\(\int \frac {e^{n \coth ^{-1}(a x)}}{x} \, dx\) [160]

Optimal result

Integrand size = 12, antiderivative size = 127 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x} \, dx=-\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{n}+\frac {2^{1+\frac {n}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{n} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.12 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )+e^{2 \coth ^{-1}(a x)} n \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )-(2+n) \left (\operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )-\operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{n (2+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6721, 140, 79, 141}

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.

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

\(\Big \downarrow \) 6721

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 79

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

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

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 140

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

rule 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(e**(acoth(a*x)*n)/x,x)