Integrand size = 22, antiderivative size = 157 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {3 c^3 (8+a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8-a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {3 c^3 \arcsin (a x)}{a}+\frac {3 c^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{8 a} \]
1/8*c^3*(-a*x+8)*(-a^2*x^2+1)^(3/2)/a^4/x^3+1/5*c^3*(-a^2*x^2+1)^(5/2)/a^6 /x^5-3/4*c^3*(-a^2*x^2+1)^(5/2)/a^5/x^4-3*c^3*arcsin(a*x)/a+3/8*c^3*arctan h((-a^2*x^2+1)^(1/2))/a-3/8*c^3*(a*x+8)*(-a^2*x^2+1)^(1/2)/a^2/x
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.18 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3 \left (8-30 a x-24 a^2 x^2+105 a^3 x^3+24 a^4 x^4-75 a^5 x^5-8 a^6 x^6-45 a^5 x^5 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+40 a^2 x^2 \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},a^2 x^2\right )+8 a^5 x^5 \left (-1+a^2 x^2\right )^3 \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},1-a^2 x^2\right )\right )}{40 a^6 x^5 \sqrt {1-a^2 x^2}} \]
(c^3*(8 - 30*a*x - 24*a^2*x^2 + 105*a^3*x^3 + 24*a^4*x^4 - 75*a^5*x^5 - 8* a^6*x^6 - 45*a^5*x^5*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]] + 40*a^2 *x^2*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[-3/2, -3/2, -1/2, a^2*x^2] + 8*a^ 5*x^5*(-1 + a^2*x^2)^3*Hypergeometric2F1[2, 5/2, 7/2, 1 - a^2*x^2]))/(40*a ^6*x^5*Sqrt[1 - a^2*x^2])
Time = 0.59 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.91, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {6707, 6699, 540, 27, 2338, 27, 537, 27, 536, 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.
| \(\displaystyle \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx\) |
| \(\Big \downarrow \) 6707 |
| \(\displaystyle -\frac {c^3 \int \frac {e^{-3 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^3}{x^6}dx}{a^6}\) |
| \(\Big \downarrow \) 6699 |
| \(\displaystyle -\frac {c^3 \int \frac {(1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{x^6}dx}{a^6}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {c^3 \left (-\frac {1}{5} \int \frac {5 \left (1-a^2 x^2\right )^{3/2} \left (x^2 a^3-3 x a^2+3 a\right )}{x^5}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^3 \left (-\int \frac {\left (1-a^2 x^2\right )^{3/2} \left (x^2 a^3-3 x a^2+3 a\right )}{x^5}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}\right )}{a^6}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} \int \frac {a^2 (12-a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}+\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \int \frac {(12-a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}+\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (\frac {1}{2} a^2 \int -\frac {3 (8-a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(8-a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}+\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \int \frac {(8-a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(8-a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}+\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\int \frac {-8 x a^2-a}{x \sqrt {1-a^2 x^2}}dx-\frac {(a x+8) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(8-a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}+\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-8 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(a x+8) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(8-a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}+\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (a x+8)}{x}-8 a \arcsin (a x)\right )-\frac {(8-a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}+\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (a x+8)}{x}-8 a \arcsin (a x)\right )-\frac {(8-a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}+\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (a x+8)}{x}-8 a \arcsin (a x)\right )-\frac {(8-a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}+\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c^3 \left (\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (a x+8)}{x}-8 a \arcsin (a x)\right )-\frac {(8-a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 x^5}+\frac {3 a \left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^6}\) |
-((c^3*(-1/5*(1 - a^2*x^2)^(5/2)/x^5 + (3*a*(1 - a^2*x^2)^(5/2))/(4*x^4) + (a^2*(-1/2*((8 - a*x)*(1 - a^2*x^2)^(3/2))/x^3 - (3*a^2*(-(((8 + a*x)*Sqr t[1 - a^2*x^2])/x) - 8*a*ArcSin[a*x] + a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/ 4))/a^6)
3.7.79.3.1 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_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer Q[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), x] - Simp[2*b*(p/((m + 1)*(m + 2))) Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) *x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && GtQ[p, 0] && !ILtQ[m + 2*p + 3, 0] && IntegerQ[2*p]
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[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^p Int[x^m*((1 - a^2*x^2)^(p + n/2)/(1 - a*x)^n), x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c , 0]) && ILtQ[(n - 1)/2, 0] && !IntegerQ[p - n/2]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[d^p Int[(u/x^(2*p))*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x ] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
Time = 0.36 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(\frac {\left (152 a^{6} x^{6}-55 a^{5} x^{5}-176 a^{4} x^{4}+85 a^{3} x^{3}+16 a^{2} x^{2}-30 a x +8\right ) c^{3}}{40 x^{5} \sqrt {-a^{2} x^{2}+1}\, a^{6}}+\frac {\left (-\frac {3 a^{6} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {3 a^{5} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}-a^{5} \sqrt {-a^{2} x^{2}+1}\right ) c^{3}}{a^{6}}\) | \(144\) |
| default | \(\frac {c^{3} \left (a^{3} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )+\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{5 x^{5}}+3 a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{4 x^{4}}-\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )-3 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{3 x^{3}}-\frac {2 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-4 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x}{4}+\frac {3 \sqrt {-a^{2} x^{2}+1}\, x}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )\right )}{3}\right )\right )}{a^{6}}\) | \(295\) |
1/40*(152*a^6*x^6-55*a^5*x^5-176*a^4*x^4+85*a^3*x^3+16*a^2*x^2-30*a*x+8)/x ^5/(-a^2*x^2+1)^(1/2)*c^3/a^6+(-3*a^6/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a ^2*x^2+1)^(1/2))+3/8*a^5*arctanh(1/(-a^2*x^2+1)^(1/2))-a^5*(-a^2*x^2+1)^(1 /2))*c^3/a^6
Time = 0.26 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.98 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {240 \, a^{5} c^{3} x^{5} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 15 \, a^{5} c^{3} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 40 \, a^{5} c^{3} x^{5} - {\left (40 \, a^{5} c^{3} x^{5} + 152 \, a^{4} c^{3} x^{4} - 55 \, a^{3} c^{3} x^{3} - 24 \, a^{2} c^{3} x^{2} + 30 \, a c^{3} x - 8 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{40 \, a^{6} x^{5}} \]
1/40*(240*a^5*c^3*x^5*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 15*a^5*c^3* x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 40*a^5*c^3*x^5 - (40*a^5*c^3*x^5 + 1 52*a^4*c^3*x^4 - 55*a^3*c^3*x^3 - 24*a^2*c^3*x^2 + 30*a*c^3*x - 8*c^3)*sqr t(-a^2*x^2 + 1))/(a^6*x^5)
Result contains complex when optimal does not.
Time = 7.60 (sec) , antiderivative size = 695, normalized size of antiderivative = 4.43 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=- \frac {c^{3} \left (\begin {cases} i \sqrt {a^{2} x^{2} - 1} - \log {\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} + i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} - \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right )}{a} + \frac {3 c^{3} \left (\begin {cases} - \frac {i a^{2} x}{\sqrt {a^{2} x^{2} - 1}} + i a \operatorname {acosh}{\left (a x \right )} + \frac {i}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} x}{\sqrt {- a^{2} x^{2} + 1}} - a \operatorname {asin}{\left (a x \right )} - \frac {1}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a^{2}} - \frac {2 c^{3} \left (\begin {cases} \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{3}} - \frac {2 c^{3} \left (\begin {cases} \frac {a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right )}{a^{4}} + \frac {3 c^{3} \left (\begin {cases} \frac {a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{5}} - \frac {c^{3} \left (\begin {cases} \frac {2 i a^{4} \sqrt {a^{2} x^{2} - 1}}{15 x} + \frac {i a^{2} \sqrt {a^{2} x^{2} - 1}}{15 x^{3}} - \frac {i \sqrt {a^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {2 a^{4} \sqrt {- a^{2} x^{2} + 1}}{15 x} + \frac {a^{2} \sqrt {- a^{2} x^{2} + 1}}{15 x^{3}} - \frac {\sqrt {- a^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right )}{a^{6}} \]
-c**3*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*a sin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/ 2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a + 3*c**3*Piecewise((-I*a**2*x/ sqrt(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2 *x**2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2* x**2 + 1)), True))/a**2 - 2*c**3*Piecewise((a**2*acosh(1/(a*x))/2 + a/(2*x *sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs( a**2*x**2) > 1), (-I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2 *x), True))/a**3 - 2*c**3*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 - a*s qrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))/a**4 + 3*c **3*Piecewise((a**4*acosh(1/(a*x))/8 - a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x **2))), 1/Abs(a**2*x**2) > 1), (-I*a**4*asin(1/(a*x))/8 + I*a**3/(8*x*sqrt (1 - 1/(a**2*x**2))) - 3*I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x** 5*sqrt(1 - 1/(a**2*x**2))), True))/a**5 - c**3*Piecewise((2*I*a**4*sqrt(a* *2*x**2 - 1)/(15*x) + I*a**2*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(a**2*x **2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(-a**2*x**2 + 1)/(15*x ) + a**2*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(-a**2*x**2 + 1)/(5*x**5), T rue))/a**6
\[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{3}}{{\left (a x + 1\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (136) = 272\).
Time = 0.30 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.46 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {{\left (2 \, c^{3} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{a^{2} x} + \frac {30 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{a^{4} x^{2}} + \frac {80 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{6} x^{3}} - \frac {580 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{8} x^{4}}\right )} a^{10} x^{5}}{320 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} {\left | a \right |}} - \frac {3 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {3 \, c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} - \frac {\frac {580 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c^{3}}{x} - \frac {80 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}} - \frac {30 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{2} x^{3}} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{4} x^{4}} - \frac {2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{3}}{a^{6} x^{5}}}{320 \, a^{4} {\left | a \right |}} \]
-1/320*(2*c^3 - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^3/(a^2*x) + 30*(sqrt( -a^2*x^2 + 1)*abs(a) + a)^2*c^3/(a^4*x^2) + 80*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/(a^6*x^3) - 580*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^3/(a^8*x^4) )*a^10*x^5/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*abs(a)) - 3*c^3*arcsin(a*x)* sgn(a)/abs(a) + 3/8*c^3*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^3/a - 1/320*(580*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2*c^3/x - 80*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/x^2 - 30*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/(a^2*x^3) + 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^3/(a^4*x^4) - 2*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^3/ (a^6*x^5))/(a^4*abs(a))
Time = 3.42 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.17 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {11\,c^3\,\sqrt {1-a^2\,x^2}}{8\,a^3\,x^2}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{a}-\frac {19\,c^3\,\sqrt {1-a^2\,x^2}}{5\,a^2\,x}-\frac {3\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {3\,c^3\,\sqrt {1-a^2\,x^2}}{5\,a^4\,x^3}-\frac {3\,c^3\,\sqrt {1-a^2\,x^2}}{4\,a^5\,x^4}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{5\,a^6\,x^5}-\frac {c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,a} \]
(11*c^3*(1 - a^2*x^2)^(1/2))/(8*a^3*x^2) - (c^3*atan((1 - a^2*x^2)^(1/2)*1 i)*3i)/(8*a) - (c^3*(1 - a^2*x^2)^(1/2))/a - (19*c^3*(1 - a^2*x^2)^(1/2))/ (5*a^2*x) - (3*c^3*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) + (3*c^3*(1 - a^2*x ^2)^(1/2))/(5*a^4*x^3) - (3*c^3*(1 - a^2*x^2)^(1/2))/(4*a^5*x^4) + (c^3*(1 - a^2*x^2)^(1/2))/(5*a^6*x^5)