Integrand size = 12, antiderivative size = 130 \[ \int e^{\text {arctanh}(a+b x)} x^2 \, dx=-\frac {\left (1-2 a+2 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}-\frac {(1-4 a) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}-\frac {x \sqrt {1-a-b x} (1+a+b x)^{3/2}}{3 b^2}+\frac {\left (1-2 a+2 a^2\right ) \arcsin (a+b x)}{2 b^3} \]
1/2*(2*a^2-2*a+1)*arcsin(b*x+a)/b^3-1/6*(1-4*a)*(b*x+a+1)^(3/2)*(-b*x-a+1) ^(1/2)/b^3-1/3*x*(b*x+a+1)^(3/2)*(-b*x-a+1)^(1/2)/b^2-1/2*(2*a^2-2*a+1)*(- b*x-a+1)^(1/2)*(b*x+a+1)^(1/2)/b^3
Time = 0.11 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.22 \[ \int e^{\text {arctanh}(a+b x)} x^2 \, dx=\frac {-\sqrt {b} \sqrt {1-a^2-2 a b x-b^2 x^2} \left (4+2 a^2+3 b x+2 b^2 x^2-a (9+2 b x)\right )+6 \left (1+2 a^2\right ) \sqrt {-b} \text {arcsinh}\left (\frac {\sqrt {-b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {b}}\right )+12 a \sqrt {-b} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {-b}}\right )}{6 b^{7/2}} \]
(-(Sqrt[b]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(4 + 2*a^2 + 3*b*x + 2*b^2*x^ 2 - a*(9 + 2*b*x))) + 6*(1 + 2*a^2)*Sqrt[-b]*ArcSinh[(Sqrt[-b]*Sqrt[1 - a - b*x])/(Sqrt[2]*Sqrt[b])] + 12*a*Sqrt[-b]*ArcSinh[(Sqrt[b]*Sqrt[1 - a - b *x])/(Sqrt[2]*Sqrt[-b])])/(6*b^(7/2))
Time = 0.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6713, 101, 25, 90, 60, 62, 1090, 223}
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.
| \(\displaystyle \int x^2 e^{\text {arctanh}(a+b x)} \, dx\) |
| \(\Big \downarrow \) 6713 |
| \(\displaystyle \int \frac {x^2 \sqrt {a+b x+1}}{\sqrt {-a-b x+1}}dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {a+b x+1} \left (-a^2+(1-4 a) b x+1\right )}{\sqrt {-a-b x+1}}dx}{3 b^2}-\frac {x \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{3 b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sqrt {a+b x+1} \left (-a^2+(1-4 a) b x+1\right )}{\sqrt {-a-b x+1}}dx}{3 b^2}-\frac {x \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{3 b^2}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {\frac {3}{2} \left (2 a^2-2 a+1\right ) \int \frac {\sqrt {a+b x+1}}{\sqrt {-a-b x+1}}dx-\frac {(1-4 a) \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 b}}{3 b^2}-\frac {x \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{3 b^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {3}{2} \left (2 a^2-2 a+1\right ) \left (\int \frac {1}{\sqrt {-a-b x+1} \sqrt {a+b x+1}}dx-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )-\frac {(1-4 a) \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 b}}{3 b^2}-\frac {x \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{3 b^2}\) |
| \(\Big \downarrow \) 62 |
| \(\displaystyle \frac {\frac {3}{2} \left (2 a^2-2 a+1\right ) \left (\int \frac {1}{\sqrt {-b^2 x^2-2 a b x+(1-a) (a+1)}}dx-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )-\frac {(1-4 a) \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 b}}{3 b^2}-\frac {x \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{3 b^2}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {\frac {3}{2} \left (2 a^2-2 a+1\right ) \left (-\frac {\int \frac {1}{\sqrt {1-\frac {\left (-2 x b^2-2 a b\right )^2}{4 b^2}}}d\left (-2 x b^2-2 a b\right )}{2 b^2}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )-\frac {(1-4 a) \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 b}}{3 b^2}-\frac {x \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{3 b^2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {\frac {3}{2} \left (2 a^2-2 a+1\right ) \left (-\frac {\arcsin \left (\frac {-2 a b-2 b^2 x}{2 b}\right )}{b}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )-\frac {(1-4 a) \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 b}}{3 b^2}-\frac {x \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{3 b^2}\) |
-1/3*(x*Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/b^2 + (-1/2*((1 - 4*a)*Sqrt [1 - a - b*x]*(1 + a + b*x)^(3/2))/b + (3*(1 - 2*a + 2*a^2)*(-((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/b) - ArcSin[(-2*a*b - 2*b^2*x)/(2*b)]/b))/2)/(3 *b^2)
3.9.19.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Int[ 1/Sqrt[a*c - b*(a - c)*x - b^2*x^2], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]
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]
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]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.) , x_Symbol] :> Int[(d + e*x)^m*((1 + a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^( n/2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
Time = 0.36 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.99
| method | result | size |
| risch | \(\frac {\left (2 b^{2} x^{2}-2 a b x +2 a^{2}+3 b x -9 a +4\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{6 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {\left (2 a^{2}-2 a +1\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}\) | \(129\) |
| default | \(b \left (-\frac {x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 b^{2}}-\frac {5 a \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{3 b}+\frac {2 \left (-a^{2}+1\right ) \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{3 b^{2}}\right )+\left (1+a \right ) \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}\right )\) | \(458\) |
1/6*(2*b^2*x^2-2*a*b*x+2*a^2+3*b*x-9*a+4)*(b^2*x^2+2*a*b*x+a^2-1)/b^3/(-b^ 2*x^2-2*a*b*x-a^2+1)^(1/2)+1/2/b^2*(2*a^2-2*a+1)/(b^2)^(1/2)*arctan((b^2)^ (1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.89 \[ \int e^{\text {arctanh}(a+b x)} x^2 \, dx=-\frac {3 \, {\left (2 \, a^{2} - 2 \, a + 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (2 \, b^{2} x^{2} - {\left (2 \, a - 3\right )} b x + 2 \, a^{2} - 9 \, a + 4\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, b^{3}} \]
-1/6*(3*(2*a^2 - 2*a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (2*b^2*x^2 - (2*a - 3)*b*x + 2*a^2 - 9*a + 4)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/b^3
Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (109) = 218\).
Time = 1.68 (sec) , antiderivative size = 563, normalized size of antiderivative = 4.33 \[ \int e^{\text {arctanh}(a+b x)} x^2 \, dx=\begin {cases} \left (- \frac {x^{2}}{3 b} - \frac {x \left (1 - \frac {2 a}{3}\right )}{2 b^{2}} - \frac {- \frac {3 a \left (1 - \frac {2 a}{3}\right )}{2 b} + \frac {2 - 2 a^{2}}{3 b}}{b^{2}}\right ) \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} + \frac {\left (- \frac {a \left (- \frac {3 a \left (1 - \frac {2 a}{3}\right )}{2 b} + \frac {2 - 2 a^{2}}{3 b}\right )}{b} + \frac {\left (1 - \frac {2 a}{3}\right ) \left (1 - a^{2}\right )}{2 b^{2}}\right ) \log {\left (- 2 a b - 2 b^{2} x + 2 \sqrt {- b^{2}} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \right )}}{\sqrt {- b^{2}}} & \text {for}\: b^{2} \neq 0 \\- \frac {\frac {a^{4} \sqrt {- a^{2} - 2 a b x + 1} - 2 a^{2} \sqrt {- a^{2} - 2 a b x + 1} + \frac {\left (2 a^{2} - 2\right ) \left (- a^{2} - 2 a b x + 1\right )^{\frac {3}{2}}}{3} + \frac {\left (- a^{2} - 2 a b x + 1\right )^{\frac {5}{2}}}{5} + \sqrt {- a^{2} - 2 a b x + 1}}{2 a b^{2}} + \frac {a^{4} \sqrt {- a^{2} - 2 a b x + 1} - 2 a^{2} \sqrt {- a^{2} - 2 a b x + 1} + \frac {\left (2 a^{2} - 2\right ) \left (- a^{2} - 2 a b x + 1\right )^{\frac {3}{2}}}{3} + \frac {\left (- a^{2} - 2 a b x + 1\right )^{\frac {5}{2}}}{5} + \sqrt {- a^{2} - 2 a b x + 1}}{2 a^{2} b^{2}} - \frac {a^{6} \sqrt {- a^{2} - 2 a b x + 1} - 3 a^{4} \sqrt {- a^{2} - 2 a b x + 1} + 3 a^{2} \sqrt {- a^{2} - 2 a b x + 1} + \frac {\left (3 a^{2} - 3\right ) \left (- a^{2} - 2 a b x + 1\right )^{\frac {5}{2}}}{5} + \frac {\left (- a^{2} - 2 a b x + 1\right )^{\frac {7}{2}}}{7} + \frac {\left (- a^{2} - 2 a b x + 1\right )^{\frac {3}{2}} \cdot \left (3 a^{4} - 6 a^{2} + 3\right )}{3} - \sqrt {- a^{2} - 2 a b x + 1}}{4 a^{3} b^{2}}}{2 a b} & \text {for}\: a b \neq 0 \\\frac {\frac {a x^{3}}{3} + \frac {b x^{4}}{4} + \frac {x^{3}}{3}}{\sqrt {1 - a^{2}}} & \text {otherwise} \end {cases} \]
Piecewise(((-x**2/(3*b) - x*(1 - 2*a/3)/(2*b**2) - (-3*a*(1 - 2*a/3)/(2*b) + (2 - 2*a**2)/(3*b))/b**2)*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + (-a*( -3*a*(1 - 2*a/3)/(2*b) + (2 - 2*a**2)/(3*b))/b + (1 - 2*a/3)*(1 - a**2)/(2 *b**2))*log(-2*a*b - 2*b**2*x + 2*sqrt(-b**2)*sqrt(-a**2 - 2*a*b*x - b**2* x**2 + 1))/sqrt(-b**2), Ne(b**2, 0)), (-((a**4*sqrt(-a**2 - 2*a*b*x + 1) - 2*a**2*sqrt(-a**2 - 2*a*b*x + 1) + (2*a**2 - 2)*(-a**2 - 2*a*b*x + 1)**(3 /2)/3 + (-a**2 - 2*a*b*x + 1)**(5/2)/5 + sqrt(-a**2 - 2*a*b*x + 1))/(2*a*b **2) + (a**4*sqrt(-a**2 - 2*a*b*x + 1) - 2*a**2*sqrt(-a**2 - 2*a*b*x + 1) + (2*a**2 - 2)*(-a**2 - 2*a*b*x + 1)**(3/2)/3 + (-a**2 - 2*a*b*x + 1)**(5/ 2)/5 + sqrt(-a**2 - 2*a*b*x + 1))/(2*a**2*b**2) - (a**6*sqrt(-a**2 - 2*a*b *x + 1) - 3*a**4*sqrt(-a**2 - 2*a*b*x + 1) + 3*a**2*sqrt(-a**2 - 2*a*b*x + 1) + (3*a**2 - 3)*(-a**2 - 2*a*b*x + 1)**(5/2)/5 + (-a**2 - 2*a*b*x + 1)* *(7/2)/7 + (-a**2 - 2*a*b*x + 1)**(3/2)*(3*a**4 - 6*a**2 + 3)/3 - sqrt(-a* *2 - 2*a*b*x + 1))/(4*a**3*b**2))/(2*a*b), Ne(a*b, 0)), ((a*x**3/3 + b*x** 4/4 + x**3/3)/sqrt(1 - a**2), True))
Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (110) = 220\).
Time = 0.30 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.73 \[ \int e^{\text {arctanh}(a+b x)} x^2 \, dx=-\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x^{2}}{3 \, b} - \frac {3 \, {\left (a + 1\right )} a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{3}} + \frac {5 \, a^{3} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{3}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a + 1\right )} x}{2 \, b^{2}} + \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a x}{6 \, b^{2}} + \frac {{\left (a^{2} - 1\right )} {\left (a + 1\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{3}} - \frac {3 \, {\left (a^{2} - 1\right )} a \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{3}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a + 1\right )} a}{2 \, b^{3}} - \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{2 \, b^{3}} + \frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )}}{3 \, b^{3}} \]
-1/3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*x^2/b - 3/2*(a + 1)*a^2*arcsin(-(b ^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^3 + 5/2*a^3*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^3 - 1/2*sqrt(-b^2*x^2 - 2*a*b*x - a^ 2 + 1)*(a + 1)*x/b^2 + 5/6*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a*x/b^2 + 1/ 2*(a^2 - 1)*(a + 1)*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b ^3 - 3/2*(a^2 - 1)*a*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/ b^3 + 3/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a + 1)*a/b^3 - 5/2*sqrt(-b^2 *x^2 - 2*a*b*x - a^2 + 1)*a^2/b^3 + 2/3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1) *(a^2 - 1)/b^3
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82 \[ \int e^{\text {arctanh}(a+b x)} x^2 \, dx=-\frac {1}{6} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (x {\left (\frac {2 \, x}{b} - \frac {2 \, a b^{3} - 3 \, b^{3}}{b^{5}}\right )} + \frac {2 \, a^{2} b^{2} - 9 \, a b^{2} + 4 \, b^{2}}{b^{5}}\right )} - \frac {{\left (2 \, a^{2} - 2 \, a + 1\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{2 \, b^{2} {\left | b \right |}} \]
-1/6*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(x*(2*x/b - (2*a*b^3 - 3*b^3)/b^5) + (2*a^2*b^2 - 9*a*b^2 + 4*b^2)/b^5) - 1/2*(2*a^2 - 2*a + 1)*arcsin(-b*x - a)*sgn(b)/(b^2*abs(b))
Timed out. \[ \int e^{\text {arctanh}(a+b x)} x^2 \, dx=\int \frac {x^2\,\left (a+b\,x+1\right )}{\sqrt {1-{\left (a+b\,x\right )}^2}} \,d x \]