\(\int e^{-\text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^p \, dx\) [793]

Optimal result

Integrand size = 22, antiderivative size = 137 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x \left (1-a^2 x^2\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),\frac {1}{2}-p,\frac {1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^p x^2 \left (1-a^2 x^2\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,1-p,2-p,a^2 x^2\right )}{2 (1-p)} \] Output:

(c-c/a^2/x^2)^p*x*hypergeom([1/2-p, 1/2-p],[3/2-p],a^2*x^2)/(1-2*p)/((-a^2 
*x^2+1)^p)-1/2*a*(c-c/a^2/x^2)^p*x^2*hypergeom([1-p, 1/2-p],[2-p],a^2*x^2) 
/(1-p)/((-a^2*x^2+1)^p)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x \left (1-a^2 x^2\right )^{-p} \left (-2 (-1+p) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,\frac {1}{2}-p,\frac {3}{2}-p,a^2 x^2\right )+a (-1+2 p) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,1-p,2-p,a^2 x^2\right )\right )}{2 (-1+p) (-1+2 p)} \] Input:

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

Output:

((c - c/(a^2*x^2))^p*x*(-2*(-1 + p)*Hypergeometric2F1[1/2 - p, 1/2 - p, 3/ 
2 - p, a^2*x^2] + a*(-1 + 2*p)*x*Hypergeometric2F1[1/2 - p, 1 - p, 2 - p, 
a^2*x^2]))/(2*(-1 + p)*(-1 + 2*p)*(1 - a^2*x^2)^p)
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6710, 6699, 557, 278}

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.

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

\(\Big \downarrow \) 6710

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

\(\Big \downarrow \) 6699

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 278

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {x^{1-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),\frac {1}{2}-p,\frac {1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}-\frac {a x^{2-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,1-p,2-p,a^2 x^2\right )}{2 (1-p)}\right )\)

Input:

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

Output:

((c - c/(a^2*x^2))^p*x^(2*p)*((x^(1 - 2*p)*Hypergeometric2F1[(1 - 2*p)/2, 
1/2 - p, (3 - 2*p)/2, a^2*x^2])/(1 - 2*p) - (a*x^(2 - 2*p)*Hypergeometric2 
F1[1/2 - p, 1 - p, 2 - p, a^2*x^2])/(2*(1 - p))))/(1 - a^2*x^2)^p
 

Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 557

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c   Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e   Int[(e*x)^( 
m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
 

rule 6699

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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