\(\int e^{-\coth ^{-1}(a x)} (c-a c x)^p \, dx\) [281]

Optimal result

Integrand size = 18, antiderivative size = 94 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-\frac {1}{2}-p} \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-1-p,-\frac {1}{2}-p,-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+p} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.81 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (\frac {-1+a x}{1+a x}\right )^{-\frac {1}{2}-p} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-1-p,-\frac {1}{2}-p,-p,\frac {2}{1+a x}\right )}{1+p} \] Input:

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

Output:

(Sqrt[1 - 1/(a^2*x^2)]*x*((-1 + a*x)/(1 + a*x))^(-1/2 - p)*(c - a*c*x)^p*H 
ypergeometric2F1[-1 - p, -1/2 - p, -p, 2/(1 + a*x)])/(1 + p)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6727, 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.

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

\(\Big \downarrow \) 6727

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

\(\Big \downarrow \) 142

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

Input:

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

Output:

(((a - x^(-1))/(a + x^(-1)))^(-1/2 - p)*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x) 
]*x*(c - a*c*x)^p*Hypergeometric2F1[-1 - p, -1/2 - p, -p, 2/((a + x^(-1))* 
x)])/(1 + 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 6727

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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