\frac {-{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{ \it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) -2\,{R}^{2}\arctan \left( {\frac {x}{\sqrt {{\it h0}}\sqrt {{ \it h0}+2\,R}}} \right) {{\it h0}}^{2}-2\,{R}^{2}x\sqrt {{\it h0}} \sqrt {{\it h0}+2\,R}+i{R}^{2}{\rm arctanh} \left( {\frac {-{R}^{2}+i \sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) {x}^{2}+i{R}^{2}{\rm arctanh} \left( {\frac {{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) {x}^{2}-2 \,{R}^{2}\arctan \left( {\frac {x}{\sqrt {{\it h0}}\sqrt {{\it h0}+2\, R}}} \right) {x}^{2}-4\,Rx{{\it h0}}^{3/2}\sqrt {{\it h0}+2\,R}-2\,R \sqrt {R-x}\sqrt {R+x}\sqrt {{\it h0}}\sqrt {{\it h0}+2\,R}x-2\,x{{ \it h0}}^{5/2}\sqrt {{\it h0}+2\,R}-2\,\sqrt {R-x}\sqrt {R+x}{{\it h0} }^{3/2}\sqrt {{\it h0}+2\,R}x \right) \left( {\it h0}+2\,R \right) ^{ -3/2}{{\it h0}}^{-3/2} \left( {{\it h0}}^{2}+2\,{\it h0}\,R+{x}^{2} \right) ^{-1}$$

3. Jul 22, 2009

### g_edgar

and for the second one, Maple says:
$$1/4\, \left( -12\,i{R}^{4}{\it h0}\,{\rm arctanh} \left( {\frac {{R}^{ 2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) {x}^{2}-12\,i{R}^{4}{\it h0 }\,{\rm arctanh} \left( {\frac {-{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt { {\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) {x}^{2}-18\,i{R}^{3}{{\it h0}}^{2}{\rm arctanh} \left( { \frac {{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) {x}^{2}-18\,i{R}^{3}{{ \it h0}}^{2}{\rm arctanh} \left( {\frac {-{R}^{2}+i\sqrt {{\it h0}+2\, R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2} }}} \right) {x}^{2}-6\,i{R}^{2}{{\it h0}}^{3}{\rm arctanh} \left( { \frac {-{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) {x}^{2}-6\,i{R}^{2}{{ \it h0}}^{3}{\rm arctanh} \left( {\frac {{R}^{2}+i\sqrt {{\it h0}+2\,R }\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}} }} \right) {x}^{2}-3\,i{R}^{2}{\it h0}\,{\rm arctanh} \left( {\frac {{ R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) {x}^{4}-3\,i{R}^{2}{\it h0} \,{\rm arctanh} \left( {\frac {-{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{ \it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) {x}^{4}+6\,\sqrt {R+x}\sqrt {R-x}{{\it h0}}^{9/2}\sqrt {{\it h0}+2\,R}x+2\,\sqrt {R+x}\sqrt {R-x}{{\it h0}}^{5/2}\sqrt {{\it h0}+2 \,R}{x}^{3}+4\,{{\it h0}}^{11/2}\sqrt {{\it h0}+2\,R}x+24\,{R}^{5} \arctan \left( {\frac {x}{\sqrt {{\it h0}}\sqrt {{\it h0}+2\,R}}} \right) {{\it h0}}^{2}+30\,{R}^{3}\arctan \left( {\frac {x}{\sqrt {{ \it h0}}\sqrt {{\it h0}+2\,R}}} \right) {{\it h0}}^{4}+48\,{R}^{4} \arctan \left( {\frac {x}{\sqrt {{\it h0}}\sqrt {{\it h0}+2\,R}}} \right) {{\it h0}}^{3}+6\,{R}^{3}\arctan \left( {\frac {x}{\sqrt {{ \it h0}}\sqrt {{\it h0}+2\,R}}} \right) {x}^{4}+6\,{R}^{2}\arctan \left( {\frac {x}{\sqrt {{\it h0}}\sqrt {{\it h0}+2\,R}}} \right) {{ \it h0}}^{5}+20\,{R}^{4}{{\it h0}}^{3/2}\sqrt {{\it h0}+2\,R}x+6\,{R}^ {3}\sqrt {{\it h0}}\sqrt {{\it h0}+2\,R}{x}^{3}+46\,{R}^{3}{{\it h0}}^ {5/2}\sqrt {{\it h0}+2\,R}x+42\,{R}^{2}{{\it h0}}^{7/2}\sqrt {{\it h0} +2\,R}x+6\,{R}^{2}{{\it h0}}^{3/2}\sqrt {{\it h0}+2\,R}{x}^{3}+20\,{{ \it h0}}^{9/2}\sqrt {{\it h0}+2\,R}xR+36\,{R}^{3}\arctan \left( { \frac {x}{\sqrt {{\it h0}}\sqrt {{\it h0}+2\,R}}} \right) {{\it h0}}^{ 2}{x}^{2}+6\,{R}^{2}\arctan \left( {\frac {x}{\sqrt {{\it h0}}\sqrt {{ \it h0}+2\,R}}} \right) {\it h0}\,{x}^{4}+12\,{R}^{2}\arctan \left( { \frac {x}{\sqrt {{\it h0}}\sqrt {{\it h0}+2\,R}}} \right) {{\it h0}}^{ 3}{x}^{2}+24\,{R}^{4}\arctan \left( {\frac {x}{\sqrt {{\it h0}}\sqrt { {\it h0}+2\,R}}} \right) {\it h0}\,{x}^{2}-12\,i{R}^{5}{{\it h0}}^{2}{ \rm arctanh} \left( {\frac {-{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{ \it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) -12\,i{R}^{5}{{\it h0}}^{2}{\rm arctanh} \left( {\frac {{R}^{ 2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) -24\,i{R}^{4}{{\it h0}}^{3} {\rm arctanh} \left( {\frac {-{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{ \it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) -24\,i{R}^{4}{{\it h0}}^{3}{\rm arctanh} \left( {\frac {{R}^{ 2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) -15\,i{R}^{3}{{\it h0}}^{4} {\rm arctanh} \left( {\frac {-{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{ \it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) -15\,i{R}^{3}{{\it h0}}^{4}{\rm arctanh} \left( {\frac {{R}^{ 2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) -3\,i{R}^{3}{\rm arctanh} \left( {\frac {{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) {x}^{4}-3 \,i{R}^{3}{\rm arctanh} \left( {\frac {-{R}^{2}+i\sqrt {{\it h0}+2\,R} \sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}} } \right) {x}^{4}-3\,i{R}^{2}{{\it h0}}^{5}{\rm arctanh} \left( { \frac {-{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt {{\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) -3\,i{R}^{2}{{\it h0}} ^{5}{\rm arctanh} \left( {\frac {{R}^{2}+i\sqrt {{\it h0}+2\,R}\sqrt { {\it h0}}x}{ \left( {\it h0}+R \right) \sqrt {{R}^{2}-{x}^{2}}}} \right) +20\,{R}^{3}\sqrt {R+x}\sqrt {R-x}{{\it h0}}^{3/2}\sqrt {{ \it h0}+2\,R}x+34\,{R}^{2}\sqrt {R+x}\sqrt {R-x}{{\it h0}}^{5/2}\sqrt {{\it h0}+2\,R}x+24\,R\sqrt {R+x}\sqrt {R-x}{{\it h0}}^{7/2}\sqrt {{ \it h0}+2\,R}x+4\,R\sqrt {R+x}\sqrt {R-x}{{\it h0}}^{3/2}\sqrt {{\it h0}+2\,R}{x}^{3}+6\,{R}^{2}\sqrt {R+x}\sqrt {R-x}\sqrt {{\it h0}} \sqrt {{\it h0}+2\,R}{x}^{3} \right) \left( -x+i\sqrt {{\it h0}+2\,R} \sqrt {{\it h0}} \right) ^{-2} \left( x+i\sqrt {{\it h0}+2\,R}\sqrt {{ \it h0}} \right) ^{-2} \left( {\it h0}+2\,R \right) ^{-5/2}{{\it h0}}^ {-5/2}$$
but this window doesn't show the whole thing in either case.

4. Jul 22, 2009