\(\int \frac {e^{\coth ^{-1}(a x)}}{(c-\frac {c}{a x})^4} \, dx\) [401]

Optimal result

Integrand size = 20, antiderivative size = 188 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {824 \sqrt {1-\frac {1}{a^2 x^2}}}{105 c^4 \left (a-\frac {1}{x}\right )}-\frac {2 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x}{7 c^4 \left (a-\frac {1}{x}\right )^4}-\frac {17 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x}{35 c^4 \left (a-\frac {1}{x}\right )^3}-\frac {113 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}{105 c^4 \left (a-\frac {1}{x}\right )^2}+\frac {299 a \sqrt {1-\frac {1}{a^2 x^2}} x}{105 c^4 \left (a-\frac {1}{x}\right )}+\frac {5 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \] Output:

-824/105*(1-1/a^2/x^2)^(1/2)/c^4/(a-1/x)-2/7*a^4*(1-1/a^2/x^2)^(1/2)*x/c^4 
/(a-1/x)^4-17/35*a^3*(1-1/a^2/x^2)^(1/2)*x/c^4/(a-1/x)^3-113/105*a^2*(1-1/ 
a^2/x^2)^(1/2)*x/c^4/(a-1/x)^2+299/105*a*(1-1/a^2/x^2)^(1/2)*x/c^4/(a-1/x) 
+5*arctanh((1-1/a^2/x^2)^(1/2))/a/c^4
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.60 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {824-1947 a x+485 a^2 x^2+1812 a^3 x^3-1339 a^4 x^4+105 a^5 x^5+525 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^3} \] Input:

Integrate[E^ArcCoth[a*x]/(c - c/(a*x))^4,x]
 

Output:

(824 - 1947*a*x + 485*a^2*x^2 + 1812*a^3*x^3 - 1339*a^4*x^4 + 105*a^5*x^5 
+ 525*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)] 
])/(105*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3)
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.95, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6731, 27, 570, 532, 25, 2336, 25, 2336, 25, 2336, 27, 534, 243, 73, 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.

\(\displaystyle \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx\)

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 570

\(\displaystyle -\frac {\int \frac {\left (a+\frac {1}{x}\right )^5 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{9/2}}d\frac {1}{x}}{a^5 c^4}\)

\(\Big \downarrow \) 532

\(\displaystyle -\frac {\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {1}{7} \int -\frac {\left (7 a^5+\frac {35 a^4}{x}+\frac {61 a^3}{x^2}-\frac {7 a^2}{x^3}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}}d\frac {1}{x}}{a^5 c^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{7} \int \frac {\left (7 a^5+\frac {35 a^4}{x}+\frac {61 a^3}{x^2}-\frac {7 a^2}{x^3}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}}d\frac {1}{x}+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^5 c^4}\)

\(\Big \downarrow \) 2336

\(\displaystyle -\frac {\frac {1}{7} \left (\frac {4 a^3 \left (7 a+\frac {17}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} \int -\frac {\left (35 a^5+\frac {175 a^4}{x}+\frac {272 a^3}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^5 c^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \int \frac {\left (35 a^5+\frac {175 a^4}{x}+\frac {272 a^3}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}+\frac {4 a^3 \left (7 a+\frac {17}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^5 c^4}\)

\(\Big \downarrow \) 2336

\(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {a^3 \left (175 a+\frac {307}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} \int -\frac {\left (105 a^5+\frac {525 a^4}{x}+\frac {614 a^3}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}\right )+\frac {4 a^3 \left (7 a+\frac {17}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^5 c^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (105 a^5+\frac {525 a^4}{x}+\frac {614 a^3}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {a^3 \left (175 a+\frac {307}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {4 a^3 \left (7 a+\frac {17}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^5 c^4}\)

\(\Big \downarrow \) 2336

\(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (525 a+\frac {719}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}-\int -\frac {105 a^4 \left (a+\frac {5}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {a^3 \left (175 a+\frac {307}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {4 a^3 \left (7 a+\frac {17}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^5 c^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 a^4 \int \frac {\left (a+\frac {5}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {a^3 \left (525 a+\frac {719}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {a^3 \left (175 a+\frac {307}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {4 a^3 \left (7 a+\frac {17}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^5 c^4}\)

\(\Big \downarrow \) 534

\(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 a^4 \left (5 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^3 \left (525 a+\frac {719}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {a^3 \left (175 a+\frac {307}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {4 a^3 \left (7 a+\frac {17}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^5 c^4}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 a^4 \left (\frac {5}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^3 \left (525 a+\frac {719}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {a^3 \left (175 a+\frac {307}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {4 a^3 \left (7 a+\frac {17}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^5 c^4}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (105 a^4 \left (-5 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^3 \left (525 a+\frac {719}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {a^3 \left (175 a+\frac {307}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {4 a^3 \left (7 a+\frac {17}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^5 c^4}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {16 a^3 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {1}{7} \left (\frac {4 a^3 \left (7 a+\frac {17}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {1}{5} \left (\frac {a^3 \left (175 a+\frac {307}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {1}{3} \left (105 a^4 \left (-5 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^3 \left (525 a+\frac {719}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )\right )\right )}{a^5 c^4}\)

Input:

Int[E^ArcCoth[a*x]/(c - c/(a*x))^4,x]
 

Output:

-(((16*a^3*(a + x^(-1)))/(7*(1 - 1/(a^2*x^2))^(7/2)) + ((4*a^3*(7*a + 17/x 
))/(5*(1 - 1/(a^2*x^2))^(5/2)) + ((a^3*(175*a + 307/x))/(3*(1 - 1/(a^2*x^2 
))^(3/2)) + ((a^3*(525*a + 719/x))/Sqrt[1 - 1/(a^2*x^2)] + 105*a^4*(-(a*Sq 
rt[1 - 1/(a^2*x^2)]*x) - 5*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/3)/5)/7)/(a^5* 
c^4))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 532

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[x^m 
*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), 
x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, 
 -1] && IntegerQ[2*p]
 

rule 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 570

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

rule 2336

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema 
inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) 
^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* 
b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 6731

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.41

method	result	size
risch	\(\frac {a x -1}{a \,c^{4} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {5 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}-\frac {1024 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{105 a^{6} \left (x -\frac {1}{a}\right )}-\frac {2 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{7 a^{9} \left (x -\frac {1}{a}\right )^{4}}-\frac {57 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{35 a^{8} \left (x -\frac {1}{a}\right )^{3}}-\frac {446 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{105 a^{7} \left (x -\frac {1}{a}\right )^{2}}\right ) a^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(265\)
default	\(\frac {525 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-2625 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-420 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{3} x^{3}-2625 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}+5250 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+1076 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+5250 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-5250 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-970 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -5250 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+2625 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +299 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+2625 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -525 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )-525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{105 a \sqrt {a^{2}}\, \left (a x -1\right )^{4} c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}\)	\(523\)

Input:

int(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^4,x,method=_RETURNVERBOSE)
 

Output:

1/a*(a*x-1)/c^4/((a*x-1)/(a*x+1))^(1/2)+(5/a^4*ln(a^2*x/(a^2)^(1/2)+(a^2*x 
^2-1)^(1/2))/(a^2)^(1/2)-1024/105/a^6/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^ 
(1/2)-2/7/a^9/(x-1/a)^4*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-57/35/a^8/(x-1/a 
)^3*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-446/105/a^7/(x-1/a)^2*((x-1/a)^2*a^2 
+2*a*(x-1/a))^(1/2))*a^4/c^4/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/ 
2)/(a*x+1)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.09 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {525 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 525 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (105 \, a^{5} x^{5} - 1339 \, a^{4} x^{4} + 1812 \, a^{3} x^{3} + 485 \, a^{2} x^{2} - 1947 \, a x + 824\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^4,x, algorithm="fricas")
 

Output:

1/105*(525*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(sqrt((a*x - 1 
)/(a*x + 1)) + 1) - 525*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log( 
sqrt((a*x - 1)/(a*x + 1)) - 1) + (105*a^5*x^5 - 1339*a^4*x^4 + 1812*a^3*x^ 
3 + 485*a^2*x^2 - 1947*a*x + 824)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^4*x^4 
- 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)
 

Sympy [F]

\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {a^{4} \int \frac {x^{4}}{a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 6 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{4}} \] Input:

integrate(1/((a*x-1)/(a*x+1))**(1/2)/(c-c/a/x)**4,x)
 

Output:

a**4*Integral(x**4/(a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 4*a**3*x 
**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + 6*a**2*x**2*sqrt(a*x/(a*x + 1) - 1 
/(a*x + 1)) - 4*a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + sqrt(a*x/(a*x + 1) 
 - 1/(a*x + 1))), x)/c**4
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {1}{420} \, a {\left (\frac {\frac {111 \, {\left (a x - 1\right )}}{a x + 1} + \frac {469 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2765 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {4200 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 15}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} + \frac {2100 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {2100 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^4,x, algorithm="maxima")
 

Output:

1/420*a*((111*(a*x - 1)/(a*x + 1) + 469*(a*x - 1)^2/(a*x + 1)^2 + 2765*(a* 
x - 1)^3/(a*x + 1)^3 - 4200*(a*x - 1)^4/(a*x + 1)^4 + 15)/(a^2*c^4*((a*x - 
 1)/(a*x + 1))^(9/2) - a^2*c^4*((a*x - 1)/(a*x + 1))^(7/2)) + 2100*log(sqr 
t((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 2100*log(sqrt((a*x - 1)/(a*x + 1)) 
 - 1)/(a^2*c^4))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^4,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 13.58 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.73 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {10\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^4}-\frac {\frac {67\,{\left (a\,x-1\right )}^2}{15\,{\left (a\,x+1\right )}^2}+\frac {79\,{\left (a\,x-1\right )}^3}{3\,{\left (a\,x+1\right )}^3}-\frac {40\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {37\,\left (a\,x-1\right )}{35\,\left (a\,x+1\right )}+\frac {1}{7}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}-4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}} \] Input:

int(1/((c - c/(a*x))^4*((a*x - 1)/(a*x + 1))^(1/2)),x)
 

Output:

(10*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^4) - ((67*(a*x - 1)^2)/(15*(a 
*x + 1)^2) + (79*(a*x - 1)^3)/(3*(a*x + 1)^3) - (40*(a*x - 1)^4)/(a*x + 1) 
^4 + (37*(a*x - 1))/(35*(a*x + 1)) + 1/7)/(4*a*c^4*((a*x - 1)/(a*x + 1))^( 
7/2) - 4*a*c^4*((a*x - 1)/(a*x + 1))^(9/2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.43 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {1050 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{3} x^{3}-3150 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{2} x^{2}+3150 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -1050 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+754 \sqrt {a x -1}\, a^{3} x^{3}-2262 \sqrt {a x -1}\, a^{2} x^{2}+2262 \sqrt {a x -1}\, a x -754 \sqrt {a x -1}+105 \sqrt {a x +1}\, a^{4} x^{4}-1444 \sqrt {a x +1}\, a^{3} x^{3}+3256 \sqrt {a x +1}\, a^{2} x^{2}-2771 \sqrt {a x +1}\, a x +824 \sqrt {a x +1}}{105 \sqrt {a x -1}\, a \,c^{4} \left (a^{3} x^{3}-3 a^{2} x^{2}+3 a x -1\right )} \] Input:

int(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^4,x)
 

Output:

(1050*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**3*x**3 
 - 3150*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**2*x* 
*2 + 3150*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x - 
 1050*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 754*sqr 
t(a*x - 1)*a**3*x**3 - 2262*sqrt(a*x - 1)*a**2*x**2 + 2262*sqrt(a*x - 1)*a 
*x - 754*sqrt(a*x - 1) + 105*sqrt(a*x + 1)*a**4*x**4 - 1444*sqrt(a*x + 1)* 
a**3*x**3 + 3256*sqrt(a*x + 1)*a**2*x**2 - 2771*sqrt(a*x + 1)*a*x + 824*sq 
rt(a*x + 1))/(105*sqrt(a*x - 1)*a*c**4*(a**3*x**3 - 3*a**2*x**2 + 3*a*x - 
1))