\(\int \frac {e^{n \coth ^{-1}(a x)}}{x^5} \, dx\) [164]

Optimal result

Integrand size = 12, antiderivative size = 220 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^5} \, dx=\frac {1}{24} a^4 \left (6+2 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}-\frac {1}{12} a^4 n \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{4 x^2}+\frac {2^{-2+\frac {n}{2}} a^4 n \left (8+n^2\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{3 (2-n)} \] Output:

1/24*a^4*(n^2+2*n+6)*(1-1/a/x)^(1-1/2*n)*(1+1/a/x)^(1+1/2*n)-1/12*a^4*n*(1 
-1/a/x)^(2-1/2*n)*(1+1/a/x)^(1+1/2*n)+1/4*a^2*(1-1/a/x)^(1-1/2*n)*(1+1/a/x 
)^(1+1/2*n)/x^2+2^(-2+1/2*n)*a^4*n*(n^2+8)*(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)/(6-3*n)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.67 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^5} \, dx=-\frac {1}{24} a^4 e^{n \coth ^{-1}(a x)} \left (-6-n^2+\frac {6}{a^4 x^4}+\frac {2 n}{a^3 x^3}+\frac {n^2}{a^2 x^2}+\frac {6 n}{a x}+\frac {n^3}{a x}-\frac {e^{2 \coth ^{-1}(a x)} n^2 \left (8+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )}{2+n}+n \left (8+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )\right ) \] Input:

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

Output:

-1/24*(a^4*E^(n*ArcCoth[a*x])*(-6 - n^2 + 6/(a^4*x^4) + (2*n)/(a^3*x^3) + 
n^2/(a^2*x^2) + (6*n)/(a*x) + n^3/(a*x) - (E^(2*ArcCoth[a*x])*n^2*(8 + n^2 
)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])])/(2 + n) + n 
*(8 + n^2)*Hypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcCoth[a*x])]))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6721, 111, 25, 27, 164, 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.

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

\(\Big \downarrow \) 6721

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 164

\(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{4 x^2}-\frac {1}{4} a \left (\frac {1}{6} a^2 n \left (n^2+8\right ) \int \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2}d\frac {1}{x}-\frac {1}{6} a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (a \left (n^2+6\right )+\frac {2 n}{x}\right )\right )\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{4 x^2}-\frac {1}{4} a \left (-\frac {a^3 2^{n/2} n \left (n^2+8\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{3 (2-n)}-\frac {1}{6} a^2 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (a \left (n^2+6\right )+\frac {2 n}{x}\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}\right )\)

Input:

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

Output:

(a^2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(4*x^2) - (a*(-1/6 
*(a^2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(a*(6 + n^2) + (2* 
n)/x)) - (2^(n/2)*a^3*n*(8 + n^2)*(1 - 1/(a*x))^(1 - n/2)*Hypergeometric2F 
1[1 - n/2, -1/2*n, 2 - n/2, (a - x^(-1))/(2*a)])/(3*(2 - n))))/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 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 111

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

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{x^{5}}d x\]

Input:

int(exp(n*arccoth(a*x))/x^5,x)
 

Output:

int(exp(n*arccoth(a*x))/x^5,x)
 

Fricas [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^5} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{5}} \,d x } \] Input:

integrate(exp(n*arccoth(a*x))/x^5,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^5} \, dx=\int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x^{5}}\, dx \] Input:

integrate(exp(n*acoth(a*x))/x**5,x)
 

Output:

Integral(exp(n*acoth(a*x))/x**5, x)
 

Maxima [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^5} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{5}} \,d x } \] Input:

integrate(exp(n*arccoth(a*x))/x^5,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^5} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{5}} \,d x } \] Input:

integrate(exp(n*arccoth(a*x))/x^5,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^5} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x^5} \,d x \] Input:

int(exp(n*acoth(a*x))/x^5,x)
 

Output:

int(exp(n*acoth(a*x))/x^5, x)
 

Reduce [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^5} \, dx=\frac {-e^{\mathit {acoth} \left (a x \right ) n}+\left (\int \frac {e^{\mathit {acoth} \left (a x \right ) n}}{a^{2} n^{2} x^{6}+8 a^{2} x^{6}-n^{2} x^{4}-8 x^{4}}d x \right ) a \,n^{3} x^{4}+8 \left (\int \frac {e^{\mathit {acoth} \left (a x \right ) n}}{a^{2} n^{2} x^{6}+8 a^{2} x^{6}-n^{2} x^{4}-8 x^{4}}d x \right ) a n \,x^{4}}{4 x^{4}} \] Input:

int(exp(n*acoth(a*x))/x^5,x)
                                                                                    
                                                                                    
 

Output:

( - e**(acoth(a*x)*n) + int(e**(acoth(a*x)*n)/(a**2*n**2*x**6 + 8*a**2*x** 
6 - n**2*x**4 - 8*x**4),x)*a*n**3*x**4 + 8*int(e**(acoth(a*x)*n)/(a**2*n** 
2*x**6 + 8*a**2*x**6 - n**2*x**4 - 8*x**4),x)*a*n*x**4)/(4*x**4)