\(\int e^{\text {arctanh}(a x)} (c-\frac {c}{a x})^p \, dx\) [601]

Optimal result

Integrand size = 20, antiderivative size = 60 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (c-\frac {c}{a x}\right )^p x (1-a x)^{-p} \operatorname {AppellF1}\left (1-p,\frac {1}{2}-p,-\frac {1}{2},2-p,a x,-a x\right )}{1-p} \] Output:

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

Mathematica [F]

\[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.

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

\(\Big \downarrow \) 6684

\(\displaystyle x^p (1-a x)^{-p} \left (c-\frac {c}{a x}\right )^p \int e^{\text {arctanh}(a x)} x^{-p} (1-a x)^pdx\)

\(\Big \downarrow \) 6679

\(\displaystyle x^p (1-a x)^{-p} \left (c-\frac {c}{a x}\right )^p \int x^{-p} (1-a x)^{p-\frac {1}{2}} \sqrt {a x+1}dx\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {x (1-a x)^{-p} \operatorname {AppellF1}\left (1-p,\frac {1}{2}-p,-\frac {1}{2},2-p,a x,-a x\right ) \left (c-\frac {c}{a x}\right )^p}{1-p}\)

Input:

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

Output:

((c - c/(a*x))^p*x*AppellF1[1 - p, 1/2 - p, -1/2, 2 - p, a*x, -(a*x)])/((1 
 - p)*(1 - a*x)^p)
 

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]

\[\int \frac {\left (a x +1\right ) \left (c -\frac {c}{a x}\right )^{p}}{\sqrt {-a^{2} x^{2}+1}}d x\]

Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^p,x)
 

Output:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^p,x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**p,x)
 

Output:

Integral((-c*(-1 + 1/(a*x)))**p*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(((c - c/(a*x))^p*(a*x + 1))/(1 - a^2*x^2)^(1/2),x)
 

Output:

int(((c - c/(a*x))^p*(a*x + 1))/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

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

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^p,x)
 

Output:

(int((a*c*x - c)**p/(x**p*sqrt( - a**2*x**2 + 1)),x) + int(((a*c*x - c)**p 
*x)/(x**p*sqrt( - a**2*x**2 + 1)),x)*a)/a**p