\(\int e^{n \text {arctanh}(a x)} (c-\frac {c}{a x})^p \, dx\) [589]

Optimal result

Integrand size = 22, antiderivative size = 64 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (c-\frac {c}{a x}\right )^p x (1-a x)^{-p} \operatorname {AppellF1}\left (1-p,\frac {1}{2} (n-2 p),-\frac {n}{2},2-p,a x,-a x\right )}{1-p} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.

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

\(\Big \downarrow \) 6684

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

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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