\(\int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx\) [162]

Optimal result

Integrand size = 12, antiderivative size = 114 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}+\frac {2^{n/2} a^2 n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{2-n} \] Output:

1/2*a^2*(1-1/a/x)^(1-1/2*n)*(1+1/a/x)^(1+1/2*n)+2^(1/2*n)*a^2*n*(1-1/a/x)^ 
(1-1/2*n)*hypergeom([-1/2*n, 1-1/2*n],[2-1/2*n],1/2*(a-1/x)/a)/(2-n)
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.94 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {a^2 e^{n \coth ^{-1}(a x)} \left (-e^{2 \coth ^{-1}(a x)} n^2 \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-1+\frac {1}{a^2 x^2}+\frac {n}{a x}+n \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{2 (2+n)} \] Input:

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

Output:

-1/2*(a^2*E^(n*ArcCoth[a*x])*(-(E^(2*ArcCoth[a*x])*n^2*Hypergeometric2F1[1 
, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])]) + (2 + n)*(-1 + 1/(a^2*x^2) + n/ 
(a*x) + n*Hypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcCoth[a*x])])))/(2 + 
n)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6721, 90, 79}

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.

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

\(\Big \downarrow \) 6721

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {a^2 2^{n/2} n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{2-n}+\frac {1}{2} a^2 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}\)

Input:

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

Output:

(a^2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/2 + (2^(n/2)*a^2*n 
*(1 - 1/(a*x))^(1 - n/2)*Hypergeometric2F1[1 - n/2, -1/2*n, 2 - n/2, (a - 
x^(-1))/(2*a)])/(2 - n)
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 6721

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

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{x^{3}}d x\]

Input:

int(exp(n*arccoth(a*x))/x^3,x)
 

Output:

int(exp(n*arccoth(a*x))/x^3,x)
 

Fricas [F]

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

integrate(exp(n*arccoth(a*x))/x^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x^{3}}\, dx \] Input:

integrate(exp(n*acoth(a*x))/x**3,x)
 

Output:

Integral(exp(n*acoth(a*x))/x**3, x)
 

Maxima [F]

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

integrate(exp(n*arccoth(a*x))/x^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(exp(n*arccoth(a*x))/x^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x^3} \,d x \] Input:

int(exp(n*acoth(a*x))/x^3,x)
 

Output:

int(exp(n*acoth(a*x))/x^3, x)
 

Reduce [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=\frac {-e^{\mathit {acoth} \left (a x \right ) n}+\left (\int \frac {e^{\mathit {acoth} \left (a x \right ) n}}{a^{2} x^{4}-x^{2}}d x \right ) a n \,x^{2}}{2 x^{2}} \] Input:

int(exp(n*acoth(a*x))/x^3,x)
 

Output:

( - e**(acoth(a*x)*n) + int(e**(acoth(a*x)*n)/(a**2*x**4 - x**2),x)*a*n*x* 
*2)/(2*x**2)