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

Optimal. Leaf size=141 \[ -\frac{2^{n/2} \left (n^2+2\right ) (1-a x)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{3 a^3 (2-n)}-\frac{n (a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{6 a^3}-\frac{x (a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{3 a^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0948837, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6126, 90, 80, 69} \[ -\frac{2^{n/2} \left (n^2+2\right ) (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{3 a^3 (2-n)}-\frac{n (a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{6 a^3}-\frac{x (a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{3 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6126

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

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(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 69

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

Rubi steps

\begin{align*} \int e^{n \tanh ^{-1}(a x)} x^2 \, dx &=\int x^2 (1-a x)^{-n/2} (1+a x)^{n/2} \, dx\\ &=-\frac{x (1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{3 a^2}-\frac{\int (1-a x)^{-n/2} (1+a x)^{n/2} (-1-a n x) \, dx}{3 a^2}\\ &=-\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{\left (2+n^2\right ) \int (1-a x)^{-n/2} (1+a x)^{n/2} \, dx}{6 a^2}\\ &=-\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}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{3 a^3 (2-n)}\\ \end{align*}

Mathematica [A]  time = 0.051358, size = 96, normalized size = 0.68 \[ -\frac{(1-a x)^{1-\frac{n}{2}} \left ((n-2) (a x+1)^{\frac{n}{2}+1} (2 a x+n)-2^{\frac{n}{2}+1} \left (n^2+2\right ) \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )\right )}{6 a^3 (n-2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}{x}^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} e^{n \operatorname{atanh}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*exp(n*atanh(a*x)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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