Integrand size = 22, antiderivative size = 195 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=-\frac {c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{2 a}+\frac {3 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{2 a}+\frac {4 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{3 a}+c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x+\frac {3 c^2 \csc ^{-1}(a x)}{2 a}-\frac {c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]
4/3*c^2*(1-1/a/x)^(3/2)*(1+1/a/x)^(3/2)/a+c^2*(1-1/a/x)^(5/2)*(1+1/a/x)^(3 /2)*x+3/2*c^2*arccsc(a*x)/a-c^2*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a +3/2*c^2*(1+1/a/x)^(3/2)*(1-1/a/x)^(1/2)/a-1/2*c^2*(1-1/a/x)^(1/2)*(1+1/a/ x)^(1/2)/a
Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.48 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^2 \left (\sqrt {1-\frac {1}{a^2 x^2}} \left (-2+3 a x+8 a^2 x^2+6 a^3 x^3\right )+9 a^2 x^2 \arcsin \left (\frac {1}{a x}\right )-6 a^2 x^2 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{6 a^3 x^2} \]
(c^2*(Sqrt[1 - 1/(a^2*x^2)]*(-2 + 3*a*x + 8*a^2*x^2 + 6*a^3*x^3) + 9*a^2*x ^2*ArcSin[1/(a*x)] - 6*a^2*x^2*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a^3 *x^2)
Time = 0.38 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.94, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {6748, 108, 25, 27, 171, 27, 171, 27, 171, 25, 27, 175, 39, 103, 221, 223}
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.
| \(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^2 e^{-\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -c^2 \int \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle -c^2 \left (\int -\frac {\left (a+\frac {4}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{a^2}d\frac {1}{x}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\int \frac {\left (a+\frac {4}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{a^2}d\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\int \left (a+\frac {4}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} xd\frac {1}{x}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\frac {1}{3} a \int \frac {3 \left (a+\frac {3}{x}\right ) \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{a}d\frac {1}{x}+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\int \left (a+\frac {3}{x}\right ) \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} xd\frac {1}{x}+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\frac {1}{2} a \int \frac {\left (2 a+\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {3}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\frac {1}{2} \int \frac {\left (2 a+\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {3}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\frac {1}{2} \left (a \left (-\int -\frac {\left (2 a+\frac {3}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )-a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {3}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\frac {1}{2} \left (a \int \frac {\left (2 a+\frac {3}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {3}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\frac {1}{2} \left (\int \frac {\left (2 a+\frac {3}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {3}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\frac {1}{2} \left (2 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+3 \int \frac {1}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {3}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\frac {1}{2} \left (3 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+2 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {3}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\frac {1}{2} \left (3 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-2 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )+a \left (-\sqrt {1-\frac {1}{a x}}\right ) \sqrt {\frac {1}{a x}+1}\right )+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {3}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\frac {1}{2} \left (3 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-2 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {3}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -c^2 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{5/2}-\frac {\frac {1}{2} \left (3 a \arcsin \left (\frac {1}{a x}\right )-2 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {4}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {3}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{a^2}\right )\) |
-(c^2*(-((1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)*x) - ((3*a*Sqrt[1 - 1/(a* x)]*(1 + 1/(a*x))^(3/2))/2 + (4*a*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)) /3 + (-(a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + 3*a*ArcSin[1/(a*x)] - 2*a *ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/2)/a^2))
3.9.8.3.1 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[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Simp[-c^p Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/x^2), x], x , 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[ n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegersQ[2*p, p + n/2]
Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.81
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \left (8 a^{2} x^{2}+3 a x -2\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}}{6 x^{3} a^{4}}+\frac {\left (-\frac {a^{4} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {3 a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}+a^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{4} \left (a x -1\right )}\) | \(157\) |
| default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{2} \left (-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-9 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+3 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{3} \sqrt {a^{2}}}\) | \(224\) |
1/6*(a*x+1)*(8*a^2*x^2+3*a*x-2)/x^3*c^2/a^4*((a*x-1)/(a*x+1))^(1/2)+(-a^4* ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)+3/2*a^3*arctan(1/(a^2* x^2-1)^(1/2))+a^3*((a*x-1)*(a*x+1))^(1/2))*c^2/a^4*((a*x-1)/(a*x+1))^(1/2) *((a*x-1)*(a*x+1))^(1/2)/(a*x-1)
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.80 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=-\frac {18 \, a^{3} c^{2} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 6 \, a^{3} c^{2} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 6 \, a^{3} c^{2} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (6 \, a^{4} c^{2} x^{4} + 14 \, a^{3} c^{2} x^{3} + 11 \, a^{2} c^{2} x^{2} + a c^{2} x - 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \]
-1/6*(18*a^3*c^2*x^3*arctan(sqrt((a*x - 1)/(a*x + 1))) + 6*a^3*c^2*x^3*log (sqrt((a*x - 1)/(a*x + 1)) + 1) - 6*a^3*c^2*x^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (6*a^4*c^2*x^4 + 14*a^3*c^2*x^3 + 11*a^2*c^2*x^2 + a*c^2*x - 2* c^2)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x^3)
\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^{2} \left (\int a^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{4}}\, dx + \int \left (- \frac {2 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{2}}\right )\, dx\right )}{a^{4}} \]
c**2*(Integral(a**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(sqrt( a*x/(a*x + 1) - 1/(a*x + 1))/x**4, x) + Integral(-2*a**2*sqrt(a*x/(a*x + 1 ) - 1/(a*x + 1))/x**2, x))/a**4
Time = 0.30 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.14 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=-\frac {1}{3} \, a {\left (\frac {9 \, c^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {3 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 29 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 15 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {2 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} \]
-1/3*a*(9*c^2*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - (3*c^2*((a*x - 1)/(a*x + 1))^(7/2) + c^2*((a*x - 1)/(a*x + 1))^(5/2) + 29* c^2*((a*x - 1)/(a*x + 1))^(3/2) + 15*c^2*sqrt((a*x - 1)/(a*x + 1)))/(2*(a* x - 1)*a^2/(a*x + 1) - 2*(a*x - 1)^3*a^2/(a*x + 1)^3 - (a*x - 1)^4*a^2/(a* x + 1)^4 + a^2))
Time = 0.30 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.35 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=-\frac {3 \, c^{2} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c^{2} \mathrm {sgn}\left (a x + 1\right )}{a} - \frac {3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} c^{2} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 12 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{2} \mathrm {sgn}\left (a x + 1\right ) - 12 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{2} \mathrm {sgn}\left (a x + 1\right ) - 3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{2} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 8 \, a c^{2} \mathrm {sgn}\left (a x + 1\right )}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3} a {\left | a \right |}} \]
-3*c^2*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1)/a + c^2*log(abs( -x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + sqrt(a^2*x^2 - 1)*c^ 2*sgn(a*x + 1)/a - 1/3*(3*(x*abs(a) - sqrt(a^2*x^2 - 1))^5*c^2*abs(a)*sgn( a*x + 1) - 12*(x*abs(a) - sqrt(a^2*x^2 - 1))^4*a*c^2*sgn(a*x + 1) - 12*(x* abs(a) - sqrt(a^2*x^2 - 1))^2*a*c^2*sgn(a*x + 1) - 3*(x*abs(a) - sqrt(a^2* x^2 - 1))*c^2*abs(a)*sgn(a*x + 1) - 8*a*c^2*sgn(a*x + 1))/(((x*abs(a) - sq rt(a^2*x^2 - 1))^2 + 1)^3*a*abs(a))
Time = 3.88 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.94 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {5\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {29\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+\frac {c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{3}+c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{a+\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}-\frac {2\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}-\frac {3\,c^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
(5*c^2*((a*x - 1)/(a*x + 1))^(1/2) + (29*c^2*((a*x - 1)/(a*x + 1))^(3/2))/ 3 + (c^2*((a*x - 1)/(a*x + 1))^(5/2))/3 + c^2*((a*x - 1)/(a*x + 1))^(7/2)) /(a + (2*a*(a*x - 1))/(a*x + 1) - (2*a*(a*x - 1)^3)/(a*x + 1)^3 - (a*(a*x - 1)^4)/(a*x + 1)^4) - (3*c^2*atan(((a*x - 1)/(a*x + 1))^(1/2)))/a - (2*c^ 2*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a