\(\int e^{n \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^p \, dx\) [917]

Optimal result

Integrand size = 22, antiderivative size = 121 \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {2^{1+\frac {n}{2}+p} \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}+p} \operatorname {AppellF1}\left (1-\frac {n}{2}+p,2,-\frac {n}{2}-p,2-\frac {n}{2}+p,1-\frac {1}{a x},\frac {a-\frac {1}{x}}{2 a}\right )}{a (2-n+2 p)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.65 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6751, 6748, 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.

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

\(\Big \downarrow \) 6751

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

\(\Big \downarrow \) 6748

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

\(\Big \downarrow \) 153

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(exp(n*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))^(1/2*n), x)
 

Giac [F]

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

integrate(exp(n*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))^(1/2*n), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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