\(\int e^{n \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\) [612]

Optimal result

Integrand size = 24, antiderivative size = 54 \[ \int e^{n \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2} (-1+n),-\frac {n}{2},\frac {3}{2},a x,-a x\right )}{\sqrt {1-a x}} \] Output:

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

Mathematica [F(-1)]

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

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

Output:

$Aborted
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6684, 6679, 150}

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.

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

\(\Big \downarrow \) 6684

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

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 150

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

rule 6679

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

rule 6684

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(exp(n*arctanh(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]

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

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

Output:

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

Maxima [F]

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

integrate(exp(n*arctanh(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]

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

integrate(exp(n*arctanh(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. \[ \int e^{n \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\sqrt {c-\frac {c}{a\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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