Integrand size = 24, antiderivative size = 213 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}-\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}+\frac {2 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 \arcsin (a x)}{(1-a x)^{3/2} (1+a x)^{3/2}}+\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{2 (1-a x)^{3/2} (1+a x)^{3/2}} \]
a*(c-c/a^2/x^2)^(3/2)*x^2/(-a*x+1)-5/2*a^2*(c-c/a^2/x^2)^(3/2)*x^3/(-a*x+1 )/(a*x+1)+1/2*(c-c/a^2/x^2)^(3/2)*x*(a*x+1)/(-a*x+1)+2*a^2*(c-c/a^2/x^2)^( 3/2)*x^3*arcsin(a*x)/(-a*x+1)^(3/2)/(a*x+1)^(3/2)+1/2*a^2*(c-c/a^2/x^2)^(3 /2)*x^3*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))/(-a*x+1)^(3/2)/(a*x+1)^(3/2)
Time = 0.17 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.54 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\frac {c \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (-1-4 a x+2 a^2 x^2\right )+a^2 x^2 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )+4 a^2 x^2 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{2 a^2 x \sqrt {-1+a^2 x^2}} \]
(c*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-1 - 4*a*x + 2*a^2*x^2) + a^ 2*x^2*ArcTan[1/Sqrt[-1 + a^2*x^2]] + 4*a^2*x^2*Log[a*x + Sqrt[-1 + a^2*x^2 ]]))/(2*a^2*x*Sqrt[-1 + a^2*x^2])
Time = 0.47 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.53, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6717, 6709, 540, 25, 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.
| \(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^{3/2} e^{2 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2}dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle -\frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \int \frac {(a x+1)^2 \sqrt {1-a^2 x^2}}{x^3}dx}{\left (1-a^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (-\frac {1}{2} \int -\frac {a (a x+4) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} \int \frac {a (a x+4) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} a \int \frac {(a x+4) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle -\frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} a \left (\int \frac {a-4 a^2 x}{x \sqrt {1-a^2 x^2}}dx-\frac {(4-a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} a \left (-4 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(4-a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} a \left (a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (4-a x)}{x}-4 a \arcsin (a x)\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} a \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (4-a x)}{x}-4 a \arcsin (a x)\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} a \left (-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (4-a x)}{x}-4 a \arcsin (a x)\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} a \left (-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (4-a x)}{x}-4 a \arcsin (a x)\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\) |
-(((c - c/(a^2*x^2))^(3/2)*x^3*(-1/2*(1 - a^2*x^2)^(3/2)/x^2 + (a*(-(((4 - a*x)*Sqrt[1 - a^2*x^2])/x) - 4*a*ArcSin[a*x] - a*ArcTanh[Sqrt[1 - a^2*x^2 ]]))/2))/(1 - a^2*x^2)^(3/2))
3.9.40.3.1 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[((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^(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 = 0.55 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(\frac {\left (2 a^{4} x^{4}-4 a^{3} x^{3}-3 a^{2} x^{2}+4 a x +1\right ) c \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 x \,a^{2} \left (a^{2} x^{2}-1\right )}+\frac {\left (\frac {2 a^{3} \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}+\frac {a^{2} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{2 \sqrt {-c}}\right ) c \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {c \left (a^{2} x^{2}-1\right )}}{a^{2} \left (a^{2} x^{2}-1\right )}\) | \(202\) |
| default | \(\frac {{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}} x \left (12 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{5} c \,x^{3}-12 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{5} x +4 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{4} c \,x^{2}-{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{4} c \,x^{2}+6 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} c^{2} x^{3}-3 a^{4} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}-18 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{3} c^{2} x^{3}+18 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {5}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{2}-6 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {5}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right ) a \,x^{2}+3 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{2} c^{2} x^{2}+3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{3} x^{2}\right )}{6 a^{2} \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} c}\) | \(455\) |
1/2*(2*a^4*x^4-4*a^3*x^3-3*a^2*x^2+4*a*x+1)/x*c/a^2*(c*(a^2*x^2-1)/a^2/x^2 )^(1/2)/(a^2*x^2-1)+(2*a^3*ln(a^2*c*x/(a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/( a^2*c)^(1/2)+1/2*a^2/(-c)^(1/2)*ln((-2*c+2*(-c)^(1/2)*(a^2*c*x^2-c)^(1/2)) /x))*c/a^2*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)*x*(c*(a^2*x^2-1))^(1/2)/(a^2*x^2- 1)
Time = 0.26 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.49 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\left [-\frac {8 \, a \sqrt {-c} c x \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - a \sqrt {-c} c x \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, a^{2} x}, \frac {a c^{\frac {3}{2}} x \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 2 \, a c^{\frac {3}{2}} x \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{2} x}\right ] \]
[-1/4*(8*a*sqrt(-c)*c*x*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2* x^2))/(a^2*c*x^2 - c)) - a*sqrt(-c)*c*x*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*s qrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(2*a^2*c*x^2 - 4*a*c*x - c) *sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x), 1/2*(a*c^(3/2)*x*arctan(a*sqrt( c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 2*a*c^(3/2)*x*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) + (2* a^2*c*x^2 - 4*a*c*x - c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x)]
Result contains complex when optimal does not.
Time = 6.57 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.77 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=c \left (\begin {cases} \frac {\sqrt {c} \sqrt {a^{2} x^{2} - 1}}{a} - \frac {i \sqrt {c} \log {\left (a x \right )}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} + \frac {\sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} - \frac {i \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )}}{a} & \text {otherwise} \end {cases}\right ) + \frac {2 c \left (\begin {cases} - \frac {a \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {\sqrt {c}}{a x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i a \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - i \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {i \sqrt {c}}{a x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a} + \frac {c \left (\begin {cases} \frac {i a \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {i \sqrt {c}}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {i \sqrt {c}}{2 a^{2} x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {a \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {\sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{2}} \]
c*Piecewise((sqrt(c)*sqrt(a**2*x**2 - 1)/a - I*sqrt(c)*log(a*x)/a + I*sqrt (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*sqrt (c)*log(sqrt(-a**2*x**2 + 1) + 1)/a, True)) + 2*c*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*sqrt(c)*a sin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a + c*Piecewise((I *a*sqrt(c)*acosh(1/(a*x))/2 + I*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - I *sqrt(c)/(2*a**2*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (- a*sqrt(c)*asin(1/(a*x))/2 - sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/ a**2
\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}{a x - 1} \,d x } \]
Time = 0.42 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.25 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=-{\left (\frac {c^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {2 \, c^{\frac {3}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a {\left | a \right |}} - \frac {\sqrt {a^{2} c x^{2} - c} c \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {{\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{2} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 4 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) - {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{3} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 4 \, a c^{\frac {7}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]
-(c^(3/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x)/a^ 2 + 2*c^(3/2)*log(abs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/(a*abs (a)) - sqrt(a^2*c*x^2 - c)*c*sgn(x)/a^2 - ((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*c^2*abs(a)*sgn(x) - 4*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*a*c ^(5/2)*sgn(x) - (sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))*c^3*abs(a)*sgn(x) - 4*a*c^(7/2)*sgn(x))/(((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^2*a^2*a bs(a)))*abs(a)
Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]