\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^2} \, dx\) [827]

Optimal result

Integrand size = 22, antiderivative size = 181 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {6 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {9 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {24 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c^2 \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6329, 105, 21, 101, 157, 12, 94, 214} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^2}+\frac {x \sqrt {1-\frac {1}{a x}}}{c^2 \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {24 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \sqrt {\frac {1}{a x}+1}}+\frac {9 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {6 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (\frac {1}{a x}+1\right )^{5/2}} \]

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^2),x]

[Out]

(6*Sqrt[1 - 1/(a*x)])/(5*a*c^2*(1 + 1/(a*x))^(5/2)) + (9*Sqrt[1 - 1/(a*x)])/(5*a*c^2*(1 + 1/(a*x))^(3/2)) + (2
4*Sqrt[1 - 1/(a*x)])/(5*a*c^2*Sqrt[1 + 1/(a*x)]) + (Sqrt[1 - 1/(a*x)]*x)/(c^2*(1 + 1/(a*x))^(5/2)) - (3*ArcTan
h[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 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 157

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 + 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*
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]

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 6329

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^2} \\ & = \frac {\sqrt {1-\frac {1}{a x}} x}{c^2 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {\frac {3}{a}-\frac {3 x}{a^2}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^2} \\ & = \frac {\sqrt {1-\frac {1}{a x}} x}{c^2 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}}}{x \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{a c^2} \\ & = \frac {6 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c^2 \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {6 \text {Subst}\left (\int \frac {-\frac {5}{2}+\frac {2 x}{a}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 a c^2} \\ & = \frac {6 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {9 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c^2 \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {2 \text {Subst}\left (\int \frac {-\frac {15}{2 a}+\frac {9 x}{2 a^2}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{5 c^2} \\ & = \frac {6 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {9 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {24 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c^2 \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {(2 a) \text {Subst}\left (\int -\frac {15}{2 a^2 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 c^2} \\ & = \frac {6 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {9 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {24 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c^2 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c^2} \\ & = \frac {6 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {9 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {24 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c^2 \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {3 \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a^2 c^2} \\ & = \frac {6 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {9 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {24 \sqrt {1-\frac {1}{a x}}}{5 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c^2 \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (24+57 a x+39 a^2 x^2+5 a^3 x^3\right )}{5 (1+a x)^3}-3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^2} \]

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^2),x]

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(24 + 57*a*x + 39*a^2*x^2 + 5*a^3*x^3))/(5*(1 + a*x)^3) - 3*Log[(1 + Sqrt[1 - 1/(a
^2*x^2)])*x])/(a*c^2)

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.14

method	result	size
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{2}}+\frac {\left (-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{5 a^{8} \left (x +\frac {1}{a}\right )^{3}}-\frac {6 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{5 a^{7} \left (x +\frac {1}{a}\right )^{2}}+\frac {24 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{5 a^{6} \left (x +\frac {1}{a}\right )}\right ) a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{2} \left (a x -1\right )}\)	\(207\)
default	\(-\frac {\left (120 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-125 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}+480 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+85 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+720 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+148 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -750 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+480 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +67 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +120 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )-125 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{40 a \sqrt {a^{2}}\, \left (a x +1\right )^{2} c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\)	\(438\)

[In]

int(((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {15 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (5 \, a^{3} x^{3} + 39 \, a^{2} x^{2} + 57 \, a x + 24\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^2,x, algorithm="fricas")

[Out]

-1/5*(15*(a^2*x^2 + 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*(a^2*x^2 + 2*a*x + 1)*log(sqrt((a*x - 1
)/(a*x + 1)) - 1) - (5*a^3*x^3 + 39*a^2*x^2 + 57*a*x + 24)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c^2*x^2 + 2*a^2*c^2
*x + a*c^2)

Sympy [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {a^{4} \left (\int \left (- \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\right )\, dx + \int \frac {a x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\, dx\right )}{c^{2}} \]

[In]

integrate(((a*x-1)/(a*x+1))**(3/2)/(c-c/a**2/x**2)**2,x)

[Out]

a**4*(Integral(-x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*
x + 1), x) + Integral(a*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x
**2 + a*x + 1), x))/c**2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {1}{20} \, a {\left (\frac {40 \, \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2} c^{2}}{a x + 1} - a^{2} c^{2}} - \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 10 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 85 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2}} + \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \]

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^2,x, algorithm="maxima")

[Out]

-1/20*a*(40*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)*a^2*c^2/(a*x + 1) - a^2*c^2) - (((a*x - 1)/(a*x + 1))^(5/2) +
10*((a*x - 1)/(a*x + 1))^(3/2) + 85*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^2) + 60*log(sqrt((a*x - 1)/(a*x + 1)) +
1)/(a^2*c^2) - 60*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^2))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.33 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {3 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{c^{2} {\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} \mathrm {sgn}\left (a x + 1\right )}{a c^{2}} \]

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^2,x, algorithm="giac")

[Out]

3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/(c^2*abs(a)) + sqrt(a^2*x^2 - 1)*sgn(a*x + 1)/(a*c^2)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c^2-\frac {a\,c^2\,\left (a\,x-1\right )}{a\,x+1}}+\frac {17\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,a\,c^2}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{2\,a\,c^2}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{20\,a\,c^2}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c^2} \]

[In]

int(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a^2*x^2))^2,x)

[Out]

(2*((a*x - 1)/(a*x + 1))^(1/2))/(a*c^2 - (a*c^2*(a*x - 1))/(a*x + 1)) + (17*((a*x - 1)/(a*x + 1))^(1/2))/(4*a*
c^2) + ((a*x - 1)/(a*x + 1))^(3/2)/(2*a*c^2) + ((a*x - 1)/(a*x + 1))^(5/2)/(20*a*c^2) + (atan(((a*x - 1)/(a*x
+ 1))^(1/2)*1i)*6i)/(a*c^2)