\(\int e^{-2 \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^{5/2} \, dx\) [864]

Optimal result

Integrand size = 24, antiderivative size = 293 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=-\frac {7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}+\frac {2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \arcsin (a x)}{(1-a x)^{5/2} (1+a x)^{5/2}}-\frac {9 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}} \]

[Out]

-7/8*a^4*(c-c/a^2/x^2)^(5/2)*x^5/(-a*x+1)^2/(a*x+1)^2-1/6*a*(c-c/a^2/x^2)^(5/2)*x^2/(a*x+1)+2*a^3*(c-c/a^2/x^2
)^(5/2)*x^4/(-a*x+1)^2/(a*x+1)-7/24*a^2*(c-c/a^2/x^2)^(5/2)*x^3/(-a*x+1)/(a*x+1)+1/4*(c-c/a^2/x^2)^(5/2)*x*(-a
*x+1)/(a*x+1)+2*a^4*(c-c/a^2/x^2)^(5/2)*x^5*arcsin(a*x)/(-a*x+1)^(5/2)/(a*x+1)^(5/2)-9/8*a^4*(c-c/a^2/x^2)^(5/
2)*x^5*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))/(-a*x+1)^(5/2)/(a*x+1)^(5/2)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=-\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{6 (a x+1)}+\frac {x (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 (a x+1)}-\frac {7 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{24 (1-a x) (a x+1)}+\frac {2 a^4 x^5 \arcsin (a x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{(1-a x)^{5/2} (a x+1)^{5/2}}-\frac {9 a^4 x^5 \text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^{5/2} (a x+1)^{5/2}}-\frac {7 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 (a x+1)^2}+\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{(1-a x)^2 (a x+1)} \]

[In]

Int[(c - c/(a^2*x^2))^(5/2)/E^(2*ArcCoth[a*x]),x]

[Out]

(-7*a^4*(c - c/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(1 + a*x)^2) - (a*(c - c/(a^2*x^2))^(5/2)*x^2)/(6*(1 + a*x
)) + (2*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/((1 - a*x)^2*(1 + a*x)) - (7*a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(24*(1
- a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^(5/2)*x*(1 - a*x))/(4*(1 + a*x)) + (2*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*A
rcSin[a*x])/((1 - a*x)^(5/2)*(1 + a*x)^(5/2)) - (9*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*ArcTanh[Sqrt[1 - a*x]*Sqrt[
1 + a*x]])/(8*(1 - a*x)^(5/2)*(1 + a*x)^(5/2))

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 )^{5/2} \, dx \\ & = -\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {e^{-2 \text {arctanh}(a x)} (1-a x)^{5/2} (1+a x)^{5/2}}{x^5} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1-a x)^{7/2} (1+a x)^{3/2}}{x^5} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}} \\ & = \frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1-a x)^{5/2} \sqrt {1+a x} \left (-2 a-5 a^2 x\right )}{x^4} \, dx}{4 (1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1-a x)^{3/2} \sqrt {1+a x} \left (-7 a^2+17 a^3 x\right )}{x^3} \, dx}{12 (1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {\sqrt {1-a x} \sqrt {1+a x} \left (48 a^3-27 a^4 x\right )}{x^2} \, dx}{24 (1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {\sqrt {1+a x} \left (-27 a^4-21 a^5 x\right )}{x \sqrt {1-a x}} \, dx}{24 (1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {27 a^5+48 a^6 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{24 a (1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}+\frac {\left (9 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{8 (1-a x)^{5/2} (1+a x)^{5/2}}+\frac {\left (2 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}-\frac {\left (9 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \text {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}}+\frac {\left (2 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}} \\ & = -\frac {7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}+\frac {2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \arcsin (a x)}{(1-a x)^{5/2} (1+a x)^{5/2}}-\frac {9 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.46 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (6-16 a x-3 a^2 x^2+64 a^3 x^3+24 a^4 x^4\right )+27 a^4 x^4 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )-48 a^4 x^4 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{24 a^4 x^3 \sqrt {-1+a^2 x^2}} \]

[In]

Integrate[(c - c/(a^2*x^2))^(5/2)/E^(2*ArcCoth[a*x]),x]

[Out]

(c^2*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(6 - 16*a*x - 3*a^2*x^2 + 64*a^3*x^3 + 24*a^4*x^4) + 27*a^4*x^4
*ArcTan[1/Sqrt[-1 + a^2*x^2]] - 48*a^4*x^4*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(24*a^4*x^3*Sqrt[-1 + a^2*x^2])

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.80

method	result	size
risch	\(\frac {\left (64 a^{5} x^{5}-3 a^{4} x^{4}-80 a^{3} x^{3}+9 a^{2} x^{2}+16 a x -6\right ) c^{2} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{24 x^{3} a^{4} \left (a^{2} x^{2}-1\right )}+\frac {\left (-\frac {2 a^{5} \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}+\frac {9 a^{4} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{8 \sqrt {-c}}+\frac {a^{4} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c}\right ) c^{2} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {c \left (a^{2} x^{2}-1\right )}}{a^{4} \left (a^{2} x^{2}-1\right )}\)	\(234\)
default	\(-\frac {{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {5}{2}} x \left (-80 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{7} c \,x^{5}+80 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} a^{7} x^{3}-48 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{6} c \,x^{4}-27 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} a^{6} c \,x^{4}+60 \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^{2} x^{5}+75 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} a^{6} x^{2}+100 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{5} c^{2} x^{5}-80 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} a^{5} x +45 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{4} c^{2} x^{4}-90 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} c^{3} x^{5}-150 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c^{3} x^{5}+30 a^{4} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}+150 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {7}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{4}+90 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {7}{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^{4}-135 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c^{3} x^{4}-135 \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^{4} x^{4}\right )}{120 a^{2} \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} c}\)	\(625\)

[In]

int((c-c/a^2/x^2)^(5/2)*(a*x-1)/(a*x+1),x,method=_RETURNVERBOSE)

[Out]

1/24*(64*a^5*x^5-3*a^4*x^4-80*a^3*x^3+9*a^2*x^2+16*a*x-6)/x^3*c^2/a^4*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1
)+(-2*a^5*ln(a^2*c*x/(a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)+9/8*a^4/(-c)^(1/2)*ln((-2*c+2*(-c)^(1/2)
*(a^2*c*x^2-c)^(1/2))/x)+a^4/c*(c*(a^2*x^2-1))^(1/2))*c^2/a^4*(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 = 394, normalized size of antiderivative = 1.34 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\left [\frac {96 \, a^{3} \sqrt {-c} c^{2} x^{3} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 27 \, a^{3} \sqrt {-c} c^{2} x^{3} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (24 \, a^{4} c^{2} x^{4} + 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, a^{4} x^{3}}, \frac {27 \, a^{3} c^{\frac {5}{2}} x^{3} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 24 \, a^{3} c^{\frac {5}{2}} x^{3} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (24 \, a^{4} c^{2} x^{4} + 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, a^{4} x^{3}}\right ] \]

[In]

integrate((c-c/a^2/x^2)^(5/2)*(a*x-1)/(a*x+1),x, algorithm="fricas")

[Out]

[1/48*(96*a^3*sqrt(-c)*c^2*x^3*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 27*a
^3*sqrt(-c)*c^2*x^3*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*(24*a^4*c
^2*x^4 + 64*a^3*c^2*x^3 - 3*a^2*c^2*x^2 - 16*a*c^2*x + 6*c^2)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3), 1/24
*(27*a^3*c^(5/2)*x^3*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 24*a^3*c^(5/2)*x^3*
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) + (24*a^4*c^2*x^4 + 64*a^3*c^2*x^3 -
3*a^2*c^2*x^2 - 16*a*c^2*x + 6*c^2)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3)]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.97 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.71 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=c^{2} \left (\begin {cases} \frac {\sqrt {c} \sqrt {a^{2} x^{2} - 1}}{a} - \frac {i \sqrt {c} \log {\left (a x \right )}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} + \frac {\sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} - \frac {i \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )}}{a} & \text {otherwise} \end {cases}\right ) - \frac {2 c^{2} \left (\begin {cases} - \frac {a \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {\sqrt {c}}{a x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i a \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - i \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {i \sqrt {c}}{a x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a} + \frac {2 c^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {a^{2} \left (c - \frac {c}{a^{2} x^{2}}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right )}{a^{3}} - \frac {c^{2} \left (\begin {cases} \frac {i a^{3} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {i a^{2} \sqrt {c}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 i \sqrt {c}}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {i \sqrt {c}}{4 a^{2} x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {a^{3} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {a^{2} \sqrt {c}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 \sqrt {c}}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {\sqrt {c}}{4 a^{2} x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{4}} \]

[In]

integrate((c-c/a**2/x**2)**(5/2)*(a*x-1)/(a*x+1),x)

[Out]

c**2*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**2*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 + 2*c**2*Piecewise((0, Eq(c, 0)), (a**2*(c
- c/(a**2*x**2))**(3/2)/(3*c), True))/a**3 - c**2*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*sqrt(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

Maxima [F]

\[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\int { \frac {{\left (a x - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{a x + 1} \,d x } \]

[In]

integrate((c-c/a^2/x^2)^(5/2)*(a*x-1)/(a*x+1),x, algorithm="maxima")

[Out]

integrate((a*x - 1)*(c - c/(a^2*x^2))^(5/2)/(a*x + 1), x)

Giac [A] (verification not implemented)

none

Time = 1.98 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.42 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=-\frac {1}{12} \, {\left (\frac {27 \, c^{\frac {5}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {24 \, c^{\frac {5}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a {\left | a \right |}} - \frac {12 \, \sqrt {a^{2} c x^{2} - c} c^{2} \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{3} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 96 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 192 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{5} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 160 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 64 \, a c^{\frac {13}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{4} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]

[In]

integrate((c-c/a^2/x^2)^(5/2)*(a*x-1)/(a*x+1),x, algorithm="giac")

[Out]

-1/12*(27*c^(5/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x)/a^2 - 24*c^(5/2)*log(abs(-sqrt
(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/(a*abs(a)) - 12*sqrt(a^2*c*x^2 - c)*c^2*sgn(x)/a^2 - (3*(sqrt(a^2*c)*
x - sqrt(a^2*c*x^2 - c))^7*c^3*abs(a)*sgn(x) + 96*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^6*a*c^(7/2)*sgn(x) - 2
1*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^5*c^4*abs(a)*sgn(x) + 192*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a*c^
(9/2)*sgn(x) + 21*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*c^5*abs(a)*sgn(x) + 160*(sqrt(a^2*c)*x - sqrt(a^2*c*
x^2 - c))^2*a*c^(11/2)*sgn(x) - 3*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))*c^6*abs(a)*sgn(x) + 64*a*c^(13/2)*sgn(
x))/(((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^4*a^2*abs(a)))*abs(a)

Mupad [F(-1)]

Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]

[In]

int(((c - c/(a^2*x^2))^(5/2)*(a*x - 1))/(a*x + 1),x)

[Out]

int(((c - c/(a^2*x^2))^(5/2)*(a*x - 1))/(a*x + 1), x)