\(\int e^{-2 p \coth ^{-1}(a x)} (c-\frac {c}{a x})^p \, dx\) [565]

Optimal result

Integrand size = 23, antiderivative size = 80 \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {2^{-p} \left (1-\frac {1}{a x}\right )^{1+p} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (1+2 p,2,p,2 (1+p),1-\frac {1}{a x},\frac {a-\frac {1}{x}}{2 a}\right )}{a (1+2 p)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.62 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.

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

\(\Big \downarrow \) 6736

\(\displaystyle \left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p \int e^{-2 p \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^pdx\)

\(\Big \downarrow \) 6732

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

\(\Big \downarrow \) 153

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((a*c*x - c)**p/(x**p*e**(2*acoth(a*x)*p)),x)/a**p