Integrand size = 24, antiderivative size = 189 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {51 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {19 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {3 c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}-\left (c-\frac {c}{a x}\right )^{9/2} x+\frac {13 c^{9/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {64 \sqrt {2} c^{9/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
19/3*c^3*(c-c/a/x)^(3/2)/a+3/5*c^2*(c-c/a/x)^(5/2)/a-5/7*c*(c-c/a/x)^(7/2) /a-(c-c/a/x)^(9/2)*x+13*c^(9/2)*arctanh((c-c/a/x)^(1/2)/c^(1/2))/a-64*c^(9 /2)*arctanh(1/2*(c-c/a/x)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)/a+51*c^4*(c-c/a/x )^(1/2)/a
Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.70 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (-30+258 a x-1196 a^2 x^2+6428 a^3 x^3-105 a^4 x^4\right )}{105 a^4 x^3}+\frac {13 c^{9/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {64 \sqrt {2} c^{9/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
(c^4*Sqrt[c - c/(a*x)]*(-30 + 258*a*x - 1196*a^2*x^2 + 6428*a^3*x^3 - 105* a^4*x^4))/(105*a^4*x^3) + (13*c^(9/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/ a - (64*Sqrt[2]*c^(9/2)*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a
Time = 0.48 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {6683, 1035, 281, 899, 109, 27, 171, 27, 171, 27, 171, 27, 171, 27, 174, 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.
| \(\displaystyle \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle \int \frac {(1-a x) \left (c-\frac {c}{a x}\right )^{9/2}}{a x+1}dx\) |
| \(\Big \downarrow \) 1035 |
| \(\displaystyle \int \frac {\left (\frac {1}{x}-a\right ) \left (c-\frac {c}{a x}\right )^{9/2}}{a+\frac {1}{x}}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle -\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{11/2}}{a+\frac {1}{x}}dx}{c}\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{11/2} x^2}{a+\frac {1}{x}}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {a \left (-\frac {\int \frac {c^2 \left (13 a+\frac {5}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2} x}{2 a \left (a+\frac {1}{x}\right )}d\frac {1}{x}}{a}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \int \frac {\left (13 a+\frac {5}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2} x}{a+\frac {1}{x}}d\frac {1}{x}}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (\frac {2}{7} \int \frac {7 c \left (13 a-\frac {3}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2} x}{2 \left (a+\frac {1}{x}\right )}d\frac {1}{x}+\frac {10}{7} \left (c-\frac {c}{a x}\right )^{7/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (c \int \frac {\left (13 a-\frac {3}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2} x}{a+\frac {1}{x}}d\frac {1}{x}+\frac {10}{7} \left (c-\frac {c}{a x}\right )^{7/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (\frac {2}{5} \int \frac {5 c \left (13 a-\frac {19}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2} x}{2 \left (a+\frac {1}{x}\right )}d\frac {1}{x}-\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )+\frac {10}{7} \left (c-\frac {c}{a x}\right )^{7/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \int \frac {\left (13 a-\frac {19}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2} x}{a+\frac {1}{x}}d\frac {1}{x}-\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )+\frac {10}{7} \left (c-\frac {c}{a x}\right )^{7/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (\frac {2}{3} \int \frac {3 c \left (13 a-\frac {51}{x}\right ) \sqrt {c-\frac {c}{a x}} x}{2 \left (a+\frac {1}{x}\right )}d\frac {1}{x}-\frac {38}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )-\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )+\frac {10}{7} \left (c-\frac {c}{a x}\right )^{7/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (c \int \frac {\left (13 a-\frac {51}{x}\right ) \sqrt {c-\frac {c}{a x}} x}{a+\frac {1}{x}}d\frac {1}{x}-\frac {38}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )-\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )+\frac {10}{7} \left (c-\frac {c}{a x}\right )^{7/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (c \left (2 \int \frac {c \left (13 a-\frac {115}{x}\right ) x}{2 \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-102 \sqrt {c-\frac {c}{a x}}\right )-\frac {38}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )-\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )+\frac {10}{7} \left (c-\frac {c}{a x}\right )^{7/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (c \left (c \int \frac {\left (13 a-\frac {115}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-102 \sqrt {c-\frac {c}{a x}}\right )-\frac {38}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )-\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )+\frac {10}{7} \left (c-\frac {c}{a x}\right )^{7/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (c \left (c \left (13 \int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-128 \int \frac {1}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}\right )-102 \sqrt {c-\frac {c}{a x}}\right )-\frac {38}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )-\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )+\frac {10}{7} \left (c-\frac {c}{a x}\right )^{7/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (c \left (c \left (\frac {256 a \int \frac {1}{2 a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {26 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}\right )-102 \sqrt {c-\frac {c}{a x}}\right )-\frac {38}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )-\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )+\frac {10}{7} \left (c-\frac {c}{a x}\right )^{7/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (c \left (c \left (\frac {128 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {26 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-102 \sqrt {c-\frac {c}{a x}}\right )-\frac {38}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )-\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )+\frac {10}{7} \left (c-\frac {c}{a x}\right )^{7/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{9/2}}{a}\right )}{c}\) |
(a*(-((c*(c - c/(a*x))^(9/2)*x)/a) - (c^2*((10*(c - c/(a*x))^(7/2))/7 + c* ((-6*(c - c/(a*x))^(5/2))/5 + c*((-38*(c - c/(a*x))^(3/2))/3 + c*(-102*Sqr t[c - c/(a*x)] + c*((-26*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/Sqrt[c] + (12 8*Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/Sqrt[c]))))))/(2*a ^2)))/c
3.6.44.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.)*((e_) + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(n*(p + r))*(b + a/x^n)^p*(c + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[ mn, -n] && IntegerQ[p] && IntegerQ[r]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Int[u*(c + d/x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[n/2] && !G tQ[c, 0]
Time = 0.14 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.23
| method | result | size |
| risch | \(-\frac {\left (105 a^{5} x^{5}-6533 a^{4} x^{4}+7624 a^{3} x^{3}-1454 a^{2} x^{2}+288 a x -30\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{105 x^{3} a^{4} \left (a x -1\right )}-\frac {\left (-\frac {13 a^{4} \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 \sqrt {a^{2} c}}-\frac {32 a^{3} \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{\sqrt {c}}\right ) c^{4} \sqrt {c \left (a x -1\right ) a x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a^{4} \left (a x -1\right )}\) | \(232\) |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \left (6720 \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{5}-17430 a^{\frac {9}{2}} \sqrt {a \,x^{2}-x}\, \sqrt {\frac {1}{a}}\, x^{5}+10920 a^{\frac {7}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {\frac {1}{a}}\, x^{3}-6720 a^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) \sqrt {2}\, x^{5}+8715 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{4} x^{5}-10080 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{4} x^{5}-1936 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2} \sqrt {\frac {1}{a}}+456 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}-60 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{210 x^{4} a^{\frac {9}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}}\) | \(305\) |
-1/105*(105*a^5*x^5-6533*a^4*x^4+7624*a^3*x^3-1454*a^2*x^2+288*a*x-30)/x^3 *c^4/a^4*(c*(a*x-1)/a/x)^(1/2)/(a*x-1)-(-13/2*a^4*ln((-1/2*a*c+a^2*c*x)/(a ^2*c)^(1/2)+(a^2*c*x^2-a*c*x)^(1/2))/(a^2*c)^(1/2)-32*a^3*2^(1/2)/c^(1/2)* ln((4*c-3*(x+1/a)*a*c+2*2^(1/2)*c^(1/2)*(a^2*c*(x+1/a)^2-3*(x+1/a)*a*c+2*c )^(1/2))/(x+1/a)))*c^4/a^4*(c*(a*x-1)*a*x)^(1/2)*(c*(a*x-1)/a/x)^(1/2)/(a* x-1)
Time = 0.27 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.81 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\left [\frac {6720 \, \sqrt {2} a^{3} c^{\frac {9}{2}} x^{3} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 1365 \, a^{3} c^{\frac {9}{2}} x^{3} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (105 \, a^{4} c^{4} x^{4} - 6428 \, a^{3} c^{4} x^{3} + 1196 \, a^{2} c^{4} x^{2} - 258 \, a c^{4} x + 30 \, c^{4}\right )} \sqrt {\frac {a c x - c}{a x}}}{210 \, a^{4} x^{3}}, \frac {6720 \, \sqrt {2} a^{3} \sqrt {-c} c^{4} x^{3} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) - 1365 \, a^{3} \sqrt {-c} c^{4} x^{3} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (105 \, a^{4} c^{4} x^{4} - 6428 \, a^{3} c^{4} x^{3} + 1196 \, a^{2} c^{4} x^{2} - 258 \, a c^{4} x + 30 \, c^{4}\right )} \sqrt {\frac {a c x - c}{a x}}}{105 \, a^{4} x^{3}}\right ] \]
[1/210*(6720*sqrt(2)*a^3*c^(9/2)*x^3*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c* x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + 1365*a^3*c^(9/2)*x^3*log(-2*a*c* x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(105*a^4*c^4*x^4 - 6428 *a^3*c^4*x^3 + 1196*a^2*c^4*x^2 - 258*a*c^4*x + 30*c^4)*sqrt((a*c*x - c)/( a*x)))/(a^4*x^3), 1/105*(6720*sqrt(2)*a^3*sqrt(-c)*c^4*x^3*arctan(1/2*sqrt (2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 1365*a^3*sqrt(-c)*c^4*x^3*arctan (sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (105*a^4*c^4*x^4 - 6428*a^3*c^4*x^3 + 1196*a^2*c^4*x^2 - 258*a*c^4*x + 30*c^4)*sqrt((a*c*x - c)/(a*x)))/(a^4* x^3)]
\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=- \int \left (- \frac {5 c^{4} \sqrt {c - \frac {c}{a x}}}{a x + 1}\right )\, dx - \int \frac {10 c^{4} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2} + a x}\, dx - \int \left (- \frac {10 c^{4} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3} + a^{2} x^{2}}\right )\, dx - \int \frac {5 c^{4} \sqrt {c - \frac {c}{a x}}}{a^{4} x^{4} + a^{3} x^{3}}\, dx - \int \left (- \frac {c^{4} \sqrt {c - \frac {c}{a x}}}{a^{5} x^{5} + a^{4} x^{4}}\right )\, dx - \int \frac {a c^{4} x \sqrt {c - \frac {c}{a x}}}{a x + 1}\, dx \]
-Integral(-5*c**4*sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(10*c**4*sqrt( c - c/(a*x))/(a**2*x**2 + a*x), x) - Integral(-10*c**4*sqrt(c - c/(a*x))/( a**3*x**3 + a**2*x**2), x) - Integral(5*c**4*sqrt(c - c/(a*x))/(a**4*x**4 + a**3*x**3), x) - Integral(-c**4*sqrt(c - c/(a*x))/(a**5*x**5 + a**4*x**4 ), x) - Integral(a*c**4*x*sqrt(c - c/(a*x))/(a*x + 1), x)
\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{{\left (a x + 1\right )}^{2}} \,d x } \]
Exception generated. \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=-\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]