Integrand size = 22, antiderivative size = 138 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=-\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^5}+\frac {2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^5} \]
-2/5*(a+1/x)/a^2/c^5/(1-1/a^2/x^2)^(5/2)+1/15*(-10*a-13/x)/a^2/c^5/(1-1/a^ 2/x^2)^(3/2)+2*arctanh((1-1/a^2/x^2)^(1/2))/a/c^5+1/15*(-30*a-41/x)/a^2/c^ 5/(1-1/a^2/x^2)^(1/2)+x*(1-1/a^2/x^2)^(1/2)/c^5
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {-56+82 a x+32 a^2 x^2-76 a^3 x^3+15 a^4 x^4+30 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2} \]
(-56 + 82*a*x + 32*a^2*x^2 - 76*a^3*x^3 + 15*a^4*x^4 + 30*a*Sqrt[1 - 1/(a^ 2*x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(15*a^2*c^5*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)
Time = 0.60 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96, 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, 25, 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.
\(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx\) |
\(\Big \downarrow \) 6731 |
\(\displaystyle -\frac {\int \frac {a^2 x^2}{c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right )^2}d\frac {1}{x}}{c^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a^2 \int \frac {x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right )^2}d\frac {1}{x}}{c^5}\) |
\(\Big \downarrow \) 570 |
\(\displaystyle -\frac {\int \frac {\left (a+\frac {1}{x}\right )^2 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}}d\frac {1}{x}}{a^2 c^5}\) |
\(\Big \downarrow \) 532 |
\(\displaystyle -\frac {\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} \int -\frac {\left (5 a^2+\frac {10 a}{x}+\frac {8}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}}{a^2 c^5}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {1}{5} \int \frac {\left (5 a^2+\frac {10 a}{x}+\frac {8}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\) |
\(\Big \downarrow \) 2336 |
\(\displaystyle -\frac {\frac {1}{5} \left (\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} \int -\frac {\left (15 a^2+\frac {30 a}{x}+\frac {26}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (15 a^2+\frac {30 a}{x}+\frac {26}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\) |
\(\Big \downarrow \) 2336 |
\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}-\int -\frac {15 a \left (a+\frac {2}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a \int \frac {\left (a+\frac {2}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a \left (2 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a \left (\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a \left (-2 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a \left (-2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\) |
-(((2*(a + x^(-1)))/(5*(1 - 1/(a^2*x^2))^(5/2)) + ((10*a + 13/x)/(3*(1 - 1 /(a^2*x^2))^(3/2)) + ((30*a + 41/x)/Sqrt[1 - 1/(a^2*x^2)] + 15*a*(-(a*Sqrt [1 - 1/(a^2*x^2)]*x) - 2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/3)/5)/(a^2*c^5))
3.5.37.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(258\) vs. \(2(122)=244\).
Time = 0.24 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.88
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{5}}+\frac {\left (\frac {2 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{5} \sqrt {a^{2}}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{10 a^{9} \left (x -\frac {1}{a}\right )^{3}}-\frac {41 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{60 a^{8} \left (x -\frac {1}{a}\right )^{2}}-\frac {383 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{120 a^{7} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{8 a^{7} \left (x +\frac {1}{a}\right )}\right ) a^{5} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{5} \left (a x -1\right )}\) | \(259\) |
default | \(-\frac {\left (-75 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}-60 \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}+45 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+150 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+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^{6} x^{5}+2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+75 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}+60 \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}-64 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-300 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-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^{4} x^{3}-14 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +75 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+60 \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}+37 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+150 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +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^{2} x -75 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-60 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}}}{30 a \sqrt {a^{2}}\, c^{5} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )^{5}}\) | \(615\) |
1/a*(a*x+1)/c^5*((a*x-1)/(a*x+1))^(1/2)+(2/a^5*ln(a^2*x/(a^2)^(1/2)+(a^2*x ^2-1)^(1/2))/(a^2)^(1/2)-1/10/a^9/(x-1/a)^3*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1 /2)-41/60/a^8/(x-1/a)^2*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-383/120/a^7/(x-1 /a)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)+1/8/a^7/(x+1/a)*(a^2*(x+1/a)^2-2*a*( x+1/a))^(1/2))*a^5/c^5/(a*x-1)*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^( 1/2)
Time = 0.26 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (15 \, a^{4} x^{4} - 76 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 82 \, a x - 56\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} \]
1/15*(30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (15*a^4*x^4 - 76*a^3*x^3 + 32*a^2*x^2 + 82*a*x - 56)*sqrt((a*x - 1)/ (a*x + 1)))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5)
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {a^{5} \left (\int \left (- \frac {x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{6} x^{6} - 4 a^{5} x^{5} + 5 a^{4} x^{4} - 5 a^{2} x^{2} + 4 a x - 1}\right )\, dx + \int \frac {a x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{6} x^{6} - 4 a^{5} x^{5} + 5 a^{4} x^{4} - 5 a^{2} x^{2} + 4 a x - 1}\, dx\right )}{c^{5}} \]
a**5*(Integral(-x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**6*x**6 - 4*a**5 *x**5 + 5*a**4*x**4 - 5*a**2*x**2 + 4*a*x - 1), x) + Integral(a*x**6*sqrt( a*x/(a*x + 1) - 1/(a*x + 1))/(a**6*x**6 - 4*a**5*x**5 + 5*a**4*x**4 - 5*a* *2*x**2 + 4*a*x - 1), x))/c**5
Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {1}{120} \, a {\left (\frac {\frac {32 \, {\left (a x - 1\right )}}{a x + 1} + \frac {310 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {585 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{5}} - \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{5}} + \frac {15 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{5}}\right )} \]
1/120*a*((32*(a*x - 1)/(a*x + 1) + 310*(a*x - 1)^2/(a*x + 1)^2 - 585*(a*x - 1)^3/(a*x + 1)^3 + 3)/(a^2*c^5*((a*x - 1)/(a*x + 1))^(7/2) - a^2*c^5*((a *x - 1)/(a*x + 1))^(5/2)) + 240*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^ 5) - 240*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^5) + 15*sqrt((a*x - 1)/ (a*x + 1))/(a^2*c^5))
Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{8\,a\,c^5}-\frac {\frac {62\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {39\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {32\,\left (a\,x-1\right )}{15\,\left (a\,x+1\right )}+\frac {1}{5}}{8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}+\frac {4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^5} \]