\(\int e^{-2 \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^{7/2} \, dx\) [863]

Optimal result

Integrand size = 24, antiderivative size = 375 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {7 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}+\frac {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac {2 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \arcsin (a x)}{(1-a x)^{7/2} (1+a x)^{7/2}}+\frac {25 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}} \]

[Out]

7/16*a^6*(c-c/a^2/x^2)^(7/2)*x^7/(-a*x+1)^3/(a*x+1)^3+3/8*a^5*(c-c/a^2/x^2)^(7/2)*x^6/(-a*x+1)^3/(a*x+1)^2-1/1
5*a*(c-c/a^2/x^2)^(7/2)*x^2/(a*x+1)-19/16*a^4*(c-c/a^2/x^2)^(7/2)*x^5/(-a*x+1)^3/(a*x+1)+2/3*a^3*(c-c/a^2/x^2)
^(7/2)*x^4/(-a*x+1)^2/(a*x+1)-23/120*a^2*(c-c/a^2/x^2)^(7/2)*x^3/(-a*x+1)/(a*x+1)+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)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6302, 6294, 6264, 99, 154, 159, 163, 41, 222, 94, 214} \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=-\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{15 (a x+1)}+\frac {x (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{6 (a x+1)}-\frac {23 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{120 (1-a x) (a x+1)}-\frac {2 a^6 x^7 \arcsin (a x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{(1-a x)^{7/2} (a x+1)^{7/2}}+\frac {25 a^6 x^7 \text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right ) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^{7/2} (a x+1)^{7/2}}+\frac {7 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)^3}+\frac {3 a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{8 (1-a x)^3 (a x+1)^2}-\frac {19 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)}+\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{3 (1-a x)^2 (a x+1)} \]

[In]

Int[(c - c/(a^2*x^2))^(7/2)/E^(2*ArcCoth[a*x]),x]

[Out]

(7*a^6*(c - c/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^3*(1 + a*x)^3) + (3*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(8*(1 -
a*x)^3*(1 + a*x)^2) - (a*(c - c/(a^2*x^2))^(7/2)*x^2)/(15*(1 + a*x)) - (19*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(
16*(1 - a*x)^3*(1 + a*x)) + (2*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(3*(1 - a*x)^2*(1 + a*x)) - (23*a^2*(c - c/(a^
2*x^2))^(7/2)*x^3)/(120*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^(7/2)*x*(1 - a*x))/(6*(1 + a*x)) - (2*a^6*(c
- c/(a^2*x^2))^(7/2)*x^7*ArcSin[a*x])/((1 - a*x)^(7/2)*(1 + a*x)^(7/2)) + (25*a^6*(c - c/(a^2*x^2))^(7/2)*x^7
*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(16*(1 - a*x)^(7/2)*(1 + a*x)^(7/2))

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[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] /; FreeQ[{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]

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 214

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]

Rule 222

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]

Rule 6264

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[u*(1 + d*(x/c))^
p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 6294

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[x^(2*p)*((c + d/x^2)^p/((
1 - a*x)^p*(1 + a*x)^p)), Int[(u/x^(2*p))*(1 - a*x)^p*(1 + a*x)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d
, n, p}, x] && EqQ[c + a^2*d, 0] &&  !IntegerQ[p] && IntegerQ[n/2] &&  !GtQ[c, 0]

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx \\ & = -\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {e^{-2 \text {arctanh}(a x)} (1-a x)^{7/2} (1+a x)^{7/2}}{x^7} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}} \\ & = -\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{9/2} (1+a x)^{5/2}}{x^7} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}} \\ & = \frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{7/2} (1+a x)^{3/2} \left (-2 a-7 a^2 x\right )}{x^6} \, dx}{6 (1-a x)^{7/2} (1+a x)^{7/2}} \\ & = -\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{5/2} (1+a x)^{3/2} \left (-23 a^2+37 a^3 x\right )}{x^5} \, dx}{30 (1-a x)^{7/2} (1+a x)^{7/2}} \\ & = -\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{3/2} (1+a x)^{3/2} \left (240 a^3-125 a^4 x\right )}{x^4} \, dx}{120 (1-a x)^{7/2} (1+a x)^{7/2}} \\ & = -\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {\sqrt {1-a x} (1+a x)^{3/2} \left (-855 a^4+135 a^5 x\right )}{x^3} \, dx}{360 (1-a x)^{7/2} (1+a x)^{7/2}} \\ & = -\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1+a x)^{3/2} \left (270 a^5+585 a^6 x\right )}{x^2 \sqrt {1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}} \\ & = \frac {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {\sqrt {1+a x} \left (1125 a^6+315 a^7 x\right )}{x \sqrt {1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}} \\ & = \frac {7 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 {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {-1125 a^7-1440 a^8 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{720 a (1-a x)^{7/2} (1+a x)^{7/2}} \\ & = \frac {7 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 {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac {\left (25 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{16 (1-a x)^{7/2} (1+a x)^{7/2}}-\frac {\left (2 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}} \\ & = \frac {7 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 {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac {\left (25 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \text {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}}-\frac {\left (2 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}} \\ & = \frac {7 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 {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.40 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (-40+96 a x+70 a^2 x^2-352 a^3 x^3+105 a^4 x^4+736 a^5 x^5+240 a^6 x^6\right )+375 a^6 x^6 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )-480 a^6 x^6 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{240 a^6 x^5 \sqrt {-1+a^2 x^2}} \]

[In]

Integrate[(c - c/(a^2*x^2))^(7/2)/E^(2*ArcCoth[a*x]),x]

[Out]

(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]]))/
(240*a^6*x^5*Sqrt[-1 + a^2*x^2])

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.67

method	result	size
risch	\(\frac {\left (736 a^{7} x^{7}+105 a^{6} x^{6}-1088 a^{5} x^{5}-35 a^{4} x^{4}+448 a^{3} x^{3}-110 a^{2} x^{2}-96 a x +40\right ) c^{3} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{240 x^{5} a^{6} \left (a^{2} x^{2}-1\right )}+\frac {\left (-\frac {2 a^{7} \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}+\frac {25 a^{6} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{16 \sqrt {-c}}+\frac {a^{6} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c}\right ) c^{3} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {c \left (a^{2} x^{2}-1\right )}}{a^{6} \left (a^{2} x^{2}-1\right )}\)	\(250\)
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 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {7}{2}} \sqrt {-\frac {c}{a^{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 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} \sqrt {-\frac {c}{a^{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}\)	\(795\)

[In]

int((c-c/a^2/x^2)^(7/2)*(a*x-1)/(a*x+1),x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.17 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\left [\frac {960 \, a^{5} \sqrt {-c} c^{3} x^{5} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 375 \, a^{5} \sqrt {-c} c^{3} x^{5} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (240 \, a^{6} c^{3} x^{6} + 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} - 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} + 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{480 \, a^{6} x^{5}}, \frac {375 \, a^{5} c^{\frac {7}{2}} x^{5} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 240 \, a^{5} c^{\frac {7}{2}} x^{5} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (240 \, a^{6} c^{3} x^{6} + 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} - 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} + 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{240 \, a^{6} x^{5}}\right ] \]

[In]

integrate((c-c/a^2/x^2)^(7/2)*(a*x-1)/(a*x+1),x, algorithm="fricas")

[Out]

[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)) + 37
5*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 - 40*c^3)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5)]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 18.11 (sec) , antiderivative size = 1059, normalized size of antiderivative = 2.82 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\text {Too large to display} \]

[In]

integrate((c-c/a**2/x**2)**(7/2)*(a*x-1)/(a*x+1),x)

[Out]

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*sqrt(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
*sqrt(c)*asin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a - c**3*Piecewise((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*Piecewi
se((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*sqrt(-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*s
qrt(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*Piecewise((2*a**3*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x) + a*sqrt(c)*sqr
t(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)*sq
rt(-a**2*x**2 + 1)/(15*x) + I*a*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x**3) - I*sqrt(c)*sqrt(-a**2*x**2 + 1)/(5*a*x
**5), True))/a**5 + c**3*Piecewise((I*a**5*sqrt(c)*acosh(1/(a*x))/16 - I*a**4*sqrt(c)/(16*x*sqrt(-1 + 1/(a**2*
x**2))) + I*a**2*sqrt(c)/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) + 5*I*sqrt(c)/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) -
I*sqrt(c)/(6*a**2*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a**5*sqrt(c)*asin(1/(a*x))/16 + a*
*4*sqrt(c)/(16*x*sqrt(1 - 1/(a**2*x**2))) - a**2*sqrt(c)/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*sqrt(c)/(24*x**
5*sqrt(1 - 1/(a**2*x**2))) + sqrt(c)/(6*a**2*x**7*sqrt(1 - 1/(a**2*x**2))), True))/a**6

Maxima [F]

\[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\int { \frac {{\left (a x - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{a x + 1} \,d x } \]

[In]

integrate((c-c/a^2/x^2)^(7/2)*(a*x-1)/(a*x+1),x, algorithm="maxima")

[Out]

integrate((a*x - 1)*(c - c/(a^2*x^2))^(7/2)/(a*x + 1), x)

Giac [A] (verification not implemented)

none

Time = 46.81 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.50 \[ \int e^{-2 \coth ^{-1}(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 |} \]

[In]

integrate((c-c/a^2/x^2)^(7/2)*(a*x-1)/(a*x+1),x, algorithm="giac")

[Out]

-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(-s
qrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(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 - sqrt(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 - sqrt(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)

Mupad [F(-1)]

Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]

[In]

int(((c - c/(a^2*x^2))^(7/2)*(a*x - 1))/(a*x + 1),x)

[Out]

int(((c - c/(a^2*x^2))^(7/2)*(a*x - 1))/(a*x + 1), x)