Integrand size = 22, antiderivative size = 138 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {4 \left (a+\frac {1}{x}\right )}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {5 a+\frac {7}{x}}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {15 a+\frac {19}{x}}{5 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \]
-4/5*(a+1/x)/a^2/c^4/(1-1/a^2/x^2)^(5/2)+1/5*(-5*a-7/x)/a^2/c^4/(1-1/a^2/x ^2)^(3/2)+3*arctanh((1-1/a^2/x^2)^(1/2))/a/c^4+1/5*(-15*a-19/x)/a^2/c^4/(1 -1/a^2/x^2)^(1/2)+x*(1-1/a^2/x^2)^(1/2)/c^4
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {-24+33 a x+18 a^2 x^2-34 a^3 x^3+5 a^4 x^4+15 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{5 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2} \]
(-24 + 33*a*x + 18*a^2*x^2 - 34*a^3*x^3 + 5*a^4*x^4 + 15*a*Sqrt[1 - 1/(a^2 *x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(5*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)
Time = 0.60 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.93, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6731, 27, 570, 532, 25, 2336, 27, 2336, 27, 534, 243, 73, 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.
| \(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -\frac {\int \frac {a^3 x^2}{c^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )^3}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \int \frac {x^2}{\sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )^3}d\frac {1}{x}}{c^4}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle -\frac {\int \frac {\left (a+\frac {1}{x}\right )^3 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}}d\frac {1}{x}}{a^3 c^4}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle -\frac {\frac {4 a \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} \int -\frac {\left (5 a^3+\frac {15 a^2}{x}+\frac {16 a}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}}{a^3 c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{5} \int \frac {\left (5 a^3+\frac {15 a^2}{x}+\frac {16 a}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}+\frac {4 a \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^3 c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\frac {a \left (5 a+\frac {7}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} \int -\frac {3 \left (5 a^3+\frac {15 a^2}{x}+\frac {14 a}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^3 c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{5} \left (\int \frac {\left (5 a^3+\frac {15 a^2}{x}+\frac {14 a}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {a \left (5 a+\frac {7}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^3 c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{5} \left (-\int -\frac {5 a^2 \left (a+\frac {3}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {a \left (5 a+\frac {7}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a \left (15 a+\frac {19}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^3 c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{5} \left (5 a^2 \int \frac {\left (a+\frac {3}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {a \left (5 a+\frac {7}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a \left (15 a+\frac {19}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^3 c^4}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -\frac {\frac {1}{5} \left (5 a^2 \left (3 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a \left (5 a+\frac {7}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a \left (15 a+\frac {19}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^3 c^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {\frac {1}{5} \left (5 a^2 \left (\frac {3}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a \left (5 a+\frac {7}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a \left (15 a+\frac {19}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^3 c^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {1}{5} \left (5 a^2 \left (-3 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a \left (5 a+\frac {7}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a \left (15 a+\frac {19}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^3 c^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {1}{5} \left (5 a^2 \left (-3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a \left (5 a+\frac {7}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a \left (15 a+\frac {19}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^3 c^4}\) |
-(((4*a*(a + x^(-1)))/(5*(1 - 1/(a^2*x^2))^(5/2)) + ((a*(5*a + 7/x))/(1 - 1/(a^2*x^2))^(3/2) + (a*(15*a + 19/x))/Sqrt[1 - 1/(a^2*x^2)] + 5*a^2*(-(a* Sqrt[1 - 1/(a^2*x^2)]*x) - 3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/5)/(a^3*c^4) )
3.5.20.3.1 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_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S imp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.24 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.63
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{4}}+\frac {\left (\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{5 a^{8} \left (x -\frac {1}{a}\right )^{3}}-\frac {6 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{5 a^{7} \left (x -\frac {1}{a}\right )^{2}}-\frac {24 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{5 a^{6} \left (x -\frac {1}{a}\right )}\right ) a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \left (a x -1\right )}\) | \(225\) |
| default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-125 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}-120 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+85 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+480 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-148 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -750 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-720 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+67 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +480 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -125 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-120 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right )}{40 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, c^{4} \left (a x -1\right )^{4} \sqrt {a^{2}}}\) | \(436\) |
1/a*(a*x+1)/c^4*((a*x-1)/(a*x+1))^(1/2)+(3/a^4*ln(a^2*x/(a^2)^(1/2)+(a^2*x ^2-1)^(1/2))/(a^2)^(1/2)-1/5/a^8/(x-1/a)^3*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/ 2)-6/5/a^7/(x-1/a)^2*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-24/5/a^6/(x-1/a)*(( x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2))*a^4/c^4*((a*x-1)/(a*x+1))^(1/2)/(a*x-1)*( (a*x-1)*(a*x+1))^(1/2)
Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {15 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (5 \, a^{4} x^{4} - 34 \, a^{3} x^{3} + 18 \, a^{2} x^{2} + 33 \, a x - 24\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}} \]
1/5*(15*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (5*a^4*x^4 - 34*a^3*x^3 + 18*a^2*x^2 + 33*a*x - 24)*sqrt((a*x - 1)/(a *x + 1)))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {a^{4} \int \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 4 a^{3} x^{3} + 6 a^{2} x^{2} - 4 a x + 1}\, dx}{c^{4}} \]
a**4*Integral(x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**4*x**4 - 4*a**3*x **3 + 6*a**2*x**2 - 4*a*x + 1), x)/c**4
Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {1}{20} \, a {\left (\frac {\frac {9 \, {\left (a x - 1\right )}}{a x + 1} + \frac {75 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {125 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]
1/20*a*((9*(a*x - 1)/(a*x + 1) + 75*(a*x - 1)^2/(a*x + 1)^2 - 125*(a*x - 1 )^3/(a*x + 1)^3 + 1)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(7/2) - a^2*c^4*((a*x - 1)/(a*x + 1))^(5/2)) + 60*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 60*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^4))
Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {3 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{c^{4} {\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} \mathrm {sgn}\left (a x + 1\right )}{a c^{4}} \]
-3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/(c^4*abs(a)) + sqr t(a^2*x^2 - 1)*sgn(a*x + 1)/(a*c^4)
Time = 3.87 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^4}-\frac {\frac {15\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {25\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {9\,\left (a\,x-1\right )}{5\,\left (a\,x+1\right )}+\frac {1}{5}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]