\(\int e^{n \coth ^{-1}(a x)} (c-a^2 c x^2)^p \, dx\) [736]

Optimal result

Integrand size = 22, antiderivative size = 127 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}+p} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}+p} x \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,\frac {1}{2} (n-2 p),-2 p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+2 p} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {e^{(-2+n) \coth ^{-1}(a x)} \left (-1+e^{2 \coth ^{-1}(a x)}\right ) \left (-1+a^2 x^2\right ) \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2}-p,2-\frac {n}{2}+p,e^{-2 \coth ^{-1}(a x)}\right )}{a (n-2 (1+p))} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6746, 6750, 142}

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.

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

\(\Big \downarrow \) 6746

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

\(\Big \downarrow \) 6750

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

\(\Big \downarrow \) 142

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 142

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e 
 - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + 
b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f 
*x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 
 0] &&  !IntegerQ[n]
 

rule 6746

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

rule 6750

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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