\(\int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx\) [49]

Optimal result

Integrand size = 12, antiderivative size = 103 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=\frac {2}{3} a^4 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {3 a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{8 x}+\frac {3}{8} a^4 \csc ^{-1}(a x) \] Output:

2/3*a^4*(1-1/a^2/x^2)^(1/2)-1/4*a*(1-1/a^2/x^2)^(1/2)/x^3+1/3*a^2*(1-1/a^2 
/x^2)^(1/2)/x^2-3/8*a^3*(1-1/a^2/x^2)^(1/2)/x+3/8*a^4*arccsc(a*x)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.57 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=\frac {1}{24} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (-6+8 a x-9 a^2 x^2+16 a^3 x^3\right )}{x^3}+9 a^3 \arcsin \left (\frac {1}{a x}\right )\right ) \] Input:

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

Output:

(a*((Sqrt[1 - 1/(a^2*x^2)]*(-6 + 8*a*x - 9*a^2*x^2 + 16*a^3*x^3))/x^3 + 9* 
a^3*ArcSin[1/(a*x)]))/24
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.17, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6719, 533, 25, 27, 533, 25, 27, 533, 25, 27, 455, 223}

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.

\(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx\)

\(\Big \downarrow \) 6719

\(\displaystyle -\int \frac {1-\frac {1}{a x}}{\sqrt {1-\frac {1}{a^2 x^2}} x^3}d\frac {1}{x}\)

\(\Big \downarrow \) 533

\(\displaystyle -\frac {1}{4} a^2 \int -\frac {3 a-\frac {4}{x}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2}d\frac {1}{x}-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} a^2 \int \frac {3 a-\frac {4}{x}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2}d\frac {1}{x}-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int \frac {3 a-\frac {4}{x}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}d\frac {1}{x}-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} a^2 \int -\frac {8 a-\frac {9}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}} x}d\frac {1}{x}+\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a^2 \int \frac {8 a-\frac {9}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}} x}d\frac {1}{x}\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a \int \frac {8 a-\frac {9}{x}}{\sqrt {1-\frac {1}{a^2 x^2}} x}d\frac {1}{x}\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a \left (\frac {1}{2} a^2 \int -\frac {9 a-\frac {16}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {9 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a \left (\frac {9 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a^2 \int \frac {9 a-\frac {16}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a \left (\frac {9 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a \int \frac {9 a-\frac {16}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a \left (\frac {9 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a \left (9 a \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+16 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )\right )\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{4} \left (\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a \left (\frac {9 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a \left (9 a^2 \arcsin \left (\frac {1}{a x}\right )+16 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )\right )\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}\)

Input:

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

Output:

-1/4*(a*Sqrt[1 - 1/(a^2*x^2)])/x^3 + ((4*a^2*Sqrt[1 - 1/(a^2*x^2)])/(3*x^2 
) - (a*((9*a^2*Sqrt[1 - 1/(a^2*x^2)])/(2*x) - (a*(16*a^2*Sqrt[1 - 1/(a^2*x 
^2)] + 9*a^2*ArcSin[1/(a*x)]))/2))/3)/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 223

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 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 533

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

rule 6719

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + 
x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x 
, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.98

method	result	size
risch	\(\frac {\left (a x +1\right ) \left (16 a^{3} x^{3}-9 a^{2} x^{2}+8 a x -6\right ) \sqrt {\frac {a x -1}{a x +1}}}{24 x^{4}}+\frac {3 a^{4} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{8 \left (a x -1\right )}\)	\(101\)
default	\(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-24 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{5} x^{5}+24 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{3} x^{3}-9 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}-9 a^{4} \sqrt {a^{2}}\, x^{4} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+24 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}-24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-15 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}+8 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{24 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} \sqrt {a^{2}}}\)	\(308\)

Input:

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

Output:

1/24*(a*x+1)*(16*a^3*x^3-9*a^2*x^2+8*a*x-6)/x^4*((a*x-1)/(a*x+1))^(1/2)+3/ 
8*a^4*arctan(1/(a^2*x^2-1)^(1/2))*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1) 
)^(1/2)/(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=-\frac {18 \, a^{4} x^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (16 \, a^{4} x^{4} + 7 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x - 6\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, x^{4}} \] Input:

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

Output:

-1/24*(18*a^4*x^4*arctan(sqrt((a*x - 1)/(a*x + 1))) - (16*a^4*x^4 + 7*a^3* 
x^3 - a^2*x^2 + 2*a*x - 6)*sqrt((a*x - 1)/(a*x + 1)))/x^4
 

Sympy [F]

\[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=\int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{x^{5}}\, dx \] Input:

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

Output:

Integral(sqrt((a*x - 1)/(a*x + 1))/x**5, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (85) = 170\).

Time = 0.11 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.68 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=-\frac {1}{12} \, {\left (9 \, a^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - \frac {39 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 31 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 49 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 9 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )}}{a x + 1} + \frac {6 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 1}\right )} a \] Input:

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

Output:

-1/12*(9*a^3*arctan(sqrt((a*x - 1)/(a*x + 1))) - (39*a^3*((a*x - 1)/(a*x + 
 1))^(7/2) + 31*a^3*((a*x - 1)/(a*x + 1))^(5/2) + 49*a^3*((a*x - 1)/(a*x + 
 1))^(3/2) + 9*a^3*sqrt((a*x - 1)/(a*x + 1)))/(4*(a*x - 1)/(a*x + 1) + 6*( 
a*x - 1)^2/(a*x + 1)^2 + 4*(a*x - 1)^3/(a*x + 1)^3 + (a*x - 1)^4/(a*x + 1) 
^4 + 1))*a
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (85) = 170\).

Time = 0.13 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.50 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=-\frac {3}{4} \, a^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right ) + \frac {9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{7} a^{4} \mathrm {sgn}\left (a x + 1\right ) + 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} a^{4} \mathrm {sgn}\left (a x + 1\right ) + 48 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} a^{4} \mathrm {sgn}\left (a x + 1\right ) + 64 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} a^{4} \mathrm {sgn}\left (a x + 1\right ) + 16 \, a^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )}{12 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{4}} \] Input:

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

Output:

-3/4*a^4*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1) + 1/12*(9*(x*a 
bs(a) - sqrt(a^2*x^2 - 1))^7*a^4*sgn(a*x + 1) + 33*(x*abs(a) - sqrt(a^2*x^ 
2 - 1))^5*a^4*sgn(a*x + 1) + 48*(x*abs(a) - sqrt(a^2*x^2 - 1))^4*a^3*abs(a 
)*sgn(a*x + 1) - 33*(x*abs(a) - sqrt(a^2*x^2 - 1))^3*a^4*sgn(a*x + 1) + 64 
*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a^3*abs(a)*sgn(a*x + 1) - 9*(x*abs(a) - 
sqrt(a^2*x^2 - 1))*a^4*sgn(a*x + 1) + 16*a^3*abs(a)*sgn(a*x + 1))/((x*abs( 
a) - sqrt(a^2*x^2 - 1))^2 + 1)^4
 

Mupad [B] (verification not implemented)

Time = 23.71 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.25 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=\frac {2\,a^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3}-\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,x^4}-\frac {3\,a^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4}-\frac {a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{24\,x^2}+\frac {7\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{24\,x}+\frac {a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{12\,x^3} \] Input:

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

Output:

(2*a^4*((a*x - 1)/(a*x + 1))^(1/2))/3 - ((a*x - 1)/(a*x + 1))^(1/2)/(4*x^4 
) - (3*a^4*atan(((a*x - 1)/(a*x + 1))^(1/2)))/4 - (a^2*((a*x - 1)/(a*x + 1 
))^(1/2))/(24*x^2) + (7*a^3*((a*x - 1)/(a*x + 1))^(1/2))/(24*x) + (a*((a*x 
 - 1)/(a*x + 1))^(1/2))/(12*x^3)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.26 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=\frac {-18 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{4} x^{4}+18 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{4} x^{4}+16 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}-9 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+8 \sqrt {a x +1}\, \sqrt {a x -1}\, a x -6 \sqrt {a x +1}\, \sqrt {a x -1}-16 a^{4} x^{4}}{24 x^{4}} \] Input:

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

Output:

( - 18*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**4*x**4 + 18*atan(sqrt(a* 
x - 1) + sqrt(a*x + 1) + 1)*a**4*x**4 + 16*sqrt(a*x + 1)*sqrt(a*x - 1)*a** 
3*x**3 - 9*sqrt(a*x + 1)*sqrt(a*x - 1)*a**2*x**2 + 8*sqrt(a*x + 1)*sqrt(a* 
x - 1)*a*x - 6*sqrt(a*x + 1)*sqrt(a*x - 1) - 16*a**4*x**4)/(24*x**4)