\(\int e^{n \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx\) [293]

Optimal result

Integrand size = 20, antiderivative size = 98 \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2}{3} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (-1+n)} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{2} (-1+n),-\frac {1}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {-1+a x}{1+a x}\right )^{\frac {1}{2} (-1+n)} (1+a x) \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{2} (-1+n),-\frac {1}{2},\frac {2}{1+a x}\right )}{3 a} \] Input:

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

Output:

(2*(1 + 1/(a*x))^(n/2)*((-1 + a*x)/(1 + a*x))^((-1 + n)/2)*(1 + a*x)*Sqrt[ 
c - a*c*x]*Hypergeometric2F1[-3/2, (-1 + n)/2, -1/2, 2/(1 + a*x)])/(3*a*(1 
 - 1/(a*x))^(n/2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

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

\(\Big \downarrow \) 6727

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{n/2}}{\left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 142

\(\displaystyle \frac {2}{3} x \sqrt {c-a c x} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}-\frac {1}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {n-1}{2},-\frac {1}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )\)

Input:

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

Output:

(2*((a - x^(-1))/(a + x^(-1)))^((-1 + n)/2)*(1 - 1/(a*x))^(-1/2 + (1 - n)/ 
2)*(1 + 1/(a*x))^((2 + n)/2)*x*Sqrt[c - a*c*x]*Hypergeometric2F1[-3/2, (-1 
 + n)/2, -1/2, 2/((a + x^(-1))*x)])/3
 

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]

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

Input:

int(exp(n*arccoth(a*x))*(-a*c*x+c)^(1/2),x)
 

Output:

int(exp(n*arccoth(a*x))*(-a*c*x+c)^(1/2),x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate(exp(n*acoth(a*x))*(-a*c*x+c)**(1/2),x)
 

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(exp(n*acoth(a*x))*(c - a*c*x)^(1/2),x)
 

Output:

int(exp(n*acoth(a*x))*(c - a*c*x)^(1/2), x)
 

Reduce [F]

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

int(exp(n*acoth(a*x))*(-a*c*x+c)^(1/2),x)
 

Output:

sqrt(c)*int(e**(acoth(a*x)*n)*sqrt( - a*x + 1),x)