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

3.9.27.1 Optimal result

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

output

-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)
 

3.9.27.2 Mathematica [A] (verified)

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

input

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

output

((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)
 

3.9.27.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {6748, 114, 27, 35, 110, 27, 169, 27, 169, 27, 103, 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.

\(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx\)

\(\Big \downarrow \) 6748

\(\displaystyle -\frac {\int \frac {x^2}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}}{c^2}\)

\(\Big \downarrow \) 114

\(\displaystyle -\frac {-\int \frac {3 \left (a-\frac {1}{x}\right ) x}{a^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{5/2}}}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 \int \frac {\left (a-\frac {1}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}}{a^2}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{5/2}}}{c^2}\)

\(\Big \downarrow \) 35

\(\displaystyle -\frac {-\frac {3 \int \frac {\sqrt {1-\frac {1}{a x}} x}{\left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}}{a}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{5/2}}}{c^2}\)

\(\Big \downarrow \) 110

\(\displaystyle -\frac {-\frac {3 \left (\frac {2 \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {2}{5} \int -\frac {\left (5 a-\frac {4}{x}\right ) x}{2 a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}\right )}{a}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{5/2}}}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 \left (\frac {\int \frac {\left (5 a-\frac {4}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}}{5 a}+\frac {2 \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )}{a}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{5/2}}}{c^2}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {3 \left (\frac {\frac {1}{3} a \int \frac {3 \left (5 a-\frac {3}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {3 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{5 a}+\frac {2 \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )}{a}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{5/2}}}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 \left (\frac {\int \frac {\left (5 a-\frac {3}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {3 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{5 a}+\frac {2 \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )}{a}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{5/2}}}{c^2}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {3 \left (\frac {a \int \frac {5 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {3 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{5 a}+\frac {2 \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )}{a}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{5/2}}}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 \left (\frac {5 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {3 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{5 a}+\frac {2 \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )}{a}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{5/2}}}{c^2}\)

\(\Big \downarrow \) 103

\(\displaystyle -\frac {-\frac {3 \left (\frac {-5 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )+\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {3 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{5 a}+\frac {2 \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )}{a}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{5/2}}}{c^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {3 \left (\frac {-5 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {8 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {3 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{5 a}+\frac {2 \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )}{a}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{5/2}}}{c^2}\)

input

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

output

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

3.9.27.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 35

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

rule 103

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ 
))), x_] :> Simp[b*f   Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq 
rt[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 110

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

rule 114

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])
 

rule 169

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]
 

rule 221

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 6748

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> 
 Simp[-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] &&  !IntegerQ[ 
n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegersQ[2*p, p + n/2]
 

3.9.27.4 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	\(-\frac {\left (120 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-125 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}+480 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+85 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+720 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+148 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -750 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+480 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +67 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +120 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )-125 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{40 a \sqrt {a^{2}}\, \left (a x +1\right )^{2} c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\)	\(438\)

input

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

output

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)
 

3.9.27.5 Fricas [A] (verification not implemented)

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

input

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

output

-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)
 

3.9.27.6 Sympy [F]

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

input

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

output

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
 

3.9.27.7 Maxima [A] (verification not implemented)

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

input

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

output

-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*s 
qrt((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))
 

3.9.27.8 Giac [A] (verification not implemented)

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

input

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

output

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)
 

3.9.27.9 Mupad [B] (verification not implemented)

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

input

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

output

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