\(\int e^{-3 \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^p \, dx\) [779]

Optimal result

Integrand size = 22, antiderivative size = 216 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x}{(1-2 p) \sqrt {1-a^2 x^2}}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^p x^2}{\sqrt {1-a^2 x^2}}+\frac {3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^p x^3 \left (1-a^2 x^2\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),\frac {3}{2}-p,\frac {1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}-\frac {a (5-2 p) \left (c-\frac {c}{a^2 x^2}\right )^p x^2 \left (1-a^2 x^2\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,\frac {3}{2}-p,2-p,a^2 x^2\right )}{2 (1-p)} \] Output:

(c-c/a^2/x^2)^p*x/(1-2*p)/(-a^2*x^2+1)^(1/2)+a*(c-c/a^2/x^2)^p*x^2/(-a^2*x 
^2+1)^(1/2)+3*a^2*(c-c/a^2/x^2)^p*x^3*hypergeom([3/2-p, 3/2-p],[5/2-p],a^2 
*x^2)/(3-2*p)/((-a^2*x^2+1)^p)-1/2*a*(5-2*p)*(c-c/a^2/x^2)^p*x^2*hypergeom 
([1-p, 3/2-p],[2-p],a^2*x^2)/(1-p)/((-a^2*x^2+1)^p)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.80 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {1}{2} \left (c-\frac {c}{a^2 x^2}\right )^p x \left (1-a^2 x^2\right )^{-p} \left (\frac {2 \left (1-a^2 x^2\right )^{-\frac {1}{2}+p}}{1-2 p}+\frac {3 a x \operatorname {Hypergeometric2F1}\left (1-p,\frac {3}{2}-p,2-p,a^2 x^2\right )}{-1+p}+\frac {6 a^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,\frac {3}{2}-p,\frac {5}{2}-p,a^2 x^2\right )}{3-2 p}+\frac {a^3 x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,2-p,3-p,a^2 x^2\right )}{-2+p}\right ) \] Input:

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

Output:

((c - c/(a^2*x^2))^p*x*((2*(1 - a^2*x^2)^(-1/2 + p))/(1 - 2*p) + (3*a*x*Hy 
pergeometric2F1[1 - p, 3/2 - p, 2 - p, a^2*x^2])/(-1 + p) + (6*a^2*x^2*Hyp 
ergeometric2F1[3/2 - p, 3/2 - p, 5/2 - p, a^2*x^2])/(3 - 2*p) + (a^3*x^3*H 
ypergeometric2F1[3/2 - p, 2 - p, 3 - p, a^2*x^2])/(-2 + p)))/(2*(1 - a^2*x 
^2)^p)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.93, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6710, 6699, 559, 25, 2339, 27, 278, 545, 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.

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

\(\Big \downarrow \) 6710

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

\(\Big \downarrow \) 6699

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

\(\Big \downarrow \) 559

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}-\frac {\int -x^{-2 p} \left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 x^2 a^4-(5-2 p) x a^3+a^2\right )dx}{a^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {\int x^{-2 p} \left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 x^2 a^4-(5-2 p) x a^3+a^2\right )dx}{a^2}+a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\)

\(\Big \downarrow \) 2339

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {\int a^2 x^{-2 p} (1-a (5-2 p) x) \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx+3 a^4 \int x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx}{a^2}+a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {a^2 \int x^{-2 p} (1-a (5-2 p) x) \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx+3 a^4 \int x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx}{a^2}+a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\)

\(\Big \downarrow \) 278

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {a^2 \int x^{-2 p} (1-a (5-2 p) x) \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx+\frac {3 a^4 x^{3-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),\frac {3}{2}-p,\frac {1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}}{a^2}+a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\)

\(\Big \downarrow \) 545

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {a^2 \left (\frac {x^{1-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}-a (5-2 p) \int x^{1-2 p} \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx\right )+\frac {3 a^4 x^{3-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),\frac {3}{2}-p,\frac {1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}}{a^2}+a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\)

\(\Big \downarrow \) 278

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}+\frac {a^2 \left (\frac {x^{1-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}-\frac {a (5-2 p) x^{2-2 p} \operatorname {Hypergeometric2F1}\left (1-p,\frac {3}{2}-p,2-p,a^2 x^2\right )}{2 (1-p)}\right )+\frac {3 a^4 x^{3-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),\frac {3}{2}-p,\frac {1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}}{a^2}\right )\)

Input:

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

Output:

((c - c/(a^2*x^2))^p*x^(2*p)*(a*x^(2 - 2*p)*(1 - a^2*x^2)^(-1/2 + p) + ((3 
*a^4*x^(3 - 2*p)*Hypergeometric2F1[(3 - 2*p)/2, 3/2 - p, (5 - 2*p)/2, a^2* 
x^2])/(3 - 2*p) + a^2*((x^(1 - 2*p)*(1 - a^2*x^2)^(-1/2 + p))/(1 - 2*p) - 
(a*(5 - 2*p)*x^(2 - 2*p)*Hypergeometric2F1[1 - p, 3/2 - p, 2 - p, a^2*x^2] 
)/(2*(1 - p))))/a^2))/(1 - a^2*x^2)^p
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 545

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[(-c)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1))), 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] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 559

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( 
m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1))   Int[(e*x)^m*(a + b* 
x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 
)*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && IGtQ[n, 1] &&  !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

rule 2339

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> With 
[{q = Expon[Pq, x]}, Simp[Coeff[Pq, x, q]/c^q   Int[(c*x)^(m + q)*(a + b*x^ 
2)^p, x], x] + Simp[1/c^q   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[c^q*Pq - 
Coeff[Pq, x, q]*(c*x)^q, x], x], x] /; EqQ[q, 1] || EqQ[m + q + 2*p + 1, 0] 
] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] &&  !(IGtQ[m, 0] && ILtQ[p + 
 1/2, 0])
 

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]

\[\int \frac {\left (c -\frac {c}{a^{2} x^{2}}\right )^{p} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{\left (a x +1\right )^{3}}d x\]

Input:

int((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
 

Output:

int((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
 

Fricas [F]

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

integrate((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

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

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

Output:

Integral((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))** 
p/(a*x + 1)**3, x)
 

Maxima [F]

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

integrate((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxim 
a")
 

Output:

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

Giac [F]

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

integrate((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac" 
)
 

Output:

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

Mupad [F(-1)]

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

int(((c - c/(a^2*x^2))^p*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3,x)
 

Output:

int(((c - c/(a^2*x^2))^p*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3, x)
 

Reduce [F]

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

int((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
 

Output:

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