Integrand size = 24, antiderivative size = 372 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=-\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}+\frac {57 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}-\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}-\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}+\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {2 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \arcsin (a x)}{(1-a x)^{7/2} (1+a x)^{7/2}}-\frac {25 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}} \]
-11/30*a^3*(c-c/a^2/x^2)^(7/2)*x^4/(-a*x+1)^3+57/16*a^6*(c-c/a^2/x^2)^(7/2 )*x^7/(-a*x+1)^3/(a*x+1)^3-41/24*a^5*(c-c/a^2/x^2)^(7/2)*x^6/(-a*x+1)^3/(a *x+1)^2-57/80*a^4*(c-c/a^2/x^2)^(7/2)*x^5/(-a*x+1)^3/(a*x+1)+13/40*a^2*(c- c/a^2/x^2)^(7/2)*x^3*(a*x+1)/(-a*x+1)^3-1/15*a*(c-c/a^2/x^2)^(7/2)*x^2*(a* x+1)/(-a*x+1)^2-1/6*(c-c/a^2/x^2)^(7/2)*x*(a*x+1)/(-a*x+1)-2*a^6*(c-c/a^2/ x^2)^(7/2)*x^7*arcsin(a*x)/(-a*x+1)^(7/2)/(a*x+1)^(7/2)-25/16*a^6*(c-c/a^2 /x^2)^(7/2)*x^7*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))/(-a*x+1)^(7/2)/(a*x+ 1)^(7/2)
Time = 0.16 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.40 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (-40-96 a x+70 a^2 x^2+352 a^3 x^3+105 a^4 x^4-736 a^5 x^5+240 a^6 x^6\right )+375 a^6 x^6 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )+480 a^6 x^6 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{240 a^6 x^5 \sqrt {-1+a^2 x^2}} \]
-1/240*(c^3*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-40 - 96*a*x + 70*a ^2*x^2 + 352*a^3*x^3 + 105*a^4*x^4 - 736*a^5*x^5 + 240*a^6*x^6) + 375*a^6* x^6*ArcTan[1/Sqrt[-1 + a^2*x^2]] + 480*a^6*x^6*Log[a*x + Sqrt[-1 + a^2*x^2 ]]))/(a^6*x^5*Sqrt[-1 + a^2*x^2])
Time = 0.46 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.49, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6709, 540, 25, 27, 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.
| \(\displaystyle \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \int \frac {(a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{x^7}dx}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6} \int -\frac {a (5 a x+12) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} \int \frac {a (5 a x+12) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} a \int \frac {(5 a x+12) \left (1-a^2 x^2\right )^{5/2}}{x^6}dx-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} a \left (\frac {1}{4} a^2 \int -\frac {(25 a x+48) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(25 a x+48) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} a \left (-\frac {1}{4} a^2 \int \frac {(25 a x+48) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx-\frac {(25 a x+48) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (\frac {1}{2} a^2 \int -\frac {3 (25 a x+32) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(25 a x+32) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(25 a x+48) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \int \frac {(25 a x+32) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(25 a x+32) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(25 a x+48) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\int \frac {25 a-32 a^2 x}{x \sqrt {1-a^2 x^2}}dx-\frac {(32-25 a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(25 a x+32) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(25 a x+48) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-32 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+25 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(32-25 a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(25 a x+32) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(25 a x+48) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (25 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (32-25 a x)}{x}-32 a \arcsin (a x)\right )-\frac {(25 a x+32) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(25 a x+48) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (\frac {25}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (32-25 a x)}{x}-32 a \arcsin (a x)\right )-\frac {(25 a x+32) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(25 a x+48) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-\frac {25 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (32-25 a x)}{x}-32 a \arcsin (a x)\right )-\frac {(25 a x+32) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(25 a x+48) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6} a \left (-\frac {1}{4} a^2 \left (-\frac {3}{2} a^2 \left (-25 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (32-25 a x)}{x}-32 a \arcsin (a x)\right )-\frac {(25 a x+32) \left (1-a^2 x^2\right )^{3/2}}{2 x^3}\right )-\frac {(25 a x+48) \left (1-a^2 x^2\right )^{5/2}}{20 x^5}\right )-\frac {\left (1-a^2 x^2\right )^{7/2}}{6 x^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}\) |
((c - c/(a^2*x^2))^(7/2)*x^7*(-1/6*(1 - a^2*x^2)^(7/2)/x^6 + (a*(-1/20*((4 8 + 25*a*x)*(1 - a^2*x^2)^(5/2))/x^5 - (a^2*(-1/2*((32 + 25*a*x)*(1 - a^2* x^2)^(3/2))/x^3 - (3*a^2*(-(((32 - 25*a*x)*Sqrt[1 - a^2*x^2])/x) - 32*a*Ar cSin[a*x] - 25*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/4))/6))/(1 - a^2*x^2)^(7 /2)
3.7.96.3.1 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^(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 = 0.20 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(\frac {\left (736 a^{7} x^{7}-105 a^{6} x^{6}-1088 a^{5} x^{5}+35 a^{4} x^{4}+448 a^{3} x^{3}+110 a^{2} x^{2}-96 a x -40\right ) c^{3} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{240 x^{5} a^{6} \left (a^{2} x^{2}-1\right )}-\frac {\left (\frac {2 a^{7} \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}+\frac {25 a^{6} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{16 \sqrt {-c}}+\frac {a^{6} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c}\right ) c^{3} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {c \left (a^{2} x^{2}-1\right )}}{a^{6} \left (a^{2} x^{2}-1\right )}\) | \(251\) |
| default | \(\frac {{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}} x \left (-2016 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{9} c \,x^{7}+2016 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{9} x^{5}+375 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{8} c \,x^{6}-480 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {7}{2}} a^{8} c \,x^{6}+105 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{8} x^{4}+2352 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{7} c^{2} x^{7}-560 \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^{7} c^{2} x^{7}+224 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{7} x^{3}-525 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{6} c^{2} x^{6}-2940 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{5} c^{3} x^{7}+700 \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^{3} x^{7}+630 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{6} x^{2}+875 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{4} c^{3} x^{6}+672 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{5} x +4410 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{3} c^{4} x^{7}-1050 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} c^{4} x^{7}+280 a^{4} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}-2625 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{2} c^{4} x^{6}-4410 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {9}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{6}+1050 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {9}{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^{6}-2625 \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^{5} x^{6}\right )}{1680 a^{2} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, c}\) | \(795\) |
1/240*(736*a^7*x^7-105*a^6*x^6-1088*a^5*x^5+35*a^4*x^4+448*a^3*x^3+110*a^2 *x^2-96*a*x-40)/x^5*c^3/a^6*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1)-(2*a ^7*ln(a^2*c*x/(a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)+25/16*a^6/( -c)^(1/2)*ln((-2*c+2*(-c)^(1/2)*(a^2*c*x^2-c)^(1/2))/x)+a^6/c*(c*(a^2*x^2- 1))^(1/2))*c^3/a^6*(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 = 0.29 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.18 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\left [\frac {960 \, a^{5} \sqrt {-c} c^{3} x^{5} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 375 \, a^{5} \sqrt {-c} c^{3} x^{5} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (240 \, a^{6} c^{3} x^{6} - 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} + 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} - 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{480 \, a^{6} x^{5}}, -\frac {375 \, a^{5} c^{\frac {7}{2}} x^{5} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 240 \, a^{5} c^{\frac {7}{2}} x^{5} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (240 \, a^{6} c^{3} x^{6} - 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} + 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} - 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{240 \, a^{6} x^{5}}\right ] \]
[1/480*(960*a^5*sqrt(-c)*c^3*x^5*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 375*a^5*sqrt(-c)*c^3*x^5*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*(240*a^6 *c^3*x^6 - 736*a^5*c^3*x^5 + 105*a^4*c^3*x^4 + 352*a^3*c^3*x^3 + 70*a^2*c^ 3*x^2 - 96*a*c^3*x - 40*c^3)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5), - 1/240*(375*a^5*c^(7/2)*x^5*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^ 2))/(a^2*c*x^2 - c)) - 240*a^5*c^(7/2)*x^5*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) + (240*a^6*c^3*x^6 - 736*a^5*c^3 *x^5 + 105*a^4*c^3*x^4 + 352*a^3*c^3*x^3 + 70*a^2*c^3*x^2 - 96*a*c^3*x - 4 0*c^3)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5)]
Result contains complex when optimal does not.
Time = 16.22 (sec) , antiderivative size = 1059, normalized size of antiderivative = 2.85 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\text {Too large to display} \]
-c**3*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**3*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 + c**3*Pi ecewise((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 + 4*c**3*Piecewise((0, Eq(c, 0)), (a**2*(c - c/(a**2*x**2)) **(3/2)/(3*c), True))/a**3 + c**3*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*s qrt(-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*sqrt(1 - 1/(a**2*x**2))), True))/a**4 - 2*c**3*Piece wise((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(-...
\[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{a^{2} x^{2} - 1} \,d x } \]
Time = 41.43 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.51 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {1}{120} \, {\left (\frac {375 \, c^{\frac {7}{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 {240 \, c^{\frac {7}{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 {120 \, \sqrt {a^{2} c x^{2} - c} c^{3} \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{11} c^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 1440 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{10} a c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} c^{5} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 4320 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{8} a c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 7360 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {13}{2}} \mathrm {sgn}\left (x\right ) + 150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{7} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 6720 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {15}{2}} \mathrm {sgn}\left (x\right ) - 595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{8} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 2976 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {17}{2}} \mathrm {sgn}\left (x\right ) - 105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{9} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 736 \, a c^{\frac {19}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{6} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]
1/120*(375*c^(7/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))* sgn(x)/a^2 + 240*c^(7/2)*log(abs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sg n(x)/(a*abs(a)) - 120*sqrt(a^2*c*x^2 - c)*c^3*sgn(x)/a^2 + (105*(sqrt(a^2* c)*x - sqrt(a^2*c*x^2 - c))^11*c^4*abs(a)*sgn(x) + 1440*(sqrt(a^2*c)*x - s qrt(a^2*c*x^2 - c))^10*a*c^(9/2)*sgn(x) + 595*(sqrt(a^2*c)*x - sqrt(a^2*c* x^2 - c))^9*c^5*abs(a)*sgn(x) + 4320*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c)) ^8*a*c^(11/2)*sgn(x) - 150*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^7*c^6*abs (a)*sgn(x) + 7360*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^6*a*c^(13/2)*sgn(x ) + 150*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^5*c^7*abs(a)*sgn(x) + 6720*( sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a*c^(15/2)*sgn(x) - 595*(sqrt(a^2*c )*x - sqrt(a^2*c*x^2 - c))^3*c^8*abs(a)*sgn(x) + 2976*(sqrt(a^2*c)*x - sqr t(a^2*c*x^2 - c))^2*a*c^(17/2)*sgn(x) - 105*(sqrt(a^2*c)*x - sqrt(a^2*c*x^ 2 - c))*c^9*abs(a)*sgn(x) + 736*a*c^(19/2)*sgn(x))/(((sqrt(a^2*c)*x - sqrt (a^2*c*x^2 - c))^2 + c)^6*a^2*abs(a)))*abs(a)
Timed out. \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\int -\frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]