Integrand size = 24, antiderivative size = 146 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {11}{7 a \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11}{5 a c \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {11}{a c^3 \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}} \]
11/7/a/(c-c/a/x)^(7/2)+11/5/a/c/(c-c/a/x)^(5/2)+11/3/a/c^2/(c-c/a/x)^(3/2) -x/(c-c/a/x)^(7/2)-11*arctanh((c-c/a/x)^(1/2)/c^(1/2))/a/c^(7/2)+11/a/c^3/ (c-c/a/x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.32 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {-7 x+\frac {11 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},1-\frac {1}{a x}\right )}{a}}{7 \left (c-\frac {c}{a x}\right )^{7/2}} \]
Time = 0.39 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6683, 1035, 281, 899, 87, 61, 61, 61, 61, 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.
| \(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle \int \frac {a x+1}{(1-a x) \left (c-\frac {c}{a x}\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 1035 |
| \(\displaystyle \int \frac {a+\frac {1}{x}}{\left (\frac {1}{x}-a\right ) \left (c-\frac {c}{a x}\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle -\frac {c \int \frac {a+\frac {1}{x}}{\left (c-\frac {c}{a x}\right )^{9/2}}dx}{a}\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle \frac {c \int \frac {\left (a+\frac {1}{x}\right ) x^2}{\left (c-\frac {c}{a x}\right )^{9/2}}d\frac {1}{x}}{a}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {c \left (\frac {11}{2} \int \frac {x}{\left (c-\frac {c}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {c \left (\frac {11}{2} \left (\frac {\int \frac {x}{\left (c-\frac {c}{a x}\right )^{7/2}}d\frac {1}{x}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {c \left (\frac {11}{2} \left (\frac {\frac {\int \frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{c}+\frac {2}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {c \left (\frac {11}{2} \left (\frac {\frac {\frac {\int \frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {2}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {c \left (\frac {11}{2} \left (\frac {\frac {\frac {\frac {\int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{c}+\frac {2}{c \sqrt {c-\frac {c}{a x}}}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {2}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {c \left (\frac {11}{2} \left (\frac {\frac {\frac {\frac {2}{c \sqrt {c-\frac {c}{a x}}}-\frac {2 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c^2}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {2}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {c \left (\frac {11}{2} \left (\frac {\frac {\frac {\frac {2}{c \sqrt {c-\frac {c}{a x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{c^{3/2}}}{c}+\frac {2}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {2}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{c}+\frac {2}{7 c \left (c-\frac {c}{a x}\right )^{7/2}}\right )-\frac {a x}{c \left (c-\frac {c}{a x}\right )^{7/2}}\right )}{a}\) |
(c*(-((a*x)/(c*(c - c/(a*x))^(7/2))) + (11*(2/(7*c*(c - c/(a*x))^(7/2)) + (2/(5*c*(c - c/(a*x))^(5/2)) + (2/(3*c*(c - c/(a*x))^(3/2)) + (2/(c*Sqrt[c - c/(a*x)]) - (2*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/c^(3/2))/c)/c)/c))/2 ))/a
3.6.26.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(292\) vs. \(2(124)=248\).
Time = 0.12 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.01
| method | result | size |
| risch | \(-\frac {a x -1}{a \,c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}-\frac {\left (\frac {11 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 a^{4} \sqrt {a^{2} c}}-\frac {4 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{7 a^{9} c \left (x -\frac {1}{a}\right )^{4}}-\frac {102 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{35 a^{8} c \left (x -\frac {1}{a}\right )^{3}}-\frac {712 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{105 a^{7} c \left (x -\frac {1}{a}\right )^{2}}-\frac {1516 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{105 a^{6} c \left (x -\frac {1}{a}\right )}\right ) a^{3} \sqrt {c \left (a x -1\right ) a x}}{c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\) | \(293\) |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-2310 \sqrt {\left (a x -1\right ) x}\, a^{\frac {11}{2}} x^{5}+2100 \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{3}-1155 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{5} x^{5}+11550 \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}} x^{4}-5368 \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{2}+5775 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{4} x^{4}-23100 \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}} x^{3}+4928 \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} a^{\frac {5}{2}} x -11550 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{3}+23100 \sqrt {\left (a x -1\right ) x}\, a^{\frac {5}{2}} x^{2}-1540 a^{\frac {3}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}+11550 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2} x^{2}-11550 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} x -5775 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a x +2310 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+1155 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{210 \sqrt {\left (a x -1\right ) x}\, c^{4} \sqrt {a}\, \left (a x -1\right )^{5}}\) | \(396\) |
-1/a*(a*x-1)/c^3/(c*(a*x-1)/a/x)^(1/2)-(11/2/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)-4/7/a^9/c/(x-1/a)^4*(a^2 *c*(x-1/a)^2+(x-1/a)*a*c)^(1/2)-102/35/a^8/c/(x-1/a)^3*(a^2*c*(x-1/a)^2+(x -1/a)*a*c)^(1/2)-712/105/a^7/c/(x-1/a)^2*(a^2*c*(x-1/a)^2+(x-1/a)*a*c)^(1/ 2)-1516/105/a^6/c/(x-1/a)*(a^2*c*(x-1/a)^2+(x-1/a)*a*c)^(1/2))*a^3/c^3*(c* (a*x-1)*a*x)^(1/2)/(c*(a*x-1)/a/x)^(1/2)/x
Time = 0.27 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.37 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\left [\frac {1155 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{210 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac {1155 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{105 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \]
[1/210*(1155*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*log(-2* a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(105*a^5*x^5 - 1936 *a^4*x^4 + 4466*a^3*x^3 - 3850*a^2*x^2 + 1155*a*x)*sqrt((a*c*x - c)/(a*x)) )/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4), 1/1 05*(1155*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*arctan(sqr t(-c)*sqrt((a*c*x - c)/(a*x))/c) - (105*a^5*x^5 - 1936*a^4*x^4 + 4466*a^3* x^3 - 3850*a^2*x^2 + 1155*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^5*c^4*x^4 - 4*a ^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)]
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=- \int \frac {a x}{a c^{3} x \sqrt {c - \frac {c}{a x}} - 4 c^{3} \sqrt {c - \frac {c}{a x}} + \frac {6 c^{3} \sqrt {c - \frac {c}{a x}}}{a x} - \frac {4 c^{3} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}} + \frac {c^{3} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3}}}\, dx - \int \frac {1}{a c^{3} x \sqrt {c - \frac {c}{a x}} - 4 c^{3} \sqrt {c - \frac {c}{a x}} + \frac {6 c^{3} \sqrt {c - \frac {c}{a x}}}{a x} - \frac {4 c^{3} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}} + \frac {c^{3} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3}}}\, dx \]
-Integral(a*x/(a*c**3*x*sqrt(c - c/(a*x)) - 4*c**3*sqrt(c - c/(a*x)) + 6*c **3*sqrt(c - c/(a*x))/(a*x) - 4*c**3*sqrt(c - c/(a*x))/(a**2*x**2) + c**3* sqrt(c - c/(a*x))/(a**3*x**3)), x) - Integral(1/(a*c**3*x*sqrt(c - c/(a*x) ) - 4*c**3*sqrt(c - c/(a*x)) + 6*c**3*sqrt(c - c/(a*x))/(a*x) - 4*c**3*sqr t(c - c/(a*x))/(a**2*x**2) + c**3*sqrt(c - c/(a*x))/(a**3*x**3)), x)
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 392 vs. \(2 (124) = 248\).
Time = 0.60 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.68 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=-\frac {11 \, \log \left (c^{4} {\left | a \right |} {\left | c \right |}\right ) \mathrm {sgn}\left (x\right )}{18 \, a c^{\frac {7}{2}}} + \frac {11 \, \log \left ({\left | 2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{9} \sqrt {c} {\left | a \right |} - 17 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{8} a c + 64 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{7} c^{\frac {3}{2}} {\left | a \right |} - 140 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{6} a c^{2} + 196 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} c^{\frac {5}{2}} {\left | a \right |} - 182 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a c^{3} + 112 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} c^{\frac {7}{2}} {\left | a \right |} - 44 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a c^{4} + 10 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} c^{\frac {9}{2}} {\left | a \right |} - a c^{5} \right |}\right ) \mathrm {sgn}\left (x\right )}{18 \, a c^{\frac {7}{2}}} - \frac {\sqrt {a^{2} c x^{2} - a c x} {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{2} c^{4}} \]
-11/18*log(c^4*abs(a)*abs(c))*sgn(x)/(a*c^(7/2)) + 11/18*log(abs(2*(sqrt(a ^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^9*sqrt(c)*abs(a) - 17*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^8*a*c + 64*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c* x))^7*c^(3/2)*abs(a) - 140*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^6*a*c ^2 + 196*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^5*c^(5/2)*abs(a) - 182* (sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^4*a*c^3 + 112*(sqrt(a^2*c)*x - s qrt(a^2*c*x^2 - a*c*x))^3*c^(7/2)*abs(a) - 44*(sqrt(a^2*c)*x - sqrt(a^2*c* x^2 - a*c*x))^2*a*c^4 + 10*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))*c^(9/ 2)*abs(a) - a*c^5))*sgn(x)/(a*c^(7/2)) - sqrt(a^2*c*x^2 - a*c*x)*abs(a)*sg n(x)/(a^2*c^4)
Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int -\frac {{\left (a\,x+1\right )}^2}{{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,\left (a^2\,x^2-1\right )} \,d x \]