\(\int \frac {e^{n \text {arctanh}(a x)}}{(c-\frac {c}{a x})^{3/2}} \, dx\) [614]

Optimal result

Integrand size = 24, antiderivative size = 56 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {2 x (1-a x)^{3/2} \operatorname {AppellF1}\left (\frac {5}{2},\frac {3+n}{2},-\frac {n}{2},\frac {7}{2},a x,-a x\right )}{5 \left (c-\frac {c}{a x}\right )^{3/2}} \] Output:

2/5*x*(-a*x+1)^(3/2)*AppellF1(5/2,3/2+1/2*n,-1/2*n,7/2,a*x,-a*x)/(c-c/a/x) 
^(3/2)
 

Mathematica [F(-1)]

Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\text {\$Aborted} \] Input:

Integrate[E^(n*ArcTanh[a*x])/(c - c/(a*x))^(3/2),x]
 

Output:

$Aborted
 

Rubi [A] (verified)

Time = 0.67 (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.

\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6684

\(\displaystyle \frac {(1-a x)^{3/2} \int \frac {e^{n \text {arctanh}(a x)} x^{3/2}}{(1-a x)^{3/2}}dx}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

\(\Big \downarrow \) 6679

\(\displaystyle \frac {(1-a x)^{3/2} \int x^{3/2} (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{n/2}dx}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {2 x (1-a x)^{3/2} \operatorname {AppellF1}\left (\frac {5}{2},\frac {n+3}{2},-\frac {n}{2},\frac {7}{2},a x,-a x\right )}{5 \left (c-\frac {c}{a x}\right )^{3/2}}\)

Input:

Int[E^(n*ArcTanh[a*x])/(c - c/(a*x))^(3/2),x]
 

Output:

(2*x*(1 - a*x)^(3/2)*AppellF1[5/2, (3 + n)/2, -1/2*n, 7/2, a*x, -(a*x)])/( 
5*(c - c/(a*x))^(3/2))
 

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]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (c -\frac {c}{a x}\right )^{\frac {3}{2}}}d x\]

Input:

int(exp(n*arctanh(a*x))/(c-c/a/x)^(3/2),x)
 

Output:

int(exp(n*arctanh(a*x))/(c-c/a/x)^(3/2),x)
 

Fricas [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(exp(n*atanh(a*x))/(c-c/a/x)**(3/2),x)
 

Output:

Integral(exp(n*atanh(a*x))/(-c*(-1 + 1/(a*x)))**(3/2), x)
 

Maxima [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}} \,d x \] Input:

int(exp(n*atanh(a*x))/(c - c/(a*x))^(3/2),x)
 

Output:

int(exp(n*atanh(a*x))/(c - c/(a*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {x}\, e^{\mathit {atanh} \left (a x \right ) n} x}{\sqrt {a x -1}\, a x -\sqrt {a x -1}}d x \right ) a}{\sqrt {c}\, c} \] Input:

int(exp(n*atanh(a*x))/(c-c/a/x)^(3/2),x)
 

Output:

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