\(\int e^{n \coth ^{-1}(a x)} (c-\frac {c}{a x})^p \, dx\) [563]

Optimal result

Integrand size = 22, antiderivative size = 101 \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {2^{1+\frac {n}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (1-\frac {n}{2}+p,2,-\frac {n}{2},2-\frac {n}{2}+p,1-\frac {1}{a x},\frac {a-\frac {1}{x}}{2 a}\right )}{a (2-n+2 p)} \] Output:

2^(1+1/2*n)*(1-1/a/x)^(1-1/2*n)*(c-c/a/x)^p*AppellF1(1-1/2*n+p,-1/2*n,2,2- 
1/2*n+p,1/2*(a-1/x)/a,1-1/a/x)/a/(2-n+2*p)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.64 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6736, 6732, 153}

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.

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

\(\Big \downarrow \) 6736

\(\displaystyle \left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^pdx\)

\(\Big \downarrow \) 6732

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

\(\Big \downarrow \) 153

\(\displaystyle -\frac {2^{-\frac {n}{2}+p+1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (\frac {n+2}{2},\frac {1}{2} (n-2 p),2,\frac {n+4}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (n+2)}\)

Input:

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

Output:

-((2^(1 - n/2 + p)*(1 + 1/(a*x))^((2 + n)/2)*(c - c/(a*x))^p*AppellF1[(2 + 
 n)/2, (n - 2*p)/2, 2, (4 + n)/2, (a + x^(-1))/(2*a), 1 + 1/(a*x)])/(a*(2 
+ n)*(1 - 1/(a*x))^p))
 

Defintions of rubi rules used

rule 153

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(b*e - a*f)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*Simp 
lify[b/(b*c - a*d)]^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c 
 - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, 
n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[p] && GtQ[Simplify[b/( 
b*c - a*d)], 0] &&  !(GtQ[Simplify[d/(d*a - c*b)], 0] && SimplerQ[c + d*x, 
a + b*x])
 

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])
 

rule 6736

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((e**(acoth(a*x)*n)*(a*c*x - c)**p)/x**p,x)/a**p