\(\int e^{n \text {arctanh}(a x)} x^2 \, dx\) [150]

Optimal result

Integrand size = 12, antiderivative size = 141 \[ \int e^{n \text {arctanh}(a x)} x^2 \, dx=-\frac {n (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{6 a^3}-\frac {x (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{3 a^2}-\frac {2^{n/2} \left (2+n^2\right ) (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{3 a^3 (2-n)} \] Output:

-1/6*n*(-a*x+1)^(1-1/2*n)*(a*x+1)^(1+1/2*n)/a^3-1/3*x*(-a*x+1)^(1-1/2*n)*( 
a*x+1)^(1+1/2*n)/a^2-1/3*2^(1/2*n)*(n^2+2)*(-a*x+1)^(1-1/2*n)*hypergeom([- 
1/2*n, 1-1/2*n],[2-1/2*n],-1/2*a*x+1/2)/a^3/(2-n)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.68 \[ \int e^{n \text {arctanh}(a x)} x^2 \, dx=-\frac {(1-a x)^{1-\frac {n}{2}} \left ((-2+n) (1+a x)^{1+\frac {n}{2}} (n+2 a x)-2^{1+\frac {n}{2}} \left (2+n^2\right ) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )\right )}{6 a^3 (-2+n)} \] Input:

Integrate[E^(n*ArcTanh[a*x])*x^2,x]
 

Output:

-1/6*((1 - a*x)^(1 - n/2)*((-2 + n)*(1 + a*x)^(1 + n/2)*(n + 2*a*x) - 2^(1 
 + n/2)*(2 + n^2)*Hypergeometric2F1[1 - n/2, -1/2*n, 2 - n/2, (1 - a*x)/2] 
))/(a^3*(-2 + n))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6676, 101, 25, 90, 79}

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.

\(\displaystyle \int x^2 e^{n \text {arctanh}(a x)} \, dx\)

\(\Big \downarrow \) 6676

\(\displaystyle \int x^2 (1-a x)^{-n/2} (a x+1)^{n/2}dx\)

\(\Big \downarrow \) 101

\(\displaystyle -\frac {\int -(1-a x)^{-n/2} (a x+1)^{n/2} (a n x+1)dx}{3 a^2}-\frac {x (a x+1)^{\frac {n+2}{2}} (1-a x)^{1-\frac {n}{2}}}{3 a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int (1-a x)^{-n/2} (a x+1)^{n/2} (a n x+1)dx}{3 a^2}-\frac {x (1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n+2}{2}}}{3 a^2}\)

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 79

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

Input:

Int[E^(n*ArcTanh[a*x])*x^2,x]
 

Output:

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

Defintions of rubi rules used

rule 25

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

rule 79

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

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 6676

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) 
^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] &&  !Int 
egerQ[(n - 1)/2]
 

Maple [F]

\[\int {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{2}d x\]

Input:

int(exp(n*arctanh(a*x))*x^2,x)
 

Output:

int(exp(n*arctanh(a*x))*x^2,x)
 

Fricas [F]

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

integrate(exp(n*arctanh(a*x))*x^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int e^{n \text {arctanh}(a x)} x^2 \, dx=\int x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \] Input:

integrate(exp(n*atanh(a*x))*x**2,x)
 

Output:

Integral(x**2*exp(n*atanh(a*x)), x)
 

Maxima [F]

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

integrate(exp(n*arctanh(a*x))*x^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(exp(n*arctanh(a*x))*x^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int e^{n \text {arctanh}(a x)} x^2 \, dx=\int x^2\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )} \,d x \] Input:

int(x^2*exp(n*atanh(a*x)),x)
 

Output:

int(x^2*exp(n*atanh(a*x)), x)
 

Reduce [F]

\[ \int e^{n \text {arctanh}(a x)} x^2 \, dx=\int e^{\mathit {atanh} \left (a x \right ) n} x^{2}d x \] Input:

int(exp(n*atanh(a*x))*x^2,x)
 

Output:

int(e**(atanh(a*x)*n)*x**2,x)