Integrand size = 22, antiderivative size = 253 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^3} \]
-3*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^3-2/a/c^3/(1+1/a/x)^(7/2)/ (1-1/a/x)^(1/2)+x/c^3/(1+1/a/x)^(7/2)/(1-1/a/x)^(1/2)+11/7*(1-1/a/x)^(1/2) /a/c^3/(1+1/a/x)^(7/2)+54/35*(1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(5/2)+71/35*( 1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(3/2)+176/35*(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 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-176-423 a x-125 a^2 x^2+368 a^3 x^3+286 a^4 x^4+35 a^5 x^5\right )}{35 (-1+a x) (1+a x)^4}-3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^3} \]
((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-176 - 423*a*x - 125*a^2*x^2 + 368*a^3*x^3 + 286*a^4*x^4 + 35*a^5*x^5))/(35*(-1 + a*x)*(1 + a*x)^4) - 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^3)
Time = 0.41 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.97, 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.
| \(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -\frac {\int \frac {x^2}{\left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}}{c^3}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {-\int \frac {\left (3 a-\frac {5}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (3 a-\frac {5}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {a \left (-\int -\frac {\left (3 a-\frac {8}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}\right )-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {a \int \frac {\left (3 a-\frac {8}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (3 a-\frac {8}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{7} a \int \frac {3 \left (7 a-\frac {11}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {11 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {3}{7} \int \frac {\left (7 a-\frac {11}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {11 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {3}{7} \left (\frac {1}{5} a \int \frac {\left (35 a-\frac {36}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {18 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {11 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {3}{7} \left (\frac {1}{5} \int \frac {\left (35 a-\frac {36}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {18 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {11 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} a \int \frac {\left (105 a-\frac {71}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {71 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {18 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {11 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (105 a-\frac {71}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {71 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {18 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {11 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (a \int \frac {105 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {176 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {71 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {18 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {11 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {176 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {71 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {18 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {11 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {-\frac {\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {176 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-105 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )\right )+\frac {71 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {18 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {11 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {176 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-105 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )\right )+\frac {71 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {18 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {11 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^3}\) |
-((-(x/(Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))) - ((-2*a)/(Sqrt[1 - 1/(a*x )]*(1 + 1/(a*x))^(7/2)) + (11*a*Sqrt[1 - 1/(a*x)])/(7*(1 + 1/(a*x))^(7/2)) + (3*((18*a*Sqrt[1 - 1/(a*x)])/(5*(1 + 1/(a*x))^(5/2)) + ((71*a*Sqrt[1 - 1/(a*x)])/(3*(1 + 1/(a*x))^(3/2)) + ((176*a*Sqrt[1 - 1/(a*x)])/Sqrt[1 + 1/ (a*x)] - 105*a*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/3)/5))/7)/a^2 )/c^3)
3.9.28.3.1 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 = 0.23 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.11
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{3}}+\frac {\left (-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{6} \sqrt {a^{2}}}-\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{14 a^{11} \left (x +\frac {1}{a}\right )^{4}}+\frac {71 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{140 a^{10} \left (x +\frac {1}{a}\right )^{3}}-\frac {477 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{280 a^{9} \left (x +\frac {1}{a}\right )^{2}}+\frac {2931 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{560 a^{8} \left (x +\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{16 a^{8} \left (x -\frac {1}{a}\right )}\right ) a^{6} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{3} \left (a x -1\right )}\) | \(281\) |
| default | \(-\frac {\left (-3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{7} x^{7}+3360 \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}+2555 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}-11025 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}+10080 \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}+1873 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+3360 \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}-4426 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+18375 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}-16800 \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}-3350 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+18375 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-16800 \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}+2511 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -3675 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+3360 \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}+1957 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-11025 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +10080 \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 -3675 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+3360 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 ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{1120 a \sqrt {a^{2}}\, \left (a x +1\right )^{3} c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )^{3}}\) | \(714\) |
1/a*(a*x+1)/c^3*((a*x-1)/(a*x+1))^(1/2)+(-3/a^6*ln(a^2*x/(a^2)^(1/2)+(a^2* x^2-1)^(1/2))/(a^2)^(1/2)-1/14/a^11/(x+1/a)^4*(a^2*(x+1/a)^2-2*a*(x+1/a))^ (1/2)+71/140/a^10/(x+1/a)^3*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)-477/280/a^9/ (x+1/a)^2*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)+2931/560/a^8/(x+1/a)*(a^2*(x+1 /a)^2-2*a*(x+1/a))^(1/2)-1/16/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 = 179, normalized size of antiderivative = 0.71 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=-\frac {105 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (35 \, a^{5} x^{5} + 286 \, a^{4} x^{4} + 368 \, a^{3} x^{3} - 125 \, a^{2} x^{2} - 423 \, a x - 176\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
-1/35*(105*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 105*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1) ) - 1) - (35*a^5*x^5 + 286*a^4*x^4 + 368*a^3*x^3 - 125*a^2*x^2 - 423*a*x - 176)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3* x - a*c^3)
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {a^{6} \left (\int \left (- \frac {x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{7} x^{7} + a^{6} x^{6} - 3 a^{5} x^{5} - 3 a^{4} x^{4} + 3 a^{3} x^{3} + 3 a^{2} x^{2} - a x - 1}\right )\, dx + \int \frac {a x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{7} x^{7} + a^{6} x^{6} - 3 a^{5} x^{5} - 3 a^{4} x^{4} + 3 a^{3} x^{3} + 3 a^{2} x^{2} - a x - 1}\, dx\right )}{c^{3}} \]
a**6*(Integral(-x**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**7*x**7 + a**6*x **6 - 3*a**5*x**5 - 3*a**4*x**4 + 3*a**3*x**3 + 3*a**2*x**2 - a*x - 1), x) + Integral(a*x**7*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**7*x**7 + a**6*x** 6 - 3*a**5*x**5 - 3*a**4*x**4 + 3*a**3*x**3 + 3*a**2*x**2 - a*x - 1), x))/ c**3
Time = 0.20 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=-\frac {1}{560} \, a {\left (\frac {35 \, {\left (\frac {33 \, {\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {5 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 56 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 350 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 2520 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \]
-1/560*a*(35*(33*(a*x - 1)/(a*x + 1) - 1)/(a^2*c^3*((a*x - 1)/(a*x + 1))^( 3/2) - a^2*c^3*sqrt((a*x - 1)/(a*x + 1))) - (5*((a*x - 1)/(a*x + 1))^(7/2) + 56*((a*x - 1)/(a*x + 1))^(5/2) + 350*((a*x - 1)/(a*x + 1))^(3/2) + 2520 *sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^3) + 1680*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 1680*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^3))
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{3}} \,d x } \]
Time = 3.86 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.72 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {\frac {33\,\left (a\,x-1\right )}{a\,x+1}-1}{16\,a\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}-16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}+\frac {9\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a\,c^3}+\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{8\,a\,c^3}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{10\,a\,c^3}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{112\,a\,c^3}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c^3} \]
((33*(a*x - 1))/(a*x + 1) - 1)/(16*a*c^3*((a*x - 1)/(a*x + 1))^(1/2) - 16* a*c^3*((a*x - 1)/(a*x + 1))^(3/2)) + (9*((a*x - 1)/(a*x + 1))^(1/2))/(2*a* c^3) + (5*((a*x - 1)/(a*x + 1))^(3/2))/(8*a*c^3) + ((a*x - 1)/(a*x + 1))^( 5/2)/(10*a*c^3) + ((a*x - 1)/(a*x + 1))^(7/2)/(112*a*c^3) + (atan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*6i)/(a*c^3)