Integrand size = 22, antiderivative size = 255 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^3} \]
-4/3/a/c^3/(1-1/a/x)^(3/2)/(1+1/a/x)^(5/2)+x/c^3/(1-1/a/x)^(3/2)/(1+1/a/x) ^(5/2)-arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^3-13/3/a/c^3/(1+1/a/x) ^(5/2)/(1-1/a/x)^(1/2)+14/5*(1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(5/2)+11/5*(1- 1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(3/2)+16/5*(1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(1 /2)
Time = 0.44 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.40 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (48+33 a x-87 a^2 x^2-52 a^3 x^3+38 a^4 x^4+15 a^5 x^5\right )}{15 (-1+a x)^2 (1+a x)^3}-\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^3} \]
((a*Sqrt[1 - 1/(a^2*x^2)]*x*(48 + 33*a*x - 87*a^2*x^2 - 52*a^3*x^3 + 38*a^ 4*x^4 + 15*a^5*x^5))/(15*(-1 + a*x)^2*(1 + a*x)^3) - Log[(1 + Sqrt[1 - 1/( a^2*x^2)])*x])/(a*c^3)
Time = 0.41 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.95, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {6748, 114, 27, 169, 25, 27, 169, 27, 169, 27, 169, 27, 169, 27, 103, 221}
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.
| \(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -\frac {\int \frac {x^2}{\left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}}{c^3}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {-\int \frac {\left (a-\frac {5}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (a-\frac {5}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {-\frac {1}{3} a \int -\frac {\left (3 a-\frac {16}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} a \int \frac {\left (3 a-\frac {16}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \int \frac {\left (3 a-\frac {16}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (a \left (-\int -\frac {3 \left (a-\frac {13}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}\right )-\frac {13 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (3 \int \frac {\left (a-\frac {13}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}-\frac {13 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (3 \left (\frac {1}{5} a \int \frac {\left (5 a-\frac {28}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {14 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {13 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (3 \left (\frac {1}{5} \int \frac {\left (5 a-\frac {28}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {14 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {13 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (3 \left (\frac {1}{5} \left (\frac {1}{3} a \int \frac {3 \left (5 a-\frac {11}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {11 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {14 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {13 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (3 \left (\frac {1}{5} \left (\int \frac {\left (5 a-\frac {11}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {11 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {14 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {13 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (3 \left (\frac {1}{5} \left (a \int \frac {5 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {16 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {11 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {14 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {13 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (3 \left (\frac {1}{5} \left (5 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {16 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {11 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {14 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {13 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (3 \left (\frac {1}{5} \left (-5 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )+\frac {16 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {11 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {14 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {13 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (3 \left (\frac {1}{5} \left (-5 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {16 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {11 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {14 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {13 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^3}\) |
-((-(x/((1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2))) - ((-4*a)/(3*(1 - 1/(a*x ))^(3/2)*(1 + 1/(a*x))^(5/2)) + ((-13*a)/(Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^ (5/2)) + 3*((14*a*Sqrt[1 - 1/(a*x)])/(5*(1 + 1/(a*x))^(5/2)) + ((11*a*Sqrt [1 - 1/(a*x)])/(1 + 1/(a*x))^(3/2) + (16*a*Sqrt[1 - 1/(a*x)])/Sqrt[1 + 1/( a*x)] - 5*a*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/5))/3)/a^2)/c^3)
3.9.12.3.1 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[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[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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[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 = 0.24 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{3}}+\frac {\left (-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{6} \sqrt {a^{2}}}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{20 a^{10} \left (x +\frac {1}{a}\right )^{3}}-\frac {23 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{60 a^{9} \left (x +\frac {1}{a}\right )^{2}}+\frac {493 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{240 a^{8} \left (x +\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{24 a^{9} \left (x -\frac {1}{a}\right )^{2}}-\frac {25 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{48 a^{8} \left (x -\frac {1}{a}\right )}\right ) a^{6} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{3} \left (a x -1\right )}\) | \(287\) |
| default | \(-\frac {\left (-525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{7} x^{7}+240 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{8} x^{7}+285 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}-525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}+240 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{7} x^{6}-83 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+1575 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-720 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}-218 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+1575 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}-720 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+342 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-1575 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+720 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-3 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -1575 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+720 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-243 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+525 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -240 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +525 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-240 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) \sqrt {\frac {a x -1}{a x +1}}}{240 a \left (a x +1\right )^{3} \sqrt {a^{2}}\, \left (a x -1\right )^{3} c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}\) | \(714\) |
1/a*(a*x+1)/c^3*((a*x-1)/(a*x+1))^(1/2)+(-1/a^6*ln(a^2*x/(a^2)^(1/2)+(a^2* x^2-1)^(1/2))/(a^2)^(1/2)+1/20/a^10/(x+1/a)^3*(a^2*(x+1/a)^2-2*a*(x+1/a))^ (1/2)-23/60/a^9/(x+1/a)^2*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)+493/240/a^8/(x +1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)-1/24/a^9/(x-1/a)^2*((x-1/a)^2*a^2+ 2*(x-1/a)*a)^(1/2)-25/48/a^8/(x-1/a)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2))*a^ 6/c^3*((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 = 161, normalized size of antiderivative = 0.63 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=-\frac {15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (15 \, a^{5} x^{5} + 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} - 87 \, a^{2} x^{2} + 33 \, a x + 48\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
-1/15*(15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 1 5*(a^4*x^4 - 2*a^2*x^2 + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (15*a^5*x ^5 + 38*a^4*x^4 - 52*a^3*x^3 - 87*a^2*x^2 + 33*a*x + 48)*sqrt((a*x - 1)/(a *x + 1)))/(a^5*c^3*x^4 - 2*a^3*c^3*x^2 + a*c^3)
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {a^{6} \int \frac {x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \]
a**6*Integral(x**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**6*x**6 - 3*a**4*x **4 + 3*a**2*x**2 - 1), x)/c**3
Time = 0.19 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {1}{240} \, a {\left (\frac {5 \, {\left (\frac {23 \, {\left (a x - 1\right )}}{a x + 1} - \frac {120 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1\right )}}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 40 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 450 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} - \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} + \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \]
1/240*a*(5*(23*(a*x - 1)/(a*x + 1) - 120*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2 *c^3*((a*x - 1)/(a*x + 1))^(5/2) - a^2*c^3*((a*x - 1)/(a*x + 1))^(3/2)) + (3*((a*x - 1)/(a*x + 1))^(5/2) + 40*((a*x - 1)/(a*x + 1))^(3/2) + 450*sqrt ((a*x - 1)/(a*x + 1)))/(a^2*c^3) - 240*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/ (a^2*c^3) + 240*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^3))
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{3}} \,d x } \]
Time = 0.06 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {15\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8\,a\,c^3}-\frac {\frac {23\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {40\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{6\,a\,c^3}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{80\,a\,c^3}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,c^3} \]
(15*((a*x - 1)/(a*x + 1))^(1/2))/(8*a*c^3) - ((23*(a*x - 1))/(3*(a*x + 1)) - (40*(a*x - 1)^2)/(a*x + 1)^2 + 1/3)/(16*a*c^3*((a*x - 1)/(a*x + 1))^(3/ 2) - 16*a*c^3*((a*x - 1)/(a*x + 1))^(5/2)) + ((a*x - 1)/(a*x + 1))^(3/2)/( 6*a*c^3) + ((a*x - 1)/(a*x + 1))^(5/2)/(80*a*c^3) + (atan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*2i)/(a*c^3)