Integrand size = 24, antiderivative size = 148 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {4}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7}{2 a c^3 \sqrt {c-\frac {c}{a x}}}-\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a c^{7/2}} \] Output:
4/3/a/c^2/(c-c/a/x)^(3/2)+7/2/a/c^3/(c-c/a/x)^(1/2)-x/c^2/(c-c/a/x)^(3/2)- 3*arctanh((c-c/a/x)^(1/2)/c^(1/2))/a/c^(7/2)-1/4*arctanh(1/2*(c-c/a/x)^(1/ 2)*2^(1/2)/c^(1/2))*2^(1/2)/a/c^(7/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.52 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {x \left (-3 a x+\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a-\frac {1}{x}}{2 a}\right )+3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1-\frac {1}{a x}\right )\right )}{3 c^3 \sqrt {c-\frac {c}{a x}} (-1+a x)} \] Input:
Integrate[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(7/2)),x]
Output:
(x*(-3*a*x + Hypergeometric2F1[-3/2, 1, -1/2, (a - x^(-1))/(2*a)] + 3*Hype rgeometric2F1[-3/2, 1, -1/2, 1 - 1/(a*x)]))/(3*c^3*Sqrt[c - c/(a*x)]*(-1 + a*x))
Time = 0.72 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6683, 1035, 281, 899, 114, 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.
| \(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle \int \frac {1-a x}{(a x+1) \left (c-\frac {c}{a x}\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 1035 |
| \(\displaystyle \int \frac {\frac {1}{x}-a}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle -\frac {a \int \frac {1}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}dx}{c}\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle \frac {a \int \frac {x^2}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {a \left (-\frac {\int -\frac {c \left (3 a+\frac {5}{x}\right ) x}{2 a \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{a c}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {\int \frac {\left (3 a+\frac {5}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{c}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {a \left (\frac {\frac {8}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\int -\frac {3 c \left (3 a+\frac {4}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}}{3 c^2}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {\frac {\int \frac {\left (3 a+\frac {4}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}}{c}+\frac {8}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{c}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {7}{c \sqrt {c-\frac {c}{a x}}}-\frac {\int -\frac {c \left (6 a+\frac {7}{x}\right ) x}{2 \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{c^2}}{c}+\frac {8}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {\int \frac {\left (6 a+\frac {7}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 c}+\frac {7}{c \sqrt {c-\frac {c}{a x}}}}{c}+\frac {8}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {\int \frac {1}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}+6 \int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 c}+\frac {7}{c \sqrt {c-\frac {c}{a x}}}}{c}+\frac {8}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {-\frac {12 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {2 a \int \frac {1}{2 a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}}{2 c}+\frac {7}{c \sqrt {c-\frac {c}{a x}}}}{c}+\frac {8}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {-\frac {12 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}}{2 c}+\frac {7}{c \sqrt {c-\frac {c}{a x}}}}{c}+\frac {8}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}}{2 a^2}-\frac {x}{a c \left (c-\frac {c}{a x}\right )^{3/2}}\right )}{c}\) |
Input:
Int[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(7/2)),x]
Output:
(a*(-(x/(a*c*(c - c/(a*x))^(3/2))) + (8/(3*c*(c - c/(a*x))^(3/2)) + (7/(c* Sqrt[c - c/(a*x)]) + ((-12*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/Sqrt[c] - ( Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/Sqrt[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]
Leaf count of result is larger than twice the leaf count of optimal. \(272\) vs. \(2(121)=242\).
Time = 0.17 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.84
| method | result | size |
| risch | \(-\frac {a x -1}{a \,c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}-\frac {\left (-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +\left (x -\frac {1}{a}\right ) a c}}{3 a^{7} c \left (x -\frac {1}{a}\right )^{2}}-\frac {17 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +\left (x -\frac {1}{a}\right ) a c}}{6 a^{6} c \left (x -\frac {1}{a}\right )}+\frac {3 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 a^{4} \sqrt {a^{2} c}}-\frac {\sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{8 a^{5} \sqrt {c}}\right ) a^{3} \sqrt {c \left (a x -1\right ) a x}}{c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\) | \(273\) |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-84 \sqrt {x \left (a x -1\right )}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{3}+60 \left (x \left (a x -1\right )\right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x -36 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{4} x^{3}+3 a^{\frac {7}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{3}+252 \sqrt {x \left (a x -1\right )}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{2}-52 \left (x \left (a x -1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} \sqrt {\frac {1}{a}}+108 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{2}-9 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{2}-252 \sqrt {x \left (a x -1\right )}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x -108 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x +9 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x +84 \sqrt {x \left (a x -1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+36 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-3 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}\right )}{24 a^{\frac {3}{2}} \sqrt {x \left (a x -1\right )}\, c^{4} \left (a x -1\right )^{3} \sqrt {\frac {1}{a}}}\) | \(497\) |
Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(7/2),x,method=_RETURNVERBOSE)
Output:
-1/a*(a*x-1)/c^3/(c*(a*x-1)/a/x)^(1/2)-(-1/3/a^7/c/(x-1/a)^2*((x-1/a)^2*a^ 2*c+(x-1/a)*a*c)^(1/2)-17/6/a^6/c/(x-1/a)*((x-1/a)^2*a^2*c+(x-1/a)*a*c)^(1 /2)+3/2/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)-1/8/a^5*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^3/c^3*(c*(a*x-1) *a*x)^(1/2)/(c*(a*x-1)/a/x)^(1/2)/x
Time = 0.13 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.52 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\left [\frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 36 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 4 \, {\left (6 \, a^{3} x^{3} - 29 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{24 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + 36 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) - 2 \, {\left (6 \, a^{3} x^{3} - 29 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{12 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(7/2),x, algorithm="fricas")
Output:
[1/24*(3*sqrt(2)*(a^2*x^2 - 2*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)) + 36*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 4*( 6*a^3*x^3 - 29*a^2*x^2 + 21*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^4*x^2 - 2 *a^2*c^4*x + a*c^4), 1/12*(3*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan (sqrt(2)*a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)) + 36*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)) - 2*(6*a^3*x^3 - 29*a^2*x^2 + 21*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^ 4*x^2 - 2*a^2*c^4*x + a*c^4)]
\[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=- \int \frac {a x}{a c^{3} x \sqrt {c - \frac {c}{a x}} - 2 c^{3} \sqrt {c - \frac {c}{a x}} + \frac {2 c^{3} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}} - \frac {c^{3} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3}}}\, dx - \int \left (- \frac {1}{a c^{3} x \sqrt {c - \frac {c}{a x}} - 2 c^{3} \sqrt {c - \frac {c}{a x}} + \frac {2 c^{3} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}} - \frac {c^{3} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3}}}\right )\, dx \] Input:
integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x)**(7/2),x)
Output:
-Integral(a*x/(a*c**3*x*sqrt(c - c/(a*x)) - 2*c**3*sqrt(c - c/(a*x)) + 2*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)) - 2*c**3*sqrt(c - c/(a*x)) + 2 *c**3*sqrt(c - c/(a*x))/(a**2*x**2) - c**3*sqrt(c - c/(a*x))/(a**3*x**3)), x)
\[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int { -\frac {a^{2} x^{2} - 1}{{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(7/2),x, algorithm="maxima")
Output:
-integrate((a^2*x^2 - 1)/((a*x + 1)^2*(c - c/(a*x))^(7/2)), x)
Exception generated. \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(7/2),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. \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=-\int \frac {a^2\,x^2-1}{{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,{\left (a\,x+1\right )}^2} \,d x \] Input:
int(-(a^2*x^2 - 1)/((c - c/(a*x))^(7/2)*(a*x + 1)^2),x)
Output:
-int((a^2*x^2 - 1)/((c - c/(a*x))^(7/2)*(a*x + 1)^2), x)
Time = 0.16 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.05 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (-3 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right ) a x +3 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right )-3 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right ) a x +3 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right )+3 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right ) a x -3 \sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right )-72 \sqrt {a x -1}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right ) a x +72 \sqrt {a x -1}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right )-16 \sqrt {a x -1}\, a x +16 \sqrt {a x -1}-24 \sqrt {x}\, \sqrt {a}\, a^{2} x^{2}+116 \sqrt {x}\, \sqrt {a}\, a x -84 \sqrt {x}\, \sqrt {a}\right )}{24 \sqrt {a x -1}\, a \,c^{4} \left (a x -1\right )} \] Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(7/2),x)
Output:
(sqrt(c)*( - 3*sqrt(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) - sqrt(2)*i + i)*a*x + 3*sqrt(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)* sqrt(a) - sqrt(2)*i + i) - 3*sqrt(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqr t(x)*sqrt(a) + sqrt(2)*i - i)*a*x + 3*sqrt(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) + sqrt(2)*i - i) + 3*sqrt(a*x - 1)*sqrt(2)*log(2*sqr t(x)*sqrt(a)*sqrt(a*x - 1) + 2*sqrt(2) + 2*a*x + 2)*a*x - 3*sqrt(a*x - 1)* sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 2*sqrt(2) + 2*a*x + 2) - 72* sqrt(a*x - 1)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a))*a*x + 72*sqrt(a*x - 1)* log(sqrt(a*x - 1) + sqrt(x)*sqrt(a)) - 16*sqrt(a*x - 1)*a*x + 16*sqrt(a*x - 1) - 24*sqrt(x)*sqrt(a)*a**2*x**2 + 116*sqrt(x)*sqrt(a)*a*x - 84*sqrt(x) *sqrt(a)))/(24*sqrt(a*x - 1)*a*c**4*(a*x - 1))