3.13.27 \(\int \frac {e^{-\text {arctanh}(a x)} (c-a^2 c x^2)^p}{x^2} \, dx\) [1227]

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

3.13.27.2 Mathematica [A] (verified)

Time = 0.04 (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
 

3.13.27.3 Rubi [A] (verified)

Time = 0.42 (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
 

3.13.27.3.1 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]
 

3.13.27.4 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)
 

3.13.27.5 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)
 

3.13.27.6 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)
 

3.13.27.7 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)
 

3.13.27.8 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
 

3.13.27.9 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)