\(\int e^{n \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\) [560]

Optimal result

Integrand size = 24, antiderivative size = 104 \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {2^{1+\frac {n}{2}} \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \sqrt {c-\frac {c}{a x}} \operatorname {AppellF1}\left (\frac {3-n}{2},2,-\frac {n}{2},\frac {5-n}{2},1-\frac {1}{a x},\frac {a-\frac {1}{x}}{2 a}\right )}{a (3-n)} \] Output:

2^(1+1/2*n)*(1-1/a/x)^(1-1/2*n)*(c-c/a/x)^(1/2)*AppellF1(3/2-1/2*n,-1/2*n, 
2,5/2-1/2*n,1/2*(a-1/x)/a,1-1/a/x)/a/(3-n)
 

Mathematica [F(-1)]

Timed out. \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\text {\$Aborted} \] Input:

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

Output:

$Aborted
 

Rubi [A] (warning: unable to verify)

Time = 0.68 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6736, 6732, 153}

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.

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

\(\Big \downarrow \) 6736

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

\(\Big \downarrow \) 6732

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

\(\Big \downarrow \) 153

\(\displaystyle -\frac {2^{\frac {3}{2}-\frac {n}{2}} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \operatorname {AppellF1}\left (\frac {n+2}{2},\frac {n-1}{2},2,\frac {n+4}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (n+2) \sqrt {1-\frac {1}{a x}}}\)

Input:

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

Output:

-((2^(3/2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*Sqrt[c - c/(a*x)]*AppellF1[(2 + 
 n)/2, (-1 + n)/2, 2, (4 + n)/2, (a + x^(-1))/(2*a), 1 + 1/(a*x)])/(a*(2 + 
 n)*Sqrt[1 - 1/(a*x)]))
 

Defintions of rubi rules used

rule 153

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(b*e - a*f)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*Simp 
lify[b/(b*c - a*d)]^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c 
 - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, 
n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[p] && GtQ[Simplify[b/( 
b*c - a*d)], 0] &&  !(GtQ[Simplify[d/(d*a - c*b)], 0] && SimplerQ[c + d*x, 
a + b*x])
 

rule 6732

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

rule 6736

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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