\(\int e^{-\coth ^{-1}(a x)} (c-\frac {c}{a x})^p \, dx\) [568]

Optimal result

Integrand size = 22, antiderivative size = 82 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\sqrt {2} \left (1-\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (\frac {3}{2}+p,2,\frac {1}{2},\frac {5}{2}+p,1-\frac {1}{a x},\frac {a-\frac {1}{x}}{2 a}\right )}{a (3+2 p)} \] Output:

2^(1/2)*(1-1/a/x)^(3/2)*(c-c/a/x)^p*AppellF1(3/2+p,1/2,2,5/2+p,1/2*(a-1/x) 
/a,1-1/a/x)/a/(3+2*p)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.07, 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.

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

\(\Big \downarrow \) 6736

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

\(\Big \downarrow \) 6732

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

\(\Big \downarrow \) 153

\(\displaystyle -\frac {2^{p+\frac {3}{2}} \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p-\frac {1}{2},2,\frac {3}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right ) \left (c-\frac {c}{a x}\right )^p}{a}\)

Input:

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

Output:

-((2^(3/2 + p)*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^p*AppellF1[1/2, -1/2 - p, 2 
, 3/2, (a + x^(-1))/(2*a), 1 + 1/(a*x)])/(a*(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]

\[\int \left (c -\frac {c}{a x}\right )^{p} \sqrt {\frac {a x -1}{a x +1}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((c-c/a/x)^p*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

Integral(sqrt((a*x - 1)/(a*x + 1))*(-c*(-1 + 1/(a*x)))**p, x)
 

Maxima [F]

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

integrate((c-c/a/x)^p*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((c-c/a/x)^p*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(a*x - 1)*(a*c*x - c)**p)/(x**p*sqrt(a*x + 1)),x)/a**p