\(\int e^{-\coth ^{-1}(a x)} (c-a^2 c x^2)^p \, dx\) [746]

Optimal result

Integrand size = 22, antiderivative size = 85 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (\frac {-1+a x}{1+a x}\right )^{-\frac {1}{2}-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-\frac {1}{2}-p,-2 p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+2 p} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {4^{1+p} e^{\coth ^{-1}(a x)} \left (1-e^{2 \coth ^{-1}(a x)}\right )^{2 p} \left (\frac {e^{\coth ^{-1}(a x)}}{-1+e^{2 \coth ^{-1}(a x)}}\right )^{2 p} \left (a \sqrt {1-\frac {1}{a^2 x^2}} x\right )^{-2 p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+p,2+2 p,\frac {3}{2}+p,e^{2 \coth ^{-1}(a x)}\right )}{a+2 a p} \] Input:

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

Output:

-((4^(1 + p)*E^ArcCoth[a*x]*(1 - E^(2*ArcCoth[a*x]))^(2*p)*(E^ArcCoth[a*x] 
/(-1 + E^(2*ArcCoth[a*x])))^(2*p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[1/2 
+ p, 2 + 2*p, 3/2 + p, E^(2*ArcCoth[a*x])])/((a + 2*a*p)*(a*Sqrt[1 - 1/(a^ 
2*x^2)]*x)^(2*p)))
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39, 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.

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

\(\Big \downarrow \) 6746

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

\(\Big \downarrow \) 6750

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

\(\Big \downarrow \) 142

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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