\(\int e^{\text {arctanh}(a x)} x (1-a^2 x^2)^p \, dx\) [1029]

Optimal result

Integrand size = 20, antiderivative size = 58 \[ \int e^{\text {arctanh}(a x)} x \left (1-a^2 x^2\right )^p \, dx=-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p}}{a^2 (1+2 p)}+\frac {1}{3} a x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-p,\frac {5}{2},a^2 x^2\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03 \[ \int e^{\text {arctanh}(a x)} x \left (1-a^2 x^2\right )^p \, dx=-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p}}{2 a^2 \left (\frac {1}{2}+p\right )}+\frac {1}{3} a x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-p,\frac {5}{2},a^2 x^2\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6698, 542, 241, 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.

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 542

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

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 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]
 

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.15 (sec) , antiderivative size = 243, normalized size of antiderivative = 4.19 \[ \int e^{\text {arctanh}(a x)} x \left (1-a^2 x^2\right )^p \, dx=- \frac {a a^{2 p} x^{2 p + 3} e^{i \pi p} \Gamma \left (- p - \frac {3}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p - \frac {3}{2} \\ \frac {1}{2}, - p - \frac {1}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + 1\right )} + \begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\\frac {\begin {cases} \frac {\sqrt {- a^{2} x^{2} + 1} \left (- a^{2} x^{2} + 1\right )^{p}}{- 2 p - 1} & \text {for}\: p \neq - \frac {1}{2} \\\sqrt {- a^{2} x^{2} + 1} \left (- a^{2} x^{2} + 1\right )^{p} \log {\left (\frac {1}{\sqrt {- a^{2} x^{2} + 1}} \right )} & \text {otherwise} \end {cases}}{a^{2}} & \text {otherwise} \end {cases} - \frac {a^{2 p + 3} x^{2 p + 3} e^{i \pi p} \Gamma \left (- p - \frac {3}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p + \frac {3}{2} \\ p + 1, p + \frac {5}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } a^{2} \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + 1\right )} \] Input:

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

Output:

-a*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)) + Piecewise((x**2/2, Eq(a**2, 0)), (Piecewise((sqrt(-a* 
*2*x**2 + 1)*(-a**2*x**2 + 1)**p/(-2*p - 1), Ne(p, -1/2)), (sqrt(-a**2*x** 
2 + 1)*(-a**2*x**2 + 1)**p*log(1/sqrt(-a**2*x**2 + 1)), True))/a**2, True) 
) - a**(2*p + 3)*x**(2*p + 3)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*h 
yper((1/2, 1, p + 3/2), (p + 1, p + 5/2), a**2*x**2*exp_polar(2*I*pi))/(2* 
sqrt(pi)*a**2*gamma(-p - 1/2)*gamma(p + 1))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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