\(\int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx\) [280]

Optimal result

Integrand size = 16, antiderivative size = 94 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2}-p} \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-1-p,\frac {1}{2}-p,-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(1+p) \sqrt {1-\frac {1}{a x}}} \] Output:

((a-1/x)/(a+1/x))^(1/2-p)*(1+1/a/x)^(3/2)*x*(-a*c*x+c)^p*hypergeom([-1-p, 
1/2-p],[-p],2/(a+1/x)/x)/(p+1)/(1-1/a/x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.39 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\sqrt {1+\frac {1}{a x}} \left (\frac {-1+a x}{1+a x}\right )^{-p} (c-a c x)^p \left (p (-1+a x) \left (\frac {-1+a x}{1+a x}\right )^p+\sqrt {\frac {-1+a x}{1+a x}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,1-p,\frac {2}{1+a x}\right )\right )}{a p (1+p) \sqrt {1-\frac {1}{a x}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.87, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6727, 105, 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

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 142

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 105

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)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

Output:

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