Integrand size = 19, antiderivative size = 162 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^3} \, dx=\frac {8 a^2 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^2 (10+13 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^2 (135+164 a x)}{15 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {4 a \sqrt {1-a^2 x^2}}{c^3 x}-\frac {19 a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 c^3} \]
8/5*a^2*(a*x+1)/c^3/(-a^2*x^2+1)^(5/2)+4/15*a^2*(13*a*x+10)/c^3/(-a^2*x^2+ 1)^(3/2)-19/2*a^2*arctanh((-a^2*x^2+1)^(1/2))/c^3+1/15*a^2*(164*a*x+135)/c ^3/(-a^2*x^2+1)^(1/2)-1/2*(-a^2*x^2+1)^(1/2)/c^3/x^2-4*a*(-a^2*x^2+1)^(1/2 )/c^3/x
Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^3} \, dx=\frac {-15-90 a x+638 a^2 x^2-346 a^3 x^3-611 a^4 x^4+448 a^5 x^5-285 a^2 x^2 (-1+a x)^2 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{30 c^3 x^2 (-1+a x)^2 \sqrt {1-a^2 x^2}} \]
(-15 - 90*a*x + 638*a^2*x^2 - 346*a^3*x^3 - 611*a^4*x^4 + 448*a^5*x^5 - 28 5*a^2*x^2*(-1 + a*x)^2*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(30*c ^3*x^2*(-1 + a*x)^2*Sqrt[1 - a^2*x^2])
Time = 0.69 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.98, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.842, Rules used = {6678, 27, 570, 532, 25, 2336, 25, 2336, 27, 2338, 25, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^3} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int \frac {\sqrt {1-a^2 x^2}}{c^4 x^3 (1-a x)^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {1-a^2 x^2}}{x^3 (1-a x)^4}dx}{c^3}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \frac {\int \frac {(a x+1)^4}{x^3 \left (1-a^2 x^2\right )^{7/2}}dx}{c^3}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{5} \int -\frac {32 a^3 x^3+35 a^2 x^2+20 a x+5}{x^3 \left (1-a^2 x^2\right )^{5/2}}dx}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{5} \int \frac {32 a^3 x^3+35 a^2 x^2+20 a x+5}{x^3 \left (1-a^2 x^2\right )^{5/2}}dx+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {4 a^2 (13 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {104 a^3 x^3+120 a^2 x^2+60 a x+15}{x^3 \left (1-a^2 x^2\right )^{3/2}}dx\right )+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {104 a^3 x^3+120 a^2 x^2+60 a x+15}{x^3 \left (1-a^2 x^2\right )^{3/2}}dx+\frac {4 a^2 (13 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {a^2 (164 a x+135)}{\sqrt {1-a^2 x^2}}-\int -\frac {15 \left (9 a^2 x^2+4 a x+1\right )}{x^3 \sqrt {1-a^2 x^2}}dx\right )+\frac {4 a^2 (13 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (15 \int \frac {9 a^2 x^2+4 a x+1}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {a^2 (164 a x+135)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^2 (13 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (15 \left (-\frac {1}{2} \int -\frac {a (19 a x+8)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (164 a x+135)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^2 (13 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (15 \left (\frac {1}{2} \int \frac {a (19 a x+8)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (164 a x+135)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^2 (13 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (15 \left (\frac {1}{2} a \int \frac {19 a x+8}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (164 a x+135)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^2 (13 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (15 \left (\frac {1}{2} a \left (19 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (164 a x+135)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^2 (13 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (15 \left (\frac {1}{2} a \left (\frac {19}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (164 a x+135)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^2 (13 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (15 \left (\frac {1}{2} a \left (-\frac {19 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (164 a x+135)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^2 (13 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (15 \left (\frac {1}{2} a \left (-19 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {8 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (164 a x+135)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^2 (13 a x+10)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^2 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
((8*a^2*(1 + a*x))/(5*(1 - a^2*x^2)^(5/2)) + ((4*a^2*(10 + 13*a*x))/(3*(1 - a^2*x^2)^(3/2)) + ((a^2*(135 + 164*a*x))/Sqrt[1 - a^2*x^2] + 15*(-1/2*Sq rt[1 - a^2*x^2]/x^2 + (a*((-8*Sqrt[1 - a^2*x^2])/x - 19*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/3)/5)/c^3
3.4.52.3.1 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[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Time = 0.26 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(-\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}+\frac {19 a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {4 a \sqrt {-a^{2} x^{2}+1}}{x}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {29 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}+\frac {9 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{x -\frac {1}{a}}}{c^{3}}\) | \(223\) |
| risch | \(\frac {8 a^{3} x^{3}+a^{2} x^{2}-8 a x -1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}\, c^{3}}+\frac {a^{2} \left (-19 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {4 \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{a^{2}}+\frac {\frac {10 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {10 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}}{a}-\frac {18 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a \left (x -\frac {1}{a}\right )}\right )}{2 c^{3}}\) | \(323\) |
-1/c^3*(1/2*(-a^2*x^2+1)^(1/2)/x^2+19/2*a^2*arctanh(1/(-a^2*x^2+1)^(1/2))+ 4*a*(-a^2*x^2+1)^(1/2)/x+2/5/a/(x-1/a)^3*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2 )-29/5*a*(1/3/a/(x-1/a)^2*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)-1/3/(x-1/a)*( -(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2))+9*a/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a )^(1/2))
Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^3} \, dx=\frac {398 \, a^{5} x^{5} - 1194 \, a^{4} x^{4} + 1194 \, a^{3} x^{3} - 398 \, a^{2} x^{2} + 285 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (448 \, a^{4} x^{4} - 1059 \, a^{3} x^{3} + 713 \, a^{2} x^{2} - 75 \, a x - 15\right )} \sqrt {-a^{2} x^{2} + 1}}{30 \, {\left (a^{3} c^{3} x^{5} - 3 \, a^{2} c^{3} x^{4} + 3 \, a c^{3} x^{3} - c^{3} x^{2}\right )}} \]
1/30*(398*a^5*x^5 - 1194*a^4*x^4 + 1194*a^3*x^3 - 398*a^2*x^2 + 285*(a^5*x ^5 - 3*a^4*x^4 + 3*a^3*x^3 - a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (4 48*a^4*x^4 - 1059*a^3*x^3 + 713*a^2*x^2 - 75*a*x - 15)*sqrt(-a^2*x^2 + 1)) /(a^3*c^3*x^5 - 3*a^2*c^3*x^4 + 3*a*c^3*x^3 - c^3*x^2)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^3} \, dx=- \frac {\int \frac {a x}{a^{3} x^{6} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a x^{4} \sqrt {- a^{2} x^{2} + 1} - x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{3} x^{6} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a x^{4} \sqrt {- a^{2} x^{2} + 1} - x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
-(Integral(a*x/(a**3*x**6*sqrt(-a**2*x**2 + 1) - 3*a**2*x**5*sqrt(-a**2*x* *2 + 1) + 3*a*x**4*sqrt(-a**2*x**2 + 1) - x**3*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**3*x**6*sqrt(-a**2*x**2 + 1) - 3*a**2*x**5*sqrt(-a**2*x**2 + 1) + 3*a*x**4*sqrt(-a**2*x**2 + 1) - x**3*sqrt(-a**2*x**2 + 1)), x))/c** 3
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^3} \, dx=\int { -\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}^{3} x^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (140) = 280\).
Time = 0.30 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.09 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^3} \, dx=-\frac {19 \, a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, c^{3} {\left | a \right |}} - \frac {{\left (15 \, a^{3} + \frac {165 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} - \frac {4234 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}} + \frac {14330 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{3} x^{3}} - \frac {20965 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{5} x^{4}} + \frac {14385 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{7} x^{5}} - \frac {4080 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{9} x^{6}}\right )} a^{4} x^{2}}{120 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} - \frac {\frac {16 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{3} {\left | a \right |}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3} {\left | a \right |}}{a x^{2}}}{8 \, a^{2} c^{6}} \]
-19/2*a^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/(c ^3*abs(a)) - 1/120*(15*a^3 + 165*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a/x - 423 4*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a*x^2) + 14330*(sqrt(-a^2*x^2 + 1)*ab s(a) + a)^3/(a^3*x^3) - 20965*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^5*x^4) + 14385*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^7*x^5) - 4080*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/(a^9*x^6))*a^4*x^2/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c ^3*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^5*abs(a)) - 1/8*(16*(sqrt (-a^2*x^2 + 1)*abs(a) + a)*a*c^3*abs(a)/x + (sqrt(-a^2*x^2 + 1)*abs(a) + a )^2*c^3*abs(a)/(a*x^2))/(a^2*c^6)
Time = 3.53 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 (c-a c x)^3} \, dx=\frac {29\,a^4\,\sqrt {1-a^2\,x^2}}{15\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{2\,c^3\,x^2}-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{c^3\,x}+\frac {164\,a^3\,\sqrt {1-a^2\,x^2}}{15\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {2\,a^3\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,19{}\mathrm {i}}{2\,c^3} \]
(a^2*atan((1 - a^2*x^2)^(1/2)*1i)*19i)/(2*c^3) + (29*a^4*(1 - a^2*x^2)^(1/ 2))/(15*(a^2*c^3 - 2*a^3*c^3*x + a^4*c^3*x^2)) - (1 - a^2*x^2)^(1/2)/(2*c^ 3*x^2) - (4*a*(1 - a^2*x^2)^(1/2))/(c^3*x) + (164*a^3*(1 - a^2*x^2)^(1/2)) /(15*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a)) + (2*a^3*(1 - a^2*x^2)^(1/2))/(5*(-a^2)^(1/2)*(3*c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/ 2))/a + a^2*c^3*x^3*(-a^2)^(1/2) - 3*a*c^3*x^2*(-a^2)^(1/2)))