Integrand size = 22, antiderivative size = 140 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4 \sqrt {1-a^2 x^2}}{a}-\frac {c^4 \sqrt {1-a^2 x^2}}{3 a^4 x^3}+\frac {5 c^4 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {32 c^4 \sqrt {1-a^2 x^2}}{3 a^2 x}+\frac {5 c^4 \arcsin (a x)}{a}+\frac {25 c^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \]
5*c^4*arcsin(a*x)/a+25/2*c^4*arctanh((-a^2*x^2+1)^(1/2))/a+c^4*(-a^2*x^2+1 )^(1/2)/a-1/3*c^4*(-a^2*x^2+1)^(1/2)/a^4/x^3+5/2*c^4*(-a^2*x^2+1)^(1/2)/a^ 3/x^2-32/3*c^4*(-a^2*x^2+1)^(1/2)/a^2/x
Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.19 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4 \left (-2+15 a x-62 a^2 x^2-9 a^3 x^3+64 a^4 x^4-6 a^5 x^5+75 a^3 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)+90 a^3 x^3 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+75 a^3 x^3 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{6 a^4 x^3 \sqrt {1-a^2 x^2}} \]
(c^4*(-2 + 15*a*x - 62*a^2*x^2 - 9*a^3*x^3 + 64*a^4*x^4 - 6*a^5*x^5 + 75*a ^3*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + 90*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcSin [Sqrt[1 - a*x]/Sqrt[2]] + 75*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^ 2*x^2]]))/(6*a^4*x^3*Sqrt[1 - a^2*x^2])
Time = 0.74 (sec) , antiderivative size = 134, 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 = {6681, 6678, 540, 2338, 2338, 27, 2340, 27, 538, 223, 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.
| \(\displaystyle \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx\) |
| \(\Big \downarrow \) 6681 |
| \(\displaystyle \frac {c^4 \int \frac {e^{-\text {arctanh}(a x)} (1-a x)^4}{x^4}dx}{a^4}\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle \frac {c^4 \int \frac {(1-a x)^5}{x^4 \sqrt {1-a^2 x^2}}dx}{a^4}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle \frac {c^4 \left (-\frac {1}{3} \int \frac {3 x^4 a^5-15 x^3 a^4+30 x^2 a^3-32 x a^2+15 a}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {c^4 \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {-6 x^3 a^5+30 x^2 a^4-75 x a^3+64 a^2}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {15 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {c^4 \left (\frac {1}{3} \left (\frac {1}{2} \left (-\int \frac {3 \left (2 x^2 a^5-10 x a^4+25 a^3\right )}{x \sqrt {1-a^2 x^2}}dx-\frac {64 a^2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {15 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^4 \left (\frac {1}{3} \left (\frac {1}{2} \left (-3 \int \frac {2 x^2 a^5-10 x a^4+25 a^3}{x \sqrt {1-a^2 x^2}}dx-\frac {64 a^2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {15 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle \frac {c^4 \left (\frac {1}{3} \left (\frac {1}{2} \left (-3 \left (-\frac {\int -\frac {5 a^5 (5-2 a x)}{x \sqrt {1-a^2 x^2}}dx}{a^2}-2 a^3 \sqrt {1-a^2 x^2}\right )-\frac {64 a^2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {15 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^4 \left (\frac {1}{3} \left (\frac {1}{2} \left (-3 \left (5 a^3 \int \frac {5-2 a x}{x \sqrt {1-a^2 x^2}}dx-2 a^3 \sqrt {1-a^2 x^2}\right )-\frac {64 a^2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {15 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {c^4 \left (\frac {1}{3} \left (\frac {1}{2} \left (-3 \left (5 a^3 \left (5 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-2 a \int \frac {1}{\sqrt {1-a^2 x^2}}dx\right )-2 a^3 \sqrt {1-a^2 x^2}\right )-\frac {64 a^2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {15 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {c^4 \left (\frac {1}{3} \left (\frac {1}{2} \left (-3 \left (5 a^3 \left (5 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-2 \arcsin (a x)\right )-2 a^3 \sqrt {1-a^2 x^2}\right )-\frac {64 a^2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {15 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {c^4 \left (\frac {1}{3} \left (\frac {1}{2} \left (-3 \left (5 a^3 \left (\frac {5}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-2 \arcsin (a x)\right )-2 a^3 \sqrt {1-a^2 x^2}\right )-\frac {64 a^2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {15 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {c^4 \left (\frac {1}{3} \left (\frac {1}{2} \left (-3 \left (5 a^3 \left (-\frac {5 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}-2 \arcsin (a x)\right )-2 a^3 \sqrt {1-a^2 x^2}\right )-\frac {64 a^2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {15 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {c^4 \left (\frac {1}{3} \left (\frac {15 a \sqrt {1-a^2 x^2}}{2 x^2}+\frac {1}{2} \left (-\frac {64 a^2 \sqrt {1-a^2 x^2}}{x}-3 \left (5 a^3 \left (-5 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-2 \arcsin (a x)\right )-2 a^3 \sqrt {1-a^2 x^2}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )}{a^4}\) |
(c^4*(-1/3*Sqrt[1 - a^2*x^2]/x^3 + ((15*a*Sqrt[1 - a^2*x^2])/(2*x^2) + ((- 64*a^2*Sqrt[1 - a^2*x^2])/x - 3*(-2*a^3*Sqrt[1 - a^2*x^2] + 5*a^3*(-2*ArcS in[a*x] - 5*ArcTanh[Sqrt[1 - a^2*x^2]])))/2)/3))/a^4
3.5.85.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[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[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
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[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -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]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol ] :> Simp[d^p Int[u*(1 + c*(x/d))^p*(E^(n*ArcTanh[a*x])/x^p), x], x] /; F reeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(\frac {\left (64 a^{4} x^{4}-15 a^{3} x^{3}-62 a^{2} x^{2}+15 a x -2\right ) c^{4}}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{4}}+\frac {\left (\frac {5 a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {25 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\sqrt {-a^{2} x^{2}+1}\, a^{3}\right ) c^{4}}{a^{4}}\) | \(127\) |
| default | \(\frac {c^{4} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 x^{3}}-5 a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )-15 a^{3} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+11 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {\sqrt {-a^{2} x^{2}+1}\, x}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )+16 a^{3} \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )\right )}{a^{4}}\) | \(251\) |
1/6*(64*a^4*x^4-15*a^3*x^3-62*a^2*x^2+15*a*x-2)/x^3/(-a^2*x^2+1)^(1/2)*c^4 /a^4+(5*a^4/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+25/2*a^3* arctanh(1/(-a^2*x^2+1)^(1/2))+(-a^2*x^2+1)^(1/2)*a^3)*c^4/a^4
Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {60 \, a^{3} c^{4} x^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 75 \, a^{3} c^{4} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 6 \, a^{3} c^{4} x^{3} - {\left (6 \, a^{3} c^{4} x^{3} - 64 \, a^{2} c^{4} x^{2} + 15 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}} \]
-1/6*(60*a^3*c^4*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 75*a^3*c^4*x ^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 6*a^3*c^4*x^3 - (6*a^3*c^4*x^3 - 64*a ^2*c^4*x^2 + 15*a*c^4*x - 2*c^4)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)
\[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^{4} \left (\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx + \int \left (- \frac {4 a x \sqrt {- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\right )\, dx + \int \frac {6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx + \int \left (- \frac {4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\right )\, dx + \int \frac {a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx\right )}{a^{4}} \]
c**4*(Integral(sqrt(-a**2*x**2 + 1)/(a*x**5 + x**4), x) + Integral(-4*a*x* sqrt(-a**2*x**2 + 1)/(a*x**5 + x**4), x) + Integral(6*a**2*x**2*sqrt(-a**2 *x**2 + 1)/(a*x**5 + x**4), x) + Integral(-4*a**3*x**3*sqrt(-a**2*x**2 + 1 )/(a*x**5 + x**4), x) + Integral(a**4*x**4*sqrt(-a**2*x**2 + 1)/(a*x**5 + x**4), x))/a**4
\[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}^{4}}{a x + 1} \,d x } \]
4*a*c^4*(arcsin(a*x)/a^2 + log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x))/a^2 ) + c^4*(arcsin(a*x)/a + sqrt(-a^2*x^2 + 1)/a) + integrate((6*a^2*c^4*x^2 - 4*a*c^4*x + c^4)*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a^5*x^5 + a^4*x^4), x)
Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (122) = 244\).
Time = 0.30 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.87 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {{\left (c^{4} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} + \frac {129 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} + \frac {5 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {25 \, c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} - \frac {\frac {129 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{x} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{2} x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{4} x^{3}}}{24 \, a^{2} {\left | a \right |}} \]
1/24*(c^4 - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/(a^2*x) + 129*(sqrt(-a^ 2*x^2 + 1)*abs(a) + a)^2*c^4/(a^4*x^2))*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a ) + a)^3*abs(a)) + 5*c^4*arcsin(a*x)*sgn(a)/abs(a) + 25/2*c^4*log(1/2*abs( -2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + sqrt(-a^2*x^2 + 1)*c^4/a - 1/24*(129*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/x - 15*(sqrt(-a^ 2*x^2 + 1)*abs(a) + a)^2*c^4/(a^2*x^2) + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3 *c^4/(a^4*x^3))/(a^2*abs(a))
Time = 0.05 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.97 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {5\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{a}-\frac {32\,c^4\,\sqrt {1-a^2\,x^2}}{3\,a^2\,x}+\frac {5\,c^4\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{3\,a^4\,x^3}-\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,25{}\mathrm {i}}{2\,a} \]