Integrand size = 22, antiderivative size = 393 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {35}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {35}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {7}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{16} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x^6+\frac {1}{8} a^6 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2} x^7-\frac {1}{8} a^7 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{9/2} x^9-\frac {35 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{128 a} \]
-5/48*a^5*c^4*(1-1/a/x)^(3/2)*(1+1/a/x)^(9/2)*x^6+1/8*a^6*c^4*(1-1/a/x)^(5 /2)*(1+1/a/x)^(9/2)*x^7-1/8*a^7*c^4*(1-1/a/x)^(7/2)*(1+1/a/x)^(9/2)*x^8+1/ 9*a^8*c^4*(1-1/a/x)^(9/2)*(1+1/a/x)^(9/2)*x^9-35/128*c^4*arctanh((1-1/a/x) ^(1/2)*(1+1/a/x)^(1/2))/a-35/384*a*c^4*(1+1/a/x)^(3/2)*x^2*(1-1/a/x)^(1/2) -7/192*a^2*c^4*(1+1/a/x)^(5/2)*x^3*(1-1/a/x)^(1/2)-1/64*a^3*c^4*(1+1/a/x)^ (7/2)*x^4*(1-1/a/x)^(1/2)+1/16*a^4*c^4*(1+1/a/x)^(9/2)*x^5*(1-1/a/x)^(1/2) -35/128*c^4*x*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)
Time = 0.22 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.28 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^4 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (128+837 a x-512 a^2 x^2-978 a^3 x^3+768 a^4 x^4+600 a^5 x^5-512 a^6 x^6-144 a^7 x^7+128 a^8 x^8\right )-315 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{1152 a} \]
(c^4*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(128 + 837*a*x - 512*a^2*x^2 - 978*a^3*x^3 + 768*a^4*x^4 + 600*a^5*x^5 - 512*a^6*x^6 - 144*a^7*x^7 + 128*a^8*x^8) - 315*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(1152*a)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (c-a^2 c x^2\right )^4 e^{-\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^8 c^4 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8}{a^8}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^4dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^8 c^4 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8}{a^8}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^4dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^8 c^4 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8}{a^8}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^4dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^8 c^4 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8}{a^8}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^4dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^8 c^4 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8}{a^8}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^4dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^8 c^4 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8}{a^8}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^4dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^8 c^4 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8}{a^8}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^4dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^8 c^4 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8}{a^8}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^4dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^8 c^4 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8}{a^8}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^4dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^8 c^4 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8}{a^8}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^4 x^8dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^4 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^4dx\) |
3.6.90.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symb ol] :> Simp[d^p Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && In tegerQ[p]
Time = 0.49 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.41
| method | result | size |
| risch | \(\frac {\left (128 a^{8} x^{8}-144 a^{7} x^{7}-512 a^{6} x^{6}+600 a^{5} x^{5}+768 a^{4} x^{4}-978 a^{3} x^{3}-512 a^{2} x^{2}+837 a x +128\right ) \left (a x +1\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}}{1152 a}-\frac {35 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{128 \sqrt {a^{2}}\, \left (a x -1\right )}\) | \(160\) |
| default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{4} \left (-128 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{6} x^{6}+144 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}+384 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-456 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-384 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+522 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +384 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-256 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-315 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +315 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{1152 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(279\) |
1/1152*(128*a^8*x^8-144*a^7*x^7-512*a^6*x^6+600*a^5*x^5+768*a^4*x^4-978*a^ 3*x^3-512*a^2*x^2+837*a*x+128)*(a*x+1)/a*c^4*((a*x-1)/(a*x+1))^(1/2)-35/12 8*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)*c^4*((a*x-1)/(a*x+1) )^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x-1)
Time = 0.25 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.43 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (128 \, a^{9} c^{4} x^{9} - 16 \, a^{8} c^{4} x^{8} - 656 \, a^{7} c^{4} x^{7} + 88 \, a^{6} c^{4} x^{6} + 1368 \, a^{5} c^{4} x^{5} - 210 \, a^{4} c^{4} x^{4} - 1490 \, a^{3} c^{4} x^{3} + 325 \, a^{2} c^{4} x^{2} + 965 \, a c^{4} x + 128 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{1152 \, a} \]
-1/1152*(315*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 315*c^4*log(sqrt((a* x - 1)/(a*x + 1)) - 1) - (128*a^9*c^4*x^9 - 16*a^8*c^4*x^8 - 656*a^7*c^4*x ^7 + 88*a^6*c^4*x^6 + 1368*a^5*c^4*x^5 - 210*a^4*c^4*x^4 - 1490*a^3*c^4*x^ 3 + 325*a^2*c^4*x^2 + 965*a*c^4*x + 128*c^4)*sqrt((a*x - 1)/(a*x + 1)))/a
\[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=c^{4} \left (\int \left (- 4 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx + \int 6 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- 4 a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx + \int a^{8} x^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx\right ) \]
c**4*(Integral(-4*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integr al(6*a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(-4*a**6*x* *6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(a**8*x**8*sqrt(a*x/(a* x + 1) - 1/(a*x + 1)), x) + Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) )
Time = 0.20 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.06 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {1}{1152} \, {\left (\frac {315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (315 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{2}} - 2730 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 10458 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} - 23202 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - 32768 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 23202 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 10458 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2730 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 315 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {9 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {36 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {84 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {126 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {126 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {84 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac {36 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {9 \, {\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + \frac {{\left (a x - 1\right )}^{9} a^{2}}{{\left (a x + 1\right )}^{9}} - a^{2}}\right )} a \]
-1/1152*(315*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 315*c^4*log(sqrt ((a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(315*c^4*((a*x - 1)/(a*x + 1))^(17/2) - 2730*c^4*((a*x - 1)/(a*x + 1))^(15/2) + 10458*c^4*((a*x - 1)/(a*x + 1))^( 13/2) - 23202*c^4*((a*x - 1)/(a*x + 1))^(11/2) - 32768*c^4*((a*x - 1)/(a*x + 1))^(9/2) + 23202*c^4*((a*x - 1)/(a*x + 1))^(7/2) - 10458*c^4*((a*x - 1 )/(a*x + 1))^(5/2) + 2730*c^4*((a*x - 1)/(a*x + 1))^(3/2) - 315*c^4*sqrt(( a*x - 1)/(a*x + 1)))/(9*(a*x - 1)*a^2/(a*x + 1) - 36*(a*x - 1)^2*a^2/(a*x + 1)^2 + 84*(a*x - 1)^3*a^2/(a*x + 1)^3 - 126*(a*x - 1)^4*a^2/(a*x + 1)^4 + 126*(a*x - 1)^5*a^2/(a*x + 1)^5 - 84*(a*x - 1)^6*a^2/(a*x + 1)^6 + 36*(a *x - 1)^7*a^2/(a*x + 1)^7 - 9*(a*x - 1)^8*a^2/(a*x + 1)^8 + (a*x - 1)^9*a^ 2/(a*x + 1)^9 - a^2))*a
Time = 0.30 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.50 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {35 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{128 \, {\left | a \right |}} + \frac {1}{1152} \, \sqrt {a^{2} x^{2} - 1} {\left (\frac {128 \, c^{4} \mathrm {sgn}\left (a x + 1\right )}{a} + {\left (837 \, c^{4} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (256 \, a c^{4} \mathrm {sgn}\left (a x + 1\right ) + {\left (489 \, a^{2} c^{4} \mathrm {sgn}\left (a x + 1\right ) - 4 \, {\left (96 \, a^{3} c^{4} \mathrm {sgn}\left (a x + 1\right ) + {\left (75 \, a^{4} c^{4} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (32 \, a^{5} c^{4} \mathrm {sgn}\left (a x + 1\right ) - {\left (8 \, a^{7} c^{4} x \mathrm {sgn}\left (a x + 1\right ) - 9 \, a^{6} c^{4} \mathrm {sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
35/128*c^4*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + 1 /1152*sqrt(a^2*x^2 - 1)*(128*c^4*sgn(a*x + 1)/a + (837*c^4*sgn(a*x + 1) - 2*(256*a*c^4*sgn(a*x + 1) + (489*a^2*c^4*sgn(a*x + 1) - 4*(96*a^3*c^4*sgn( a*x + 1) + (75*a^4*c^4*sgn(a*x + 1) - 2*(32*a^5*c^4*sgn(a*x + 1) - (8*a^7* c^4*x*sgn(a*x + 1) - 9*a^6*c^4*sgn(a*x + 1))*x)*x)*x)*x)*x)*x)*x)
Time = 4.43 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.92 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {\frac {35\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64}-\frac {455\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{96}+\frac {581\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{32}-\frac {1289\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{32}+\frac {512\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{9}+\frac {1289\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{32}-\frac {581\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{32}+\frac {455\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{96}-\frac {35\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/2}}{64}}{a-\frac {9\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {36\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {84\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {126\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {126\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {84\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {36\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}+\frac {9\,a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}-\frac {a\,{\left (a\,x-1\right )}^9}{{\left (a\,x+1\right )}^9}}-\frac {35\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{64\,a} \]
((35*c^4*((a*x - 1)/(a*x + 1))^(1/2))/64 - (455*c^4*((a*x - 1)/(a*x + 1))^ (3/2))/96 + (581*c^4*((a*x - 1)/(a*x + 1))^(5/2))/32 - (1289*c^4*((a*x - 1 )/(a*x + 1))^(7/2))/32 + (512*c^4*((a*x - 1)/(a*x + 1))^(9/2))/9 + (1289*c ^4*((a*x - 1)/(a*x + 1))^(11/2))/32 - (581*c^4*((a*x - 1)/(a*x + 1))^(13/2 ))/32 + (455*c^4*((a*x - 1)/(a*x + 1))^(15/2))/96 - (35*c^4*((a*x - 1)/(a* x + 1))^(17/2))/64)/(a - (9*a*(a*x - 1))/(a*x + 1) + (36*a*(a*x - 1)^2)/(a *x + 1)^2 - (84*a*(a*x - 1)^3)/(a*x + 1)^3 + (126*a*(a*x - 1)^4)/(a*x + 1) ^4 - (126*a*(a*x - 1)^5)/(a*x + 1)^5 + (84*a*(a*x - 1)^6)/(a*x + 1)^6 - (3 6*a*(a*x - 1)^7)/(a*x + 1)^7 + (9*a*(a*x - 1)^8)/(a*x + 1)^8 - (a*(a*x - 1 )^9)/(a*x + 1)^9) - (35*c^4*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(64*a)