\(\int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx\) [598]

Optimal result

Integrand size = 24, antiderivative size = 56 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {2 x \sqrt {1-a x} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1+n}{2},-\frac {n}{2},\frac {5}{2},a x,-a x\right )}{3 \sqrt {c-\frac {c}{a x}}} \] Output:

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

Mathematica [F(-1)]

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

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

Output:

$Aborted
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 56, 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.

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

\(\Big \downarrow \) 6684

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

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {2 x \sqrt {1-a x} \operatorname {AppellF1}\left (\frac {3}{2},\frac {n+1}{2},-\frac {n}{2},\frac {5}{2},a x,-a x\right )}{3 \sqrt {c-\frac {c}{a x}}}\)

Input:

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

Output:

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

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

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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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