Integrand size = 19, antiderivative size = 161 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^2} \, dx=\frac {4 a^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (21+23 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {17 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {17 a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 c^2} \]
4/3*a^3*(a*x+1)/c^2/(-a^2*x^2+1)^(3/2)-17/2*a^3*arctanh((-a^2*x^2+1)^(1/2) )/c^2+1/3*a^3*(23*a*x+21)/c^2/(-a^2*x^2+1)^(1/2)-1/3*(-a^2*x^2+1)^(1/2)/c^ 2/x^3-3/2*a*(-a^2*x^2+1)^(1/2)/c^2/x^2-17/3*a^2*(-a^2*x^2+1)^(1/2)/c^2/x
Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.69 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^2} \, dx=\frac {2+7 a x+23 a^2 x^2-91 a^3 x^3-29 a^4 x^4+80 a^5 x^5-51 a^3 x^3 (-1+a x) \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{6 c^2 x^3 (-1+a x) \sqrt {1-a^2 x^2}} \]
(2 + 7*a*x + 23*a^2*x^2 - 91*a^3*x^3 - 29*a^4*x^4 + 80*a^5*x^5 - 51*a^3*x^ 3*(-1 + a*x)*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(6*c^2*x^3*(-1 + a*x)*Sqrt[1 - a^2*x^2])
Time = 0.67 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.97, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {6678, 27, 570, 532, 25, 2336, 27, 2338, 25, 2338, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^2} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int \frac {\sqrt {1-a^2 x^2}}{c^3 x^4 (1-a x)^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {1-a^2 x^2}}{x^4 (1-a x)^3}dx}{c^2}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \frac {\int \frac {(a x+1)^3}{x^4 \left (1-a^2 x^2\right )^{5/2}}dx}{c^2}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {8 a^4 x^4+12 a^3 x^3+12 a^2 x^2+9 a x+3}{x^4 \left (1-a^2 x^2\right )^{3/2}}dx}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {8 a^4 x^4+12 a^3 x^3+12 a^2 x^2+9 a x+3}{x^4 \left (1-a^2 x^2\right )^{3/2}}dx+\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {a^3 (23 a x+21)}{\sqrt {1-a^2 x^2}}-\int -\frac {3 \left (7 a^3 x^3+5 a^2 x^2+3 a x+1\right )}{x^4 \sqrt {1-a^2 x^2}}dx\right )+\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \int \frac {7 a^3 x^3+5 a^2 x^2+3 a x+1}{x^4 \sqrt {1-a^2 x^2}}dx+\frac {a^3 (23 a x+21)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (-\frac {1}{3} \int -\frac {21 x^2 a^3+17 x a^2+9 a}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (23 a x+21)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{3} \int \frac {21 x^2 a^3+17 x a^2+9 a}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (23 a x+21)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{3} \left (-\frac {1}{2} \int -\frac {17 a^2 (3 a x+2)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {9 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (23 a x+21)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{3} \left (\frac {17}{2} a^2 \int \frac {3 a x+2}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {9 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (23 a x+21)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{3} \left (\frac {17}{2} a^2 \left (3 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {9 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (23 a x+21)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{3} \left (\frac {17}{2} a^2 \left (\frac {3}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {9 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (23 a x+21)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{3} \left (\frac {17}{2} a^2 \left (-\frac {3 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {9 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (23 a x+21)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{3} \left (\frac {17}{2} a^2 \left (-3 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {9 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (23 a x+21)}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a^3 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
((4*a^3*(1 + a*x))/(3*(1 - a^2*x^2)^(3/2)) + ((a^3*(21 + 23*a*x))/Sqrt[1 - a^2*x^2] + 3*(-1/3*Sqrt[1 - a^2*x^2]/x^3 + ((-9*a*Sqrt[1 - a^2*x^2])/(2*x ^2) + (17*a^2*((-2*Sqrt[1 - a^2*x^2])/x - 3*a*ArcTanh[Sqrt[1 - a^2*x^2]])) /2)/3))/3)/c^2
3.4.44.3.1 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 = 0.19 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.25
| method | result | size |
| risch | \(\frac {34 a^{4} x^{4}+9 a^{3} x^{3}-32 a^{2} x^{2}-9 a x -2}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {a^{3} \left (-17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\frac {4 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {4 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}}{a}-\frac {14 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a \left (x -\frac {1}{a}\right )}\right )}{2 c^{2}}\) | \(202\) |
| default | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {17 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}-7 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )+2 a^{2} \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )-\frac {7 a^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{x -\frac {1}{a}}}{c^{2}}\) | \(226\) |
1/6*(34*a^4*x^4+9*a^3*x^3-32*a^2*x^2-9*a*x-2)/x^3/(-a^2*x^2+1)^(1/2)/c^2+1 /2*a^3*(-17*arctanh(1/(-a^2*x^2+1)^(1/2))+4/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))-14 /a/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2))/c^2
Time = 0.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^2} \, dx=\frac {50 \, a^{5} x^{5} - 100 \, a^{4} x^{4} + 50 \, a^{3} x^{3} + 51 \, {\left (a^{5} x^{5} - 2 \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (80 \, a^{4} x^{4} - 109 \, a^{3} x^{3} + 18 \, a^{2} x^{2} + 5 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{2} c^{2} x^{5} - 2 \, a c^{2} x^{4} + c^{2} x^{3}\right )}} \]
1/6*(50*a^5*x^5 - 100*a^4*x^4 + 50*a^3*x^3 + 51*(a^5*x^5 - 2*a^4*x^4 + a^3 *x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (80*a^4*x^4 - 109*a^3*x^3 + 18*a^2 *x^2 + 5*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2*c^2*x^5 - 2*a*c^2*x^4 + c^2*x^3 )
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^2} \, dx=\frac {\int \frac {a x}{a^{2} x^{6} \sqrt {- a^{2} x^{2} + 1} - 2 a x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{4} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{2} x^{6} \sqrt {- a^{2} x^{2} + 1} - 2 a x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{4} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
(Integral(a*x/(a**2*x**6*sqrt(-a**2*x**2 + 1) - 2*a*x**5*sqrt(-a**2*x**2 + 1) + x**4*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**2*x**6*sqrt(-a**2*x* *2 + 1) - 2*a*x**5*sqrt(-a**2*x**2 + 1) + x**4*sqrt(-a**2*x**2 + 1)), x))/ c**2
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^2} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}^{2} x^{4}} \,d x } \]
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.31 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^2} \, dx=\frac {2 \, {\left (51 \, a^{4} \log \left (2\right ) - 102 \, a^{4} \log \left (i + 1\right ) + 160 i \, a^{4}\right )} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) + \frac {204 \, a^{4} \log \left (\sqrt {-\frac {2 \, c}{a c x - c} - 1} + 1\right )}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )} - \frac {204 \, a^{4} \log \left ({\left | \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 1 \right |}\right )}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )} - \frac {45 \, a^{4} {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 64 \, a^{4} {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} + 27 \, a^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1}}{{\left (\frac {c}{a c x - c} + 1\right )}^{3} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )} - \frac {8 \, {\left (a^{4} {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a c x - c}\right )^{2} \mathrm {sgn}\left (a\right )^{2} \mathrm {sgn}\left (c\right )^{2} + 24 \, a^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1} \mathrm {sgn}\left (\frac {1}{a c x - c}\right )^{2} \mathrm {sgn}\left (a\right )^{2} \mathrm {sgn}\left (c\right )^{2}\right )}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right )^{3} \mathrm {sgn}\left (a\right )^{3} \mathrm {sgn}\left (c\right )^{3}}}{24 \, c^{2} {\left | a \right |}} \]
1/24*(2*(51*a^4*log(2) - 102*a^4*log(I + 1) + 160*I*a^4)*sgn(1/(a*c*x - c) )*sgn(a)*sgn(c) + 204*a^4*log(sqrt(-2*c/(a*c*x - c) - 1) + 1)/(sgn(1/(a*c* x - c))*sgn(a)*sgn(c)) - 204*a^4*log(abs(sqrt(-2*c/(a*c*x - c) - 1) - 1))/ (sgn(1/(a*c*x - c))*sgn(a)*sgn(c)) - (45*a^4*(2*c/(a*c*x - c) + 1)^2*sqrt( -2*c/(a*c*x - c) - 1) - 64*a^4*(-2*c/(a*c*x - c) - 1)^(3/2) + 27*a^4*sqrt( -2*c/(a*c*x - c) - 1))/((c/(a*c*x - c) + 1)^3*sgn(1/(a*c*x - c))*sgn(a)*sg n(c)) - 8*(a^4*(-2*c/(a*c*x - c) - 1)^(3/2)*sgn(1/(a*c*x - c))^2*sgn(a)^2* sgn(c)^2 + 24*a^4*sqrt(-2*c/(a*c*x - c) - 1)*sgn(1/(a*c*x - c))^2*sgn(a)^2 *sgn(c)^2)/(sgn(1/(a*c*x - c))^3*sgn(a)^3*sgn(c)^3))/(c^2*abs(a))
Time = 3.53 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^2} \, dx=\frac {2\,a^5\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{3\,c^2\,x^3}-\frac {3\,a\,\sqrt {1-a^2\,x^2}}{2\,c^2\,x^2}-\frac {17\,a^2\,\sqrt {1-a^2\,x^2}}{3\,c^2\,x}+\frac {23\,a^4\,\sqrt {1-a^2\,x^2}}{3\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {a^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,17{}\mathrm {i}}{2\,c^2} \]
(a^3*atan((1 - a^2*x^2)^(1/2)*1i)*17i)/(2*c^2) + (2*a^5*(1 - a^2*x^2)^(1/2 ))/(3*(a^2*c^2 - 2*a^3*c^2*x + a^4*c^2*x^2)) - (1 - a^2*x^2)^(1/2)/(3*c^2* x^3) - (3*a*(1 - a^2*x^2)^(1/2))/(2*c^2*x^2) - (17*a^2*(1 - a^2*x^2)^(1/2) )/(3*c^2*x) + (23*a^4*(1 - a^2*x^2)^(1/2))/(3*(-a^2)^(1/2)*(c^2*x*(-a^2)^( 1/2) - (c^2*(-a^2)^(1/2))/a))