\(\int \frac {e^{-\coth ^{-1}(a x)}}{(c-\frac {c}{a x})^{5/2}} \, dx\) [469]

Optimal result

Integrand size = 24, antiderivative size = 219 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=-\frac {3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {3 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {9 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{2 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}} \]

[Out]

3*(1-1/a/x)^(5/2)*arctanh((1+1/a/x)^(1/2))/a/(c-c/a/x)^(5/2)-9/4*(1-1/a/x)^(5/2)*arctanh(1/2*(1+1/a/x)^(1/2)*2
^(1/2))/a/(c-c/a/x)^(5/2)*2^(1/2)-3/2*(1-1/a/x)^(5/2)*(1+1/a/x)^(1/2)/(a-1/x)/(c-c/a/x)^(5/2)+a*(1-1/a/x)^(5/2
)*x*(1+1/a/x)^(1/2)/(a-1/x)/(c-c/a/x)^(5/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6317, 6314, 105, 21, 101, 162, 65, 214, 212} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {3 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {9 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{2 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{5/2}}{\left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {3 \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{5/2}}{2 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}} \]

[In]

Int[1/(E^ArcCoth[a*x]*(c - c/(a*x))^(5/2)),x]

[Out]

(-3*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])/(2*(a - x^(-1))*(c - c/(a*x))^(5/2)) + (a*(1 - 1/(a*x))^(5/2)*Sqrt[
1 + 1/(a*x)]*x)/((a - x^(-1))*(c - c/(a*x))^(5/2)) + (3*(1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c
- c/(a*x))^(5/2)) - (9*(1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(2*Sqrt[2]*a*(c - c/(a*x))^(5/2
))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 101

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 + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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])

Rule 162

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 6314

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

Rule 6317

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-\frac {1}{a x}\right )^{5/2} \int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{5/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^2 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}} \\ & = \frac {a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {Subst}\left (\int \frac {-\frac {3}{2 a}-\frac {3 x}{2 a^2}}{x \left (1-\frac {x}{a}\right )^2 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}} \\ & = \frac {a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (3 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x \left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{2 a \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (3 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {-1-\frac {x}{2 a}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (9 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{4 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (3 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (3 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (9 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{2 a \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {3 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {9 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{2 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.56 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (2 a \sqrt {1+\frac {1}{a x}} x (-3+2 a x)+12 (-1+a x) \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )-9 \sqrt {2} (-1+a x) \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )\right )}{4 a c^2 \sqrt {c-\frac {c}{a x}} (-1+a x)} \]

[In]

Integrate[1/(E^ArcCoth[a*x]*(c - c/(a*x))^(5/2)),x]

[Out]

(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(-3 + 2*a*x) + 12*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]] - 9*Sqrt[2
]*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(4*a*c^2*Sqrt[c - c/(a*x)]*(-1 + a*x))

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.21

method	result	size
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (8 \sqrt {\left (a x +1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x -9 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x -12 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+12 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x -12 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+9 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{8 a^{\frac {3}{2}} c^{3} \left (a x -1\right )^{2} \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}}\)	\(264\)
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {3 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {9 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{8 a^{4} \sqrt {c}}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{2 a^{5} c \left (x -\frac {1}{a}\right )}\right ) a^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) a c x}}{c^{2} x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\)	\(268\)

[In]

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

[Out]

1/8*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x*(8*((a*x+1)*x)^(1/2)*a^(5/2)*(1/a)^(1/2)*x-9*a^(3/
2)*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(a*x-1))*x-12*((a*x+1)*x)^(1/2)*a^(3/2)*(1/a
)^(1/2)+12*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a^2*(1/a)^(1/2)*x-12*ln(1/2*(2*((a*x+1)*x)^(1
/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a*(1/a)^(1/2)+9*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(
a*x-1))*a^(1/2))/a^(3/2)/c^3/(a*x-1)^2/((a*x+1)*x)^(1/2)/(1/a)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 596, normalized size of antiderivative = 2.72 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [\frac {9 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 12 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (2 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}, \frac {9 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 12 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 4 \, {\left (2 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}\right ] \]

[In]

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

[Out]

[1/16*(9*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^3*
x^3 + 4*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3
*a*x - 1)) + 12*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sq
rt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(2*a^3*x^3 - a^2*x^2 - 3*a*x)*sqrt
((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3), 1/8*(9*sqrt(2)*(a^2*x^2 -
2*a*x + 1)*sqrt(-c)*arctan(2*sqrt(2)*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)
)/(3*a^2*c*x^2 - 2*a*c*x - c)) - 12*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x
- 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 4*(2*a^3*x^3 - a^2*x^2 - 3*a*x)*sqrt((a*
x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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