Integrand size = 24, antiderivative size = 455 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\frac {11 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac {39 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{64 (1-a x)^4 (1+a x)^3}-\frac {11 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2}{28 (1+a x)}-\frac {103 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{160 (1-a x)^4 (1+a x)}+\frac {629 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{960 (1-a x)^3 (1+a x)}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4}{5 (1-a x)^2 (1+a x)}+\frac {47 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3}{336 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1-a x)}{8 (1+a x)}-\frac {2 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9 \arcsin (a x)}{(1-a x)^{9/2} (1+a x)^{9/2}}+\frac {245 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9 \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{128 (1-a x)^{9/2} (1+a x)^{9/2}} \]
11/128*a^8*(c-c/a^2/x^2)^(9/2)*x^9/(-a*x+1)^4/(a*x+1)^4+39/64*a^7*(c-c/a^2 /x^2)^(9/2)*x^8/(-a*x+1)^4/(a*x+1)^3-11/640*a^6*(c-c/a^2/x^2)^(9/2)*x^7/(- a*x+1)^4/(a*x+1)^2+1/28*a*(c-c/a^2/x^2)^(9/2)*x^2/(a*x+1)-103/160*a^5*(c-c /a^2/x^2)^(9/2)*x^6/(-a*x+1)^4/(a*x+1)+629/960*a^4*(c-c/a^2/x^2)^(9/2)*x^5 /(-a*x+1)^3/(a*x+1)-2/5*a^3*(c-c/a^2/x^2)^(9/2)*x^4/(-a*x+1)^2/(a*x+1)+47/ 336*a^2*(c-c/a^2/x^2)^(9/2)*x^3/(-a*x+1)/(a*x+1)-1/8*(c-c/a^2/x^2)^(9/2)*x *(-a*x+1)/(a*x+1)-2*a^8*(c-c/a^2/x^2)^(9/2)*x^9*arcsin(a*x)/(-a*x+1)^(9/2) /(a*x+1)^(9/2)+245/128*a^8*(c-c/a^2/x^2)^(9/2)*x^9*arctanh((-a*x+1)^(1/2)* (a*x+1)^(1/2))/(-a*x+1)^(9/2)/(a*x+1)^(9/2)
Time = 0.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.36 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=-\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (1680-3840 a x-4760 a^2 x^2+16896 a^3 x^3+770 a^4 x^4-31232 a^5 x^5+14595 a^6 x^6+45056 a^7 x^7+13440 a^8 x^8\right )+25725 a^8 x^8 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )-26880 a^8 x^8 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{13440 a^8 x^7 \sqrt {-1+a^2 x^2}} \]
-1/13440*(c^4*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(1680 - 3840*a*x - 4760*a^2*x^2 + 16896*a^3*x^3 + 770*a^4*x^4 - 31232*a^5*x^5 + 14595*a^6*x^ 6 + 45056*a^7*x^7 + 13440*a^8*x^8) + 25725*a^8*x^8*ArcTan[1/Sqrt[-1 + a^2* x^2]] - 26880*a^8*x^8*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(a^8*x^7*Sqrt[-1 + a ^2*x^2])
Time = 0.54 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.47, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6709, 570, 540, 27, 537, 25, 537, 25, 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.
| \(\displaystyle \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \int \frac {\left (1-a^2 x^2\right )^{11/2}}{x^9 (a x+1)^2}dx}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \int \frac {(1-a x)^2 \left (1-a^2 x^2\right )^{7/2}}{x^9}dx}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} \int \frac {a (16-7 a x) \left (1-a^2 x^2\right )^{7/2}}{x^8}dx-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \int \frac {(16-7 a x) \left (1-a^2 x^2\right )^{7/2}}{x^8}dx-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (\frac {1}{6} a^2 \int -\frac {(96-49 a x) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (-\frac {1}{6} a^2 \int \frac {(96-49 a x) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (-\frac {1}{6} a^2 \left (\frac {1}{4} a^2 \int -\frac {(384-245 a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(384-245 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \int \frac {(384-245 a x) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(384-245 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (\frac {1}{2} a^2 \int -\frac {3 (256-245 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(256-245 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(384-245 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \int \frac {(256-245 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(256-245 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(384-245 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\int \frac {-256 x a^2-245 a}{x \sqrt {1-a^2 x^2}}dx-\frac {(245 a x+256) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(256-245 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(384-245 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-256 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-245 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(245 a x+256) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(256-245 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(384-245 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-245 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (245 a x+256)}{x}-256 a \arcsin (a x)\right )-\frac {(256-245 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(384-245 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-\frac {245}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (245 a x+256)}{x}-256 a \arcsin (a x)\right )-\frac {(256-245 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(384-245 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\frac {245 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (245 a x+256)}{x}-256 a \arcsin (a x)\right )-\frac {(256-245 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(384-245 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (-\frac {1}{8} a \left (-\frac {1}{6} a^2 \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (245 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (245 a x+256)}{x}-256 a \arcsin (a x)\right )-\frac {(256-245 a x) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(384-245 a x) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {(96-49 a x) \left (1-a^2 x^2\right )^{7/2}}{42 x^7}\right )-\frac {\left (1-a^2 x^2\right )^{9/2}}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}}\) |
((c - c/(a^2*x^2))^(9/2)*x^9*(-1/8*(1 - a^2*x^2)^(9/2)/x^8 - (a*(-1/42*((9 6 - 49*a*x)*(1 - a^2*x^2)^(7/2))/x^7 - (a^2*(-1/20*((384 - 245*a*x)*(1 - a ^2*x^2)^(5/2))/x^5 - (a^2*(-1/2*((256 - 245*a*x)*(1 - a^2*x^2)^(3/2))/x^3 - (3*a^2*(-(((256 + 245*a*x)*Sqrt[1 - a^2*x^2])/x) - 256*a*ArcSin[a*x] + 2 45*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/4))/6))/8))/(1 - a^2*x^2)^(9/2)
3.8.23.3.1 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]
Time = 0.25 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.59
| method | result | size |
| risch | \(-\frac {\left (45056 a^{9} x^{9}+14595 a^{8} x^{8}-76288 a^{7} x^{7}-13825 a^{6} x^{6}+48128 a^{5} x^{5}-5530 a^{4} x^{4}-20736 a^{3} x^{3}+6440 a^{2} x^{2}+3840 a x -1680\right ) c^{4} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{13440 x^{7} a^{8} \left (a^{2} x^{2}-1\right )}-\frac {\left (-\frac {2 a^{9} \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}+\frac {245 a^{8} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{128 \sqrt {-c}}+\frac {a^{8} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c}\right ) c^{4} \sqrt {c \left (a^{2} x^{2}-1\right )}\, x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{a^{8} \left (a^{2} x^{2}-1\right )}\) | \(267\) |
| default | \(\text {Expression too large to display}\) | \(965\) |
-1/13440*(45056*a^9*x^9+14595*a^8*x^8-76288*a^7*x^7-13825*a^6*x^6+48128*a^ 5*x^5-5530*a^4*x^4-20736*a^3*x^3+6440*a^2*x^2+3840*a*x-1680)/x^7*c^4/a^8*( c*(a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1)-(-2*a^9*ln(a^2*c*x/(a^2*c)^(1/2)+ (a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)+245/128*a^8/(-c)^(1/2)*ln((-2*c+2*(-c)^ (1/2)*(a^2*c*x^2-c)^(1/2))/x)+a^8/c*(c*(a^2*x^2-1))^(1/2))*c^4/a^8*(c*(a^2 *x^2-1))^(1/2)*x*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1)
Time = 0.30 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.06 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\left [-\frac {53760 \, a^{7} \sqrt {-c} c^{4} x^{7} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 25725 \, a^{7} \sqrt {-c} c^{4} x^{7} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (13440 \, a^{8} c^{4} x^{8} + 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} - 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} + 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} - 3840 \, a c^{4} x + 1680 \, c^{4}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{26880 \, a^{8} x^{7}}, -\frac {25725 \, a^{7} c^{\frac {9}{2}} x^{7} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 13440 \, a^{7} c^{\frac {9}{2}} x^{7} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (13440 \, a^{8} c^{4} x^{8} + 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} - 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} + 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} - 3840 \, a c^{4} x + 1680 \, c^{4}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{13440 \, a^{8} x^{7}}\right ] \]
[-1/26880*(53760*a^7*sqrt(-c)*c^4*x^7*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c* x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 25725*a^7*sqrt(-c)*c^4*x^7*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*( 13440*a^8*c^4*x^8 + 45056*a^7*c^4*x^7 + 14595*a^6*c^4*x^6 - 31232*a^5*c^4* x^5 + 770*a^4*c^4*x^4 + 16896*a^3*c^4*x^3 - 4760*a^2*c^4*x^2 - 3840*a*c^4* x + 1680*c^4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7), -1/13440*(25725* a^7*c^(9/2)*x^7*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c* x^2 - c)) - 13440*a^7*c^(9/2)*x^7*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) + (13440*a^8*c^4*x^8 + 45056*a^7*c^4*x^7 + 14595*a^6*c^4*x^6 - 31232*a^5*c^4*x^5 + 770*a^4*c^4*x^4 + 16896*a^3*c^4* x^3 - 4760*a^2*c^4*x^2 - 3840*a*c^4*x + 1680*c^4)*sqrt((a^2*c*x^2 - c)/(a^ 2*x^2)))/(a^8*x^7)]
Result contains complex when optimal does not.
Time = 47.69 (sec) , antiderivative size = 1408, normalized size of antiderivative = 3.09 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\text {Too large to display} \]
-c**4*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**4*Piecewise((-a*sqr t(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*sq rt(c)*asin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a + 2*c**4* 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 - 6*c**4*Piecewise((0, Eq(c, 0)), (a**2*(c - c/(a**2*x**2 ))**(3/2)/(3*c), True))/a**3 + 6*c**4*Piecewise((2*a**3*sqrt(c)*sqrt(a**2* x**2 - 1)/(15*x) + a*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x**3) - sqrt(c)*sqrt( a**2*x**2 - 1)/(5*a*x**5), Abs(a**2*x**2) > 1), (2*I*a**3*sqrt(c)*sqrt(-a* *2*x**2 + 1)/(15*x) + I*a*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x**3) - I*sqrt( c)*sqrt(-a**2*x**2 + 1)/(5*a*x**5), True))/a**5 - 2*c**4*Piecewise((I*a**5 *sqrt(c)*acosh(1/(a*x))/16 - I*a**4*sqrt(c)/(16*x*sqrt(-1 + 1/(a**2*x**2)) ) + I*a**2*sqrt(c)/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) + 5*I*sqrt(c)/(24*x* *5*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(6*a**2*x**7*sqrt(-1 + 1/(a**2*x* *2))), 1/Abs(a**2*x**2) > 1), (-a**5*sqrt(c)*asin(1/(a*x))/16 + a**4*sq...
\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}}}{{\left (a x + 1\right )}^{2}} \,d x } \]
Time = 39.95 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.55 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\frac {1}{6720} \, {\left (\frac {25725 \, c^{\frac {9}{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 {13440 \, c^{\frac {9}{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 {6720 \, \sqrt {a^{2} c x^{2} - c} c^{4} \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {14595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{15} c^{5} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 107520 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{14} a c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + 76055 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{13} c^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 430080 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{12} a c^{\frac {13}{2}} \mathrm {sgn}\left (x\right ) + 64435 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{11} c^{7} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 1111040 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{10} a c^{\frac {15}{2}} \mathrm {sgn}\left (x\right ) + 110495 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} c^{8} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 1576960 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{8} a c^{\frac {17}{2}} \mathrm {sgn}\left (x\right ) - 110495 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{9} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 1412096 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {19}{2}} \mathrm {sgn}\left (x\right ) - 64435 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{10} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 831488 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {21}{2}} \mathrm {sgn}\left (x\right ) - 76055 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{11} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 252928 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {23}{2}} \mathrm {sgn}\left (x\right ) - 14595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{12} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 45056 \, a c^{\frac {25}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{8} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]
1/6720*(25725*c^(9/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c ))*sgn(x)/a^2 - 13440*c^(9/2)*log(abs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 - c) ))*sgn(x)/(a*abs(a)) - 6720*sqrt(a^2*c*x^2 - c)*c^4*sgn(x)/a^2 + (14595*(s qrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^15*c^5*abs(a)*sgn(x) - 107520*(sqrt(a^ 2*c)*x - sqrt(a^2*c*x^2 - c))^14*a*c^(11/2)*sgn(x) + 76055*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^13*c^6*abs(a)*sgn(x) - 430080*(sqrt(a^2*c)*x - sqrt (a^2*c*x^2 - c))^12*a*c^(13/2)*sgn(x) + 64435*(sqrt(a^2*c)*x - sqrt(a^2*c* x^2 - c))^11*c^7*abs(a)*sgn(x) - 1111040*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^10*a*c^(15/2)*sgn(x) + 110495*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^9 *c^8*abs(a)*sgn(x) - 1576960*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^8*a*c^( 17/2)*sgn(x) - 110495*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^7*c^9*abs(a)*s gn(x) - 1412096*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^6*a*c^(19/2)*sgn(x) - 64435*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^5*c^10*abs(a)*sgn(x) - 83148 8*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a*c^(21/2)*sgn(x) - 76055*(sqrt( a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*c^11*abs(a)*sgn(x) - 252928*(sqrt(a^2*c) *x - sqrt(a^2*c*x^2 - c))^2*a*c^(23/2)*sgn(x) - 14595*(sqrt(a^2*c)*x - sqr t(a^2*c*x^2 - c))*c^12*abs(a)*sgn(x) - 45056*a*c^(25/2)*sgn(x))/(((sqrt(a^ 2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^8*a^2*abs(a)))*abs(a)
Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=-\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{9/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]