\(\int \frac {e^{-\text {arctanh}(a x)} (c-a^2 c x^2)^p}{x^2} \, dx\) [1249]

Optimal result

Integrand size = 25, antiderivative size = 112 \[ \int \frac {e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=-\frac {\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}-p,\frac {1}{2},a^2 x^2\right )}{x}+\frac {a \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{1+2 p} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}-p,\frac {1}{2},a^2 x^2\right )}{x}+\frac {a \left (1-a^2 x^2\right )^{\frac {1}{2}+p} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{1+2 p}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6703, 6699, 542, 243, 75, 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.

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

\(\Big \downarrow \) 6703

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

\(\Big \downarrow \) 6699

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

\(\Big \downarrow \) 542

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 75

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 75

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 542

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

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 6703

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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