\(\int e^{n \coth ^{-1}(a x)} x^2 \, dx\) [157]

Optimal result

Integrand size = 12, antiderivative size = 174 \[ \int e^{n \coth ^{-1}(a x)} x^2 \, dx=\frac {n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{6 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3+\frac {2 \left (2+n^2\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{3 a^3 (2-n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.68 \[ \int e^{n \coth ^{-1}(a x)} x^2 \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n \left (2+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (a n^2 x+2 a^3 x^3+n \left (-1+a^2 x^2\right )+\left (2+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{6 a^3 (2+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6721, 114, 25, 27, 168, 25, 27, 141}

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.

\(\displaystyle \int x^2 e^{n \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6721

\(\displaystyle -\int \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} x^4d\frac {1}{x}\)

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

((1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x^3)/3 - (-1/2*(a*n*(1 
- 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2) - (2*(2 + n^2)*(1 - 1/ 
(a*x))^(1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2)*Hypergeometric2F1[2, 1 - n/2, 
2 - n/2, (a - x^(-1))/(a + x^(-1))])/(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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 114

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

rule 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 0]
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 6721

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int e^{n \coth ^{-1}(a x)} x^2 \, dx=\int x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

integrate(exp(n*arccoth(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. \[ \int e^{n \coth ^{-1}(a x)} x^2 \, dx=\int x^2\,{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int e^{n \coth ^{-1}(a x)} x^2 \, dx=\int e^{\mathit {acoth} \left (a x \right ) n} x^{2}d x \] Input:

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

Output:

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