\(\int \frac {e^{n \coth ^{-1}(a x)}}{(c-\frac {c}{a x})^2} \, dx\) [558]

Optimal result

Integrand size = 22, antiderivative size = 166 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {(3+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^2 (2+n)}+\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x}{c^2}-\frac {2 (2+n) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c^2 n} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.68 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left (n (1+a x) (-3+2 a x+n (-1+a x))-2 (2+n)^2 (-1+a x) \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {-1+a x}{1+a x}\right )\right )}{a c^2 n (2+n) (-1+a x)} \] Input:

Integrate[E^(n*ArcCoth[a*x])/(c - c/(a*x))^2,x]
 

Output:

((1 + 1/(a*x))^(n/2)*(n*(1 + a*x)*(-3 + 2*a*x + n*(-1 + a*x)) - 2*(2 + n)^ 
2*(-1 + a*x)*Hypergeometric2F1[1, -1/2*n, 1 - n/2, (-1 + a*x)/(1 + a*x)])) 
/(a*c^2*n*(2 + n)*(1 - 1/(a*x))^(n/2)*(-1 + a*x))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6732, 114, 25, 27, 172, 25, 27, 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.

\(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx\)

\(\Big \downarrow \) 6732

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

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

Input:

Int[E^(n*ArcCoth[a*x])/(c - c/(a*x))^2,x]
 

Output:

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

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 114

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 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

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 172

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

rule 6732

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S 
imp[-c^p   Subst[Int[(1 + d*(x/c))^p*((1 + x/a)^(n/2)/(x^2*(1 - x/a)^(n/2)) 
), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && 
  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0])
 

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (c -\frac {c}{a x}\right )^{2}}d x\]

Input:

int(exp(n*arccoth(a*x))/(c-c/a/x)^2,x)
 

Output:

int(exp(n*arccoth(a*x))/(c-c/a/x)^2,x)
 

Fricas [F]

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

integrate(exp(n*arccoth(a*x))/(c-c/a/x)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(exp(n*acoth(a*x))/(c-c/a/x)**2,x)
 

Output:

a**2*Integral(x**2*exp(n*acoth(a*x))/(a**2*x**2 - 2*a*x + 1), x)/c**2
 

Maxima [F]

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

integrate(exp(n*arccoth(a*x))/(c-c/a/x)^2,x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(exp(n*arccoth(a*x))/(c-c/a/x)^2,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{1,[0,1,0]%%%} / %%%{1,[0,0,2]%%%} Error: Bad Argument Valu 
e
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-\frac {c}{a\,x}\right )}^2} \,d x \] Input:

int(exp(n*acoth(a*x))/(c - c/(a*x))^2,x)
 

Output:

int(exp(n*acoth(a*x))/(c - c/(a*x))^2, x)
 

Reduce [F]

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

int(exp(n*acoth(a*x))/(c-c/a/x)^2,x)
                                                                                    
                                                                                    
 

Output:

(int((e**(acoth(a*x)*n)*x**2)/(a**2*x**2 - 2*a*x + 1),x)*a**2)/c**2