\(\int e^{2 p \coth ^{-1}(a x)} (c-\frac {c}{a x})^p \, dx\) [564]

Optimal result

Integrand size = 23, antiderivative size = 67 \[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {\left (1-\frac {1}{a x}\right )^{-p} \left (1+\frac {1}{a x}\right )^{1+p} \left (c-\frac {c}{a x}\right )^p \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1+\frac {1}{a x}\right )}{a (1+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {\left (1-\frac {1}{a x}\right )^{-p} \left (1+\frac {1}{a x}\right )^{1+p} \left (c-\frac {c}{a x}\right )^p \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1+\frac {1}{a x}\right )}{a (1+p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6736, 6732, 75}

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

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

\(\Big \downarrow \) 75

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

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

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

Output:

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