Integrand size = 24, antiderivative size = 172 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx=-\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {5 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{9/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}} \] Output:
-6/5/a/c^2/(c-c/a/x)^(5/2)-11/6/a/c^3/(c-c/a/x)^(3/2)-21/4/a/c^4/(c-c/a/x) ^(1/2)+x/c^2/(c-c/a/x)^(5/2)+5*arctanh((c-c/a/x)^(1/2)/c^(1/2))/a/c^(9/2)+ 1/8*arctanh(1/2*(c-c/a/x)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)/a/c^(9/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.48 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx=\frac {a x^2 \left (5 a x-\operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\frac {a-\frac {1}{x}}{2 a}\right )-5 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},1-\frac {1}{a x}\right )\right )}{5 c^4 \sqrt {c-\frac {c}{a x}} (-1+a x)^2} \] Input:
Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(9/2)),x]
Output:
(a*x^2*(5*a*x - Hypergeometric2F1[-5/2, 1, -3/2, (a - x^(-1))/(2*a)] - 5*H ypergeometric2F1[-5/2, 1, -3/2, 1 - 1/(a*x)]))/(5*c^4*Sqrt[c - c/(a*x)]*(- 1 + a*x)^2)
Time = 0.98 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.12, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6717, 6683, 1035, 281, 899, 114, 27, 169, 27, 169, 27, 169, 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.
| \(\displaystyle \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle -\int \frac {1-a x}{\left (c-\frac {c}{a x}\right )^{9/2} (a x+1)}dx\) |
| \(\Big \downarrow \) 1035 |
| \(\displaystyle -\int \frac {\frac {1}{x}-a}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \frac {a \int \frac {1}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}dx}{c}\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\frac {a \int \frac {x^2}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {a \left (-\frac {\int -\frac {c \left (5 a+\frac {7}{x}\right ) x}{2 a \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}d\frac {1}{x}}{a c}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{5/2}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {\left (5 a+\frac {7}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}d\frac {1}{x}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{5/2}}\right )}{c}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a \left (\frac {\frac {12}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\int -\frac {5 c \left (5 a+\frac {6}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{5 c^2}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{5/2}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\int \frac {\left (5 a+\frac {6}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{c}+\frac {12}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{5/2}}\right )}{c}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {11}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\int -\frac {3 c \left (10 a+\frac {11}{x}\right ) x}{2 \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}}{3 c^2}}{c}+\frac {12}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{5/2}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\int \frac {\left (10 a+\frac {11}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}}{2 c}+\frac {11}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {12}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{5/2}}\right )}{c}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\frac {21}{c \sqrt {c-\frac {c}{a x}}}-\frac {\int -\frac {c \left (20 a+\frac {21}{x}\right ) x}{2 \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{c^2}}{2 c}+\frac {11}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {12}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{5/2}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\frac {\int \frac {\left (20 a+\frac {21}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 c}+\frac {21}{c \sqrt {c-\frac {c}{a x}}}}{2 c}+\frac {11}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {12}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{5/2}}\right )}{c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\frac {\int \frac {1}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}+20 \int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 c}+\frac {21}{c \sqrt {c-\frac {c}{a x}}}}{2 c}+\frac {11}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {12}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{5/2}}\right )}{c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\frac {-\frac {40 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {2 a \int \frac {1}{2 a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}}{2 c}+\frac {21}{c \sqrt {c-\frac {c}{a x}}}}{2 c}+\frac {11}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {12}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{5/2}}\right )}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\frac {-\frac {40 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}}{2 c}+\frac {21}{c \sqrt {c-\frac {c}{a x}}}}{2 c}+\frac {11}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{c}+\frac {12}{5 c \left (c-\frac {c}{a x}\right )^{5/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{5/2}}\right )}{c}\) |
Input:
Int[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(9/2)),x]
Output:
-((a*(-(x/(a*c*(c - c/(a*x))^(5/2))) + (12/(5*c*(c - c/(a*x))^(5/2)) + (11 /(3*c*(c - c/(a*x))^(3/2)) + (21/(c*Sqrt[c - c/(a*x)]) + ((-40*ArcTanh[Sqr t[c - c/(a*x)]/Sqrt[c]])/Sqrt[c] - (Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqr t[2]*Sqrt[c])])/Sqrt[c])/(2*c))/(2*c))/c)/(2*a^2)))/c)
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*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(314\) vs. \(2(141)=282\).
Time = 0.24 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.83
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +\left (x -\frac {1}{a}\right ) a c}}{5 a^{9} c \left (x -\frac {1}{a}\right )^{3}}-\frac {37 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +\left (x -\frac {1}{a}\right ) a c}}{30 a^{8} c \left (x -\frac {1}{a}\right )^{2}}-\frac {317 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +\left (x -\frac {1}{a}\right ) a c}}{60 a^{7} c \left (x -\frac {1}{a}\right )}+\frac {5 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 a^{5} \sqrt {a^{2} c}}-\frac {\sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{16 a^{6} \sqrt {c}}\right ) a^{4} \sqrt {c \left (a x -1\right ) a x}}{c^{4} x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(315\) |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (1260 a^{\frac {11}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, x^{4}+600 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{5} x^{4}-15 a^{\frac {9}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{4}-1020 a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, \left (x \left (a x -1\right )\right )^{\frac {3}{2}} x^{2}-5040 \sqrt {x \left (a x -1\right )}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{3}-2400 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{4} x^{3}+60 a^{\frac {7}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{3}+1792 \left (x \left (a x -1\right )\right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x +7560 \sqrt {x \left (a x -1\right )}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{2}+3600 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{2}-90 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{2}-820 \left (x \left (a x -1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} \sqrt {\frac {1}{a}}-5040 \sqrt {x \left (a x -1\right )}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x -2400 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x +60 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x +1260 \sqrt {x \left (a x -1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+600 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-15 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}\right )}{240 a^{\frac {3}{2}} \sqrt {x \left (a x -1\right )}\, c^{5} \left (a x -1\right )^{4} \sqrt {\frac {1}{a}}}\) | \(626\) |
Input:
int((a*x-1)/(a*x+1)/(c-c/a/x)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/a*(a*x-1)/c^4/(c*(a*x-1)/a/x)^(1/2)+(-1/5/a^9/c/(x-1/a)^3*((x-1/a)^2*a^2 *c+(x-1/a)*a*c)^(1/2)-37/30/a^8/c/(x-1/a)^2*((x-1/a)^2*a^2*c+(x-1/a)*a*c)^ (1/2)-317/60/a^7/c/(x-1/a)*((x-1/a)^2*a^2*c+(x-1/a)*a*c)^(1/2)+5/2/a^5*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)-1/ 16/a^6*2^(1/2)/c^(1/2)*ln((4*c-3*(x+1/a)*a*c+2*2^(1/2)*c^(1/2)*((x+1/a)^2* a^2*c-3*(x+1/a)*a*c+2*c)^(1/2))/(x+1/a)))*a^4/c^4/x/(c*(a*x-1)/a/x)^(1/2)* (c*(a*x-1)*a*x)^(1/2)
Time = 0.10 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.60 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx=\left [\frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 600 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 4 \, {\left (60 \, a^{4} x^{4} - 497 \, a^{3} x^{3} + 740 \, a^{2} x^{2} - 315 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{240 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}, -\frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + 600 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) - 2 \, {\left (60 \, a^{4} x^{4} - 497 \, a^{3} x^{3} + 740 \, a^{2} x^{2} - 315 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{120 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}\right ] \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a/x)^(9/2),x, algorithm="fricas")
Output:
[1/240*(15*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-(2*sqrt( 2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + 3*a*c*x - c)/(a*x + 1)) + 600*(a^ 3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt(( a*c*x - c)/(a*x)) + c) + 4*(60*a^4*x^4 - 497*a^3*x^3 + 740*a^2*x^2 - 315*a *x)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5), -1/120*(15*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arct an(sqrt(2)*a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)) + 600*(a^3*x^ 3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(a*sqrt(-c)*x*sqrt((a*c*x - c)/( a*x))/(a*c*x - c)) - 2*(60*a^4*x^4 - 497*a^3*x^3 + 740*a^2*x^2 - 315*a*x)* sqrt((a*c*x - c)/(a*x)))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^ 5)]
\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx=\int \frac {a x - 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {9}{2}} \left (a x + 1\right )}\, dx \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a/x)**(9/2),x)
Output:
Integral((a*x - 1)/((-c*(-1 + 1/(a*x)))**(9/2)*(a*x + 1)), x)
\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx=\int { \frac {a x - 1}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a/x)^(9/2),x, algorithm="maxima")
Output:
integrate((a*x - 1)/((a*x + 1)*(c - c/(a*x))^(9/2)), x)
Exception generated. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a/x)^(9/2),x, algorithm="giac")
Output:
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. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx=\int \frac {a\,x-1}{{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,\left (a\,x+1\right )} \,d x \] Input:
int((a*x - 1)/((c - c/(a*x))^(9/2)*(a*x + 1)),x)
Output:
int((a*x - 1)/((c - c/(a*x))^(9/2)*(a*x + 1)), x)
Time = 0.17 (sec) , antiderivative size = 475, normalized size of antiderivative = 2.76 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx=\frac {\sqrt {c}\, \left (15 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right ) a^{2} x^{2}-30 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right ) a x +15 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right )+15 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right ) a^{2} x^{2}-30 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right ) a x +15 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right )-15 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right ) a^{2} x^{2}+30 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right ) a x -15 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right )+1200 \sqrt {a x -1}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right ) a^{2} x^{2}-2400 \sqrt {a x -1}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right ) a x +1200 \sqrt {a x -1}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right )+632 \sqrt {a x -1}\, a^{2} x^{2}-1264 \sqrt {a x -1}\, a x +632 \sqrt {a x -1}+240 \sqrt {x}\, \sqrt {a}\, a^{3} x^{3}-1988 \sqrt {x}\, \sqrt {a}\, a^{2} x^{2}+2960 \sqrt {x}\, \sqrt {a}\, a x -1260 \sqrt {x}\, \sqrt {a}\right )}{240 \sqrt {a x -1}\, a \,c^{5} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:
int((a*x-1)/(a*x+1)/(c-c/a/x)^(9/2),x)
Output:
(sqrt(c)*(15*sqrt(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) - s qrt(2)*i + i)*a**2*x**2 - 30*sqrt(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqr t(x)*sqrt(a) - sqrt(2)*i + i)*a*x + 15*sqrt(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) - sqrt(2)*i + i) + 15*sqrt(a*x - 1)*sqrt(2)*log(sqr t(a*x - 1) + sqrt(x)*sqrt(a) + sqrt(2)*i - i)*a**2*x**2 - 30*sqrt(a*x - 1) *sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) + sqrt(2)*i - i)*a*x + 15*sqr t(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) + sqrt(2)*i - i) - 15*sqrt(a*x - 1)*sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 2*sqrt(2) + 2*a*x + 2)*a**2*x**2 + 30*sqrt(a*x - 1)*sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqr t(a*x - 1) + 2*sqrt(2) + 2*a*x + 2)*a*x - 15*sqrt(a*x - 1)*sqrt(2)*log(2*s qrt(x)*sqrt(a)*sqrt(a*x - 1) + 2*sqrt(2) + 2*a*x + 2) + 1200*sqrt(a*x - 1) *log(sqrt(a*x - 1) + sqrt(x)*sqrt(a))*a**2*x**2 - 2400*sqrt(a*x - 1)*log(s qrt(a*x - 1) + sqrt(x)*sqrt(a))*a*x + 1200*sqrt(a*x - 1)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a)) + 632*sqrt(a*x - 1)*a**2*x**2 - 1264*sqrt(a*x - 1)*a*x + 632*sqrt(a*x - 1) + 240*sqrt(x)*sqrt(a)*a**3*x**3 - 1988*sqrt(x)*sqrt(a )*a**2*x**2 + 2960*sqrt(x)*sqrt(a)*a*x - 1260*sqrt(x)*sqrt(a)))/(240*sqrt( a*x - 1)*a*c**5*(a**2*x**2 - 2*a*x + 1))