Integrand size = 24, antiderivative size = 293 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=-\frac {7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}+\frac {2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \arcsin (a x)}{(1-a x)^{5/2} (1+a x)^{5/2}}-\frac {9 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}} \]
-7/8*a^4*(c-c/a^2/x^2)^(5/2)*x^5/(-a*x+1)^2/(a*x+1)^2-1/6*a*(c-c/a^2/x^2)^ (5/2)*x^2/(a*x+1)+2*a^3*(c-c/a^2/x^2)^(5/2)*x^4/(-a*x+1)^2/(a*x+1)-7/24*a^ 2*(c-c/a^2/x^2)^(5/2)*x^3/(-a*x+1)/(a*x+1)+1/4*(c-c/a^2/x^2)^(5/2)*x*(-a*x +1)/(a*x+1)+2*a^4*(c-c/a^2/x^2)^(5/2)*x^5*arcsin(a*x)/(-a*x+1)^(5/2)/(a*x+ 1)^(5/2)-9/8*a^4*(c-c/a^2/x^2)^(5/2)*x^5*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1 /2))/(-a*x+1)^(5/2)/(a*x+1)^(5/2)
Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.46 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (6-16 a x-3 a^2 x^2+64 a^3 x^3+24 a^4 x^4\right )+27 a^4 x^4 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )-48 a^4 x^4 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{24 a^4 x^3 \sqrt {-1+a^2 x^2}} \]
(c^2*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(6 - 16*a*x - 3*a^2*x^2 + 6 4*a^3*x^3 + 24*a^4*x^4) + 27*a^4*x^4*ArcTan[1/Sqrt[-1 + a^2*x^2]] - 48*a^4 *x^4*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(24*a^4*x^3*Sqrt[-1 + a^2*x^2])
Time = 0.50 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.50, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6717, 6709, 570, 540, 27, 537, 25, 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.
| \(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^{5/2} e^{-2 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2}dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \int \frac {\left (1-a^2 x^2\right )^{7/2}}{x^5 (a x+1)^2}dx}{\left (1-a^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \int \frac {(1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{x^5}dx}{\left (1-a^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4} \int \frac {a (8-3 a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{\left (1-a^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4} a \int \frac {(8-3 a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{\left (1-a^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4} a \left (\frac {1}{2} a^2 \int -\frac {(16-9 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{\left (1-a^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \int \frac {(16-9 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{\left (1-a^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (\int \frac {-16 x a^2-9 a}{x \sqrt {1-a^2 x^2}}dx-\frac {(9 a x+16) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{\left (1-a^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (-16 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-9 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(9 a x+16) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{\left (1-a^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (-9 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (9 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{\left (1-a^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (-\frac {9}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (9 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{\left (1-a^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (\frac {9 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (9 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{\left (1-a^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (9 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (9 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{\left (1-a^2 x^2\right )^{5/2}}\) |
-(((c - c/(a^2*x^2))^(5/2)*x^5*(-1/4*(1 - a^2*x^2)^(5/2)/x^4 - (a*(-1/6*(( 16 - 9*a*x)*(1 - a^2*x^2)^(3/2))/x^3 - (a^2*(-(((16 + 9*a*x)*Sqrt[1 - a^2* x^2])/x) - 16*a*ArcSin[a*x] + 9*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/4))/(1 - a^2*x^2)^(5/2))
3.9.64.3.1 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[((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[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p) Int[u*((1 + a*x)^n/(x^ (2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && IntegerQ[n/2] && !GtQ[c, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.57 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.80
| method | result | size |
| risch | \(\frac {\left (64 a^{5} x^{5}-3 a^{4} x^{4}-80 a^{3} x^{3}+9 a^{2} x^{2}+16 a x -6\right ) c^{2} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{24 x^{3} a^{4} \left (a^{2} x^{2}-1\right )}+\frac {\left (-\frac {2 a^{5} \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}+\frac {9 a^{4} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{8 \sqrt {-c}}+\frac {a^{4} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c}\right ) c^{2} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {c \left (a^{2} x^{2}-1\right )}}{a^{4} \left (a^{2} x^{2}-1\right )}\) | \(234\) |
| default | \(-\frac {{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {5}{2}} x \left (-80 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{7} c \,x^{5}+80 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} a^{7} x^{3}-48 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{6} c \,x^{4}-27 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{6} c \,x^{4}+60 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{5} c^{2} x^{5}+75 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} a^{6} x^{2}+100 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{5} c^{2} x^{5}-80 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} a^{5} x +45 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{4} c^{2} x^{4}-90 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} c^{3} x^{5}-150 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c^{3} x^{5}+30 a^{4} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}+150 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {7}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{4}+90 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {7}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right ) a \,x^{4}-135 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c^{3} x^{4}-135 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{4} x^{4}\right )}{120 a^{2} \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c}\) | \(625\) |
1/24*(64*a^5*x^5-3*a^4*x^4-80*a^3*x^3+9*a^2*x^2+16*a*x-6)/x^3*c^2/a^4*(c*( a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1)+(-2*a^5*ln(a^2*c*x/(a^2*c)^(1/2)+(a^ 2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)+9/8*a^4/(-c)^(1/2)*ln((-2*c+2*(-c)^(1/2)*( a^2*c*x^2-c)^(1/2))/x)+a^4/c*(c*(a^2*x^2-1))^(1/2))*c^2/a^4*(c*(a^2*x^2-1) /a^2/x^2)^(1/2)/(a^2*x^2-1)*x*(c*(a^2*x^2-1))^(1/2)
Time = 0.27 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.34 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\left [\frac {96 \, a^{3} \sqrt {-c} c^{2} x^{3} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 27 \, a^{3} \sqrt {-c} c^{2} x^{3} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (24 \, a^{4} c^{2} x^{4} + 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, a^{4} x^{3}}, \frac {27 \, a^{3} c^{\frac {5}{2}} x^{3} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 24 \, a^{3} c^{\frac {5}{2}} x^{3} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (24 \, a^{4} c^{2} x^{4} + 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, a^{4} x^{3}}\right ] \]
[1/48*(96*a^3*sqrt(-c)*c^2*x^3*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c )/(a^2*x^2))/(a^2*c*x^2 - c)) + 27*a^3*sqrt(-c)*c^2*x^3*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) + 2*(24*a^4*c^2 *x^4 + 64*a^3*c^2*x^3 - 3*a^2*c^2*x^2 - 16*a*c^2*x + 6*c^2)*sqrt((a^2*c*x^ 2 - c)/(a^2*x^2)))/(a^4*x^3), 1/24*(27*a^3*c^(5/2)*x^3*arctan(a*sqrt(c)*x* sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 24*a^3*c^(5/2)*x^3*log( 2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) + (24 *a^4*c^2*x^4 + 64*a^3*c^2*x^3 - 3*a^2*c^2*x^2 - 16*a*c^2*x + 6*c^2)*sqrt(( a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3)]
Result contains complex when optimal does not.
Time = 9.97 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.71 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=c^{2} \left (\begin {cases} \frac {\sqrt {c} \sqrt {a^{2} x^{2} - 1}}{a} - \frac {i \sqrt {c} \log {\left (a x \right )}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} + \frac {\sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} - \frac {i \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )}}{a} & \text {otherwise} \end {cases}\right ) - \frac {2 c^{2} \left (\begin {cases} - \frac {a \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {\sqrt {c}}{a x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i a \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - i \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {i \sqrt {c}}{a x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a} + \frac {2 c^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {a^{2} \left (c - \frac {c}{a^{2} x^{2}}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right )}{a^{3}} - \frac {c^{2} \left (\begin {cases} \frac {i a^{3} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {i a^{2} \sqrt {c}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 i \sqrt {c}}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {i \sqrt {c}}{4 a^{2} x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {a^{3} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {a^{2} \sqrt {c}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 \sqrt {c}}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {\sqrt {c}}{4 a^{2} x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{4}} \]
c**2*Piecewise((sqrt(c)*sqrt(a**2*x**2 - 1)/a - I*sqrt(c)*log(a*x)/a + I*s qrt(c)*log(a**2*x**2)/(2*a) + sqrt(c)*asin(1/(a*x))/a, Abs(a**2*x**2) > 1) , (I*sqrt(c)*sqrt(-a**2*x**2 + 1)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) - I*s qrt(c)*log(sqrt(-a**2*x**2 + 1) + 1)/a, True)) - 2*c**2*Piecewise((-a*sqrt (c)*x/sqrt(a**2*x**2 - 1) + sqrt(c)*acosh(a*x) + sqrt(c)/(a*x*sqrt(a**2*x* *2 - 1)), Abs(a**2*x**2) > 1), (I*a*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - I*sqr t(c)*asin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a + 2*c**2*P iecewise((0, Eq(c, 0)), (a**2*(c - c/(a**2*x**2))**(3/2)/(3*c), True))/a** 3 - c**2*Piecewise((I*a**3*sqrt(c)*acosh(1/(a*x))/8 - I*a**2*sqrt(c)/(8*x* sqrt(-1 + 1/(a**2*x**2))) + 3*I*sqrt(c)/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(4*a**2*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a**3*sqrt(c)*asin(1/(a*x))/8 + a**2*sqrt(c)/(8*x*sqrt(1 - 1/(a**2*x**2) )) - 3*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + sqrt(c)/(4*a**2*x**5*sqr t(1 - 1/(a**2*x**2))), True))/a**4
\[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\int { \frac {{\left (a x - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{a x + 1} \,d x } \]
Time = 1.98 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.42 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=-\frac {1}{12} \, {\left (\frac {27 \, c^{\frac {5}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {24 \, c^{\frac {5}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a {\left | a \right |}} - \frac {12 \, \sqrt {a^{2} c x^{2} - c} c^{2} \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{3} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 96 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 192 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{5} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 160 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 64 \, a c^{\frac {13}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{4} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]
-1/12*(27*c^(5/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*s gn(x)/a^2 - 24*c^(5/2)*log(abs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn( x)/(a*abs(a)) - 12*sqrt(a^2*c*x^2 - c)*c^2*sgn(x)/a^2 - (3*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^7*c^3*abs(a)*sgn(x) + 96*(sqrt(a^2*c)*x - sqrt(a^2* c*x^2 - c))^6*a*c^(7/2)*sgn(x) - 21*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^ 5*c^4*abs(a)*sgn(x) + 192*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a*c^(9/2 )*sgn(x) + 21*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*c^5*abs(a)*sgn(x) + 160*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*a*c^(11/2)*sgn(x) - 3*(sqrt(a^ 2*c)*x - sqrt(a^2*c*x^2 - c))*c^6*abs(a)*sgn(x) + 64*a*c^(13/2)*sgn(x))/(( (sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^4*a^2*abs(a)))*abs(a)
Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]