\(\int e^{2 p \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^p \, dx\) [919]

Optimal result

Integrand size = 23, antiderivative size = 75 \[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (1+\frac {1}{a x}\right )^{1+2 p} \operatorname {Hypergeometric2F1}\left (2,1+2 p,2 (1+p),1+\frac {1}{a x}\right )}{a (1+2 p)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 75, 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 = {6751, 6748, 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.

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

\(\Big \downarrow \) 6751

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

\(\Big \downarrow \) 6748

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

\(\Big \downarrow \) 75

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

Input:

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

Output:

-(((c - c/(a^2*x^2))^p*(1 + 1/(a*x))^(1 + 2*p)*Hypergeometric2F1[2, 1 + 2* 
p, 2*(1 + p), 1 + 1/(a*x)])/(a*(1 + 2*p)*(1 - 1/(a^2*x^2))^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 6748

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

rule 6751

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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