3.1.56 \(\int \frac {e^{-3 \coth ^{-1}(a x)}}{x^3} \, dx\) [56]

3.1.56.1 Optimal result

Integrand size = 12, antiderivative size = 87 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {9}{2} a^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a+\frac {1}{x}\right )}-\frac {9}{2} a^2 \csc ^{-1}(a x) \]

output

-a^5*(1-1/a^2/x^2)^(5/2)/(a+1/x)^3-3/2*a^3*(1-1/a^2/x^2)^(3/2)/(a+1/x)-9/2 
*a^2*arccsc(a*x)-9/2*a^2*(1-1/a^2/x^2)^(1/2)
 

3.1.56.2 Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{2} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (1-5 a x-14 a^2 x^2\right )}{x (1+a x)}-9 a \arcsin \left (\frac {1}{a x}\right )\right ) \]

input

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

output

(a*((Sqrt[1 - 1/(a^2*x^2)]*(1 - 5*a*x - 14*a^2*x^2))/(x*(1 + a*x)) - 9*a*A 
rcSin[1/(a*x)]))/2
 

3.1.56.3 Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6719, 2164, 27, 2027, 2164, 27, 563, 25, 2346, 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.

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

\(\Big \downarrow \) 6719

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

\(\Big \downarrow \) 2164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2027

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

\(\Big \downarrow \) 2164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 563

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2346

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 223

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

input

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

output

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

3.1.56.3.1 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 563

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

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

rule 2164

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> Simp[d*e   Int[(d + e*x)^(m - 1)*PolynomialQuotient[Pq, a*e + b*d*x, x]* 
(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] 
 && EqQ[b*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + b*d*x, x], 0 
]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

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]
 

3.1.56.4 Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.37

method	result	size
risch	\(-\frac {\left (a x +1\right ) \left (6 a x -1\right ) \sqrt {\frac {a x -1}{a x +1}}}{2 x^{2}}+\frac {\left (-\frac {9 a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}-\frac {4 a \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\)	\(119\)
default	\(\frac {\left (-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-21 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}-9 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+12 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+11 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-24 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-18 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+4 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+12 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-12 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+4 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-9 a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 \sqrt {a^{2}}\, x^{2} \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}\)	\(642\)

input

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

output

-1/2*(a*x+1)*(6*a*x-1)/x^2*((a*x-1)/(a*x+1))^(1/2)+(-9/2*a^2*arctan(1/(a^2 
*x^2-1)^(1/2))-4*a/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2))/(a*x-1)*((a* 
x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)
 

3.1.56.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^3} \, dx=\frac {18 \, a^{2} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (14 \, a^{2} x^{2} + 5 \, a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, x^{2}} \]

input

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

output

1/2*(18*a^2*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) - (14*a^2*x^2 + 5*a*x - 
1)*sqrt((a*x - 1)/(a*x + 1)))/x^2
 

3.1.56.6 Sympy [F]

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

input

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

output

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

3.1.56.7 Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.29 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^3} \, dx={\left (9 \, a \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 4 \, a \sqrt {\frac {a x - 1}{a x + 1}} - \frac {7 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 5 \, a \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )}}{a x + 1} + \frac {{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}\right )} a \]

input

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

output

(9*a*arctan(sqrt((a*x - 1)/(a*x + 1))) - 4*a*sqrt((a*x - 1)/(a*x + 1)) - ( 
7*a*((a*x - 1)/(a*x + 1))^(3/2) + 5*a*sqrt((a*x - 1)/(a*x + 1)))/(2*(a*x - 
 1)/(a*x + 1) + (a*x - 1)^2/(a*x + 1)^2 + 1))*a
 

3.1.56.8 Giac [F]

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

input

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

output

undef
 

3.1.56.9 Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^3} \, dx=9\,a^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {5\,a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}+7\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {2\,\left (a\,x-1\right )}{a\,x+1}+1}-4\,a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}} \]

input

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

output

9*a^2*atan(((a*x - 1)/(a*x + 1))^(1/2)) - (5*a^2*((a*x - 1)/(a*x + 1))^(1/ 
2) + 7*a^2*((a*x - 1)/(a*x + 1))^(3/2))/((a*x - 1)^2/(a*x + 1)^2 + (2*(a*x 
 - 1))/(a*x + 1) + 1) - 4*a^2*((a*x - 1)/(a*x + 1))^(1/2)