Integrand size = 22, antiderivative size = 111 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {16 c^2 (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c^2 \sqrt {1-a^2 x^2}}{a}-\frac {c^2 \sqrt {1-a^2 x^2}}{a^2 x}-\frac {5 c^2 \arcsin (a x)}{a}+\frac {5 c^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{a} \]
-5*c^2*arcsin(a*x)/a+5*c^2*arctanh((-a^2*x^2+1)^(1/2))/a-16*c^2*(-a*x+1)/a /(-a^2*x^2+1)^(1/2)-c^2*(-a^2*x^2+1)^(1/2)/a-c^2*(-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 = 0.33 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.89 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2 \left (35 \left (-\sqrt {1+a x}+17 a x \sqrt {1+a x}-8 a^2 x^2 \sqrt {1+a x}-9 a^3 x^3 \sqrt {1+a x}+a^4 x^4 \sqrt {1+a x}+3 a x \sqrt {1-a x} (1+a x) \arcsin (a x)-26 a x \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-26 a^2 x^2 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+5 a x \sqrt {1+a x} \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )+7 \sqrt {2} a x (-1+a x)^3 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},\frac {1}{2} (1-a x)\right )-5 \sqrt {2} a x (-1+a x)^4 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{2},\frac {9}{2},\frac {1}{2} (1-a x)\right )\right )}{35 a^2 x \sqrt {1-a x} (1+a x)} \]
(c^2*(35*(-Sqrt[1 + a*x] + 17*a*x*Sqrt[1 + a*x] - 8*a^2*x^2*Sqrt[1 + a*x] - 9*a^3*x^3*Sqrt[1 + a*x] + a^4*x^4*Sqrt[1 + a*x] + 3*a*x*Sqrt[1 - a*x]*(1 + a*x)*ArcSin[a*x] - 26*a*x*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] - 26*a^2*x^2*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] + 5*a*x*Sqrt[1 + a *x]*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]) + 7*Sqrt[2]*a*x*(-1 + a* x)^3*(1 + a*x)*Hypergeometric2F1[3/2, 5/2, 7/2, (1 - a*x)/2] - 5*Sqrt[2]*a *x*(-1 + a*x)^4*(1 + a*x)*Hypergeometric2F1[3/2, 7/2, 9/2, (1 - a*x)/2]))/ (35*a^2*x*Sqrt[1 - a*x]*(1 + a*x))
Time = 0.58 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.83, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6681, 6678, 528, 2338, 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.
| \(\displaystyle \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx\) |
| \(\Big \downarrow \) 6681 |
| \(\displaystyle \frac {c^2 \int \frac {e^{-3 \text {arctanh}(a x)} (1-a x)^2}{x^2}dx}{a^2}\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle \frac {c^2 \int \frac {(1-a x)^5}{x^2 \left (1-a^2 x^2\right )^{3/2}}dx}{a^2}\) |
| \(\Big \downarrow \) 528 |
| \(\displaystyle \frac {c^2 \left (\int \frac {a^3 x^3-5 a^2 x^2-5 a x+1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {16 a (1-a x)}{\sqrt {1-a^2 x^2}}\right )}{a^2}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {c^2 \left (-\int \frac {-x^2 a^3+5 x a^2+5 a}{x \sqrt {1-a^2 x^2}}dx-\frac {16 a (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{a^2}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle \frac {c^2 \left (\frac {\int -\frac {5 a^3 (a x+1)}{x \sqrt {1-a^2 x^2}}dx}{a^2}+a \left (-\sqrt {1-a^2 x^2}\right )-\frac {16 a (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^2 \left (-5 a \int \frac {a x+1}{x \sqrt {1-a^2 x^2}}dx-a \sqrt {1-a^2 x^2}-\frac {16 a (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{a^2}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {c^2 \left (-5 a \left (a \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\int \frac {1}{x \sqrt {1-a^2 x^2}}dx\right )-a \sqrt {1-a^2 x^2}-\frac {16 a (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{a^2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {c^2 \left (-5 a \left (\int \frac {1}{x \sqrt {1-a^2 x^2}}dx+\arcsin (a x)\right )-a \sqrt {1-a^2 x^2}-\frac {16 a (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{a^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {c^2 \left (-5 a \left (\frac {1}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2+\arcsin (a x)\right )-a \sqrt {1-a^2 x^2}-\frac {16 a (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{a^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {c^2 \left (-5 a \left (\arcsin (a x)-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}\right )-a \sqrt {1-a^2 x^2}-\frac {16 a (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{a^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {c^2 \left (-5 a \left (\arcsin (a x)-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-a \sqrt {1-a^2 x^2}-\frac {16 a (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{a^2}\) |
(c^2*((-16*a*(1 - a*x))/Sqrt[1 - a^2*x^2] - a*Sqrt[1 - a^2*x^2] - Sqrt[1 - a^2*x^2]/x - 5*a*(ArcSin[a*x] - ArcTanh[Sqrt[1 - a^2*x^2]])))/a^2
3.6.3.3.1 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[((x_)^(m_)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_Sy mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b *x^2])), x] + Simp[c^2/a Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) ^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
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[(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 = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.21
| method | result | size |
| risch | \(\frac {\left (a^{2} x^{2}-1\right ) c^{2}}{x \sqrt {-a^{2} x^{2}+1}\, a^{2}}-\frac {\left (\frac {5 a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\sqrt {-a^{2} x^{2}+1}\, a -5 a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {16 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}\right ) c^{2}}{a^{2}}\) | \(134\) |
| default | \(\frac {c^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-4 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x}{4}+\frac {3 \sqrt {-a^{2} x^{2}+1}\, x}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )-5 a \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+17 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )+\frac {4 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+\frac {-\frac {4 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-8 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a}\right )}{a^{2}}\) | \(472\) |
(a^2*x^2-1)/x/(-a^2*x^2+1)^(1/2)*c^2/a^2-(5*a^2/(a^2)^(1/2)*arctan((a^2)^( 1/2)*x/(-a^2*x^2+1)^(1/2))+(-a^2*x^2+1)^(1/2)*a-5*a*arctanh(1/(-a^2*x^2+1) ^(1/2))+16/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))*c^2/a^2
Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.34 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {17 \, a^{2} c^{2} x^{2} + 17 \, a c^{2} x - 10 \, {\left (a^{2} c^{2} x^{2} + a c^{2} x\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 5 \, {\left (a^{2} c^{2} x^{2} + a c^{2} x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (a^{2} c^{2} x^{2} + 18 \, a c^{2} x + c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{a^{3} x^{2} + a^{2} x} \]
-(17*a^2*c^2*x^2 + 17*a*c^2*x - 10*(a^2*c^2*x^2 + a*c^2*x)*arctan((sqrt(-a ^2*x^2 + 1) - 1)/(a*x)) + 5*(a^2*c^2*x^2 + a*c^2*x)*log((sqrt(-a^2*x^2 + 1 ) - 1)/x) + (a^2*c^2*x^2 + 18*a*c^2*x + c^2)*sqrt(-a^2*x^2 + 1))/(a^3*x^2 + a^2*x)
\[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^{2} \left (\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\, dx + \int \left (- \frac {2 a x \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\right )\, dx + \int \frac {2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\, dx + \int \left (- \frac {a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\right )\, dx\right )}{a^{2}} \]
c**2*(Integral(sqrt(-a**2*x**2 + 1)/(a**3*x**5 + 3*a**2*x**4 + 3*a*x**3 + x**2), x) + Integral(-2*a*x*sqrt(-a**2*x**2 + 1)/(a**3*x**5 + 3*a**2*x**4 + 3*a*x**3 + x**2), x) + Integral(2*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x **5 + 3*a**2*x**4 + 3*a*x**3 + x**2), x) + Integral(-a**4*x**4*sqrt(-a**2* x**2 + 1)/(a**3*x**5 + 3*a**2*x**4 + 3*a*x**3 + x**2), x))/a**2
\[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{2}}{{\left (a x + 1\right )}^{3}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.77 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {5 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {5 \, c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} + \frac {{\left (c^{2} + \frac {65 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{a^{2} x}\right )} a^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{2 \, a^{2} x {\left | a \right |}} \]
-5*c^2*arcsin(a*x)*sgn(a)/abs(a) + 5*c^2*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^2/a + 1/2*(c^2 + 65*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^2/(a^2*x))*a^2*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a)) - 1/2 *(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^2/(a^2*x*abs(a))
Time = 3.46 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.24 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {16\,c^2\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{a}-\frac {5\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{a^2\,x}-\frac {c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{a} \]