3.11.6 \(\int e^{\text {arctanh}(a x)} x^2 (1-a^2 x^2)^p \, dx\) [1006]

3.11.6.1 Optimal result

Integrand size = 22, antiderivative size = 84 \[ \int e^{\text {arctanh}(a x)} x^2 \left (1-a^2 x^2\right )^p \, dx=-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p}}{a^3 (1+2 p)}+\frac {\left (1-a^2 x^2\right )^{\frac {3}{2}+p}}{a^3 (3+2 p)}+\frac {1}{3} x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-p,\frac {5}{2},a^2 x^2\right ) \]

output

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

3.11.6.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.90 \[ \int e^{\text {arctanh}(a x)} x^2 \left (1-a^2 x^2\right )^p \, dx=-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p} \left (2+a^2 (1+2 p) x^2\right )}{a^3 \left (3+8 p+4 p^2\right )}+\frac {1}{3} x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-p,\frac {5}{2},a^2 x^2\right ) \]

input

Integrate[E^ArcTanh[a*x]*x^2*(1 - a^2*x^2)^p,x]
 

output

-(((1 - a^2*x^2)^(1/2 + p)*(2 + a^2*(1 + 2*p)*x^2))/(a^3*(3 + 8*p + 4*p^2) 
)) + (x^3*Hypergeometric2F1[3/2, 1/2 - p, 5/2, a^2*x^2])/3
 

3.11.6.3 Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6698, 542, 243, 53, 278, 2009}

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.

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 542

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 278

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

\(\Big \downarrow \) 2009

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

input

Int[E^ArcTanh[a*x]*x^2*(1 - a^2*x^2)^p,x]
 

output

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

3.11.6.3.1 Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6698

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]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]
 

3.11.6.4 Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.56

method	result	size
meijerg	\(\frac {a \,x^{4} \operatorname {hypergeom}\left (\left [2, \frac {1}{2}-p \right ], \left [3\right ], a^{2} x^{2}\right )}{4}+\frac {x^{3} \operatorname {hypergeom}\left (\left [\frac {3}{2}, \frac {1}{2}-p \right ], \left [\frac {5}{2}\right ], a^{2} x^{2}\right )}{3}\)	\(47\)

input

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a^2*x^2+1)^p,x,method=_RETURNVERBOSE)
 

output

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

3.11.6.5 Fricas [F]

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

input

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

output

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

3.11.6.6 Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.49 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.06 \[ \int e^{\text {arctanh}(a x)} x^2 \left (1-a^2 x^2\right )^p \, dx=- \frac {a^{2 p} x^{2 p + 3} e^{i \pi p} \Gamma \left (- p - \frac {3}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p - \frac {3}{2} \\ \frac {1}{2}, - p - \frac {1}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p + 3} x^{2 p + 3} e^{i \pi p} \Gamma \left (- p - \frac {3}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p + \frac {3}{2} \\ p + 1, p + \frac {5}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } a^{3} \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + 1\right )} - \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} - p - 1, 1 & -1 \\- p - \frac {3}{2}, - p - 1 & 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{a^{2} x^{2}}} \right )} \Gamma \left (p + \frac {1}{2}\right )}{2 \pi a^{3}} - \frac {{G_{3, 3}^{1, 3}\left (\begin {matrix} -1, - p - 2, 1 & \\- p - 2 & - p - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{a^{2} x^{2}}} \right )} \Gamma \left (p + \frac {1}{2}\right )}{2 a^{3} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} \]

input

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

output

-a**(2*p)*x**(2*p + 3)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1 
, -p, -p - 3/2), (1/2, -p - 1/2), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p - 1/ 
2)*gamma(p + 1)) - a**(2*p + 3)*x**(2*p + 3)*exp(I*pi*p)*gamma(-p - 3/2)*g 
amma(p + 1/2)*hyper((1/2, 1, p + 3/2), (p + 1, p + 5/2), a**2*x**2*exp_pol 
ar(2*I*pi))/(2*sqrt(pi)*a**3*gamma(-p - 1/2)*gamma(p + 1)) - meijerg(((-p 
- 1, 1), (-1,)), ((-p - 3/2, -p - 1), (0,)), exp_polar(-I*pi)/(a**2*x**2)) 
*gamma(p + 1/2)/(2*pi*a**3) - meijerg(((-1, -p - 2, 1), ()), ((-p - 2,), ( 
-p - 3/2, 0)), exp_polar(-I*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*a**3*gamma( 
-p)*gamma(p + 1))
 

3.11.6.7 Maxima [F]

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

input

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

output

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

3.11.6.8 Giac [F]

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

input

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

output

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

3.11.6.9 Mupad [F(-1)]

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

input

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

output

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