\(\int e^{2 p \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^p \, dx\) [760]

Optimal result

Integrand size = 23, antiderivative size = 54 \[ \int e^{2 p \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x \left (1-a^2 x^2\right )^{-p} \operatorname {Hypergeometric2F1}(1-2 p,-2 p,2-2 p,-a x)}{1-2 p} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int e^{2 p \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x \left (1-a^2 x^2\right )^{-p} \operatorname {Hypergeometric2F1}(1-2 p,-2 p,2-2 p,-a x)}{1-2 p} \] Input:

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

Output:

((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[1 - 2*p, -2*p, 2 - 2*p, -(a*x)])/ 
((1 - 2*p)*(1 - a^2*x^2)^p)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6710, 6700, 74}

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.

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

\(\Big \downarrow \) 6710

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int e^{2 p \text {arctanh}(a x)} x^{-2 p} \left (1-a^2 x^2\right )^pdx\)

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 74

\(\displaystyle \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \operatorname {Hypergeometric2F1}(1-2 p,-2 p,2-2 p,-a x)}{1-2 p}\)

Input:

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

Output:

((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[1 - 2*p, -2*p, 2 - 2*p, -(a*x)])/ 
((1 - 2*p)*(1 - a^2*x^2)^p)
 

Defintions of rubi rules used

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 6700

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

rule 6710

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((e**(2*atanh(a*x)*p)*(a**2*c*x**2 - c)**p)/x**(2*p),x)/a**(2*p)