Integrand size = 14, antiderivative size = 282 \[ \int e^{\frac {3}{2} \text {arctanh}(a x)} x^2 \, dx=-\frac {17 \sqrt [4]{1-a x} (1+a x)^{3/4}}{24 a^3}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4}}{4 a^3}-\frac {x \sqrt [4]{1-a x} (1+a x)^{7/4}}{3 a^2}+\frac {17 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}-\frac {17 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}+\frac {17 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3}-\frac {17 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3} \]
-17/24*(-a*x+1)^(1/4)*(a*x+1)^(3/4)/a^3-1/4*(-a*x+1)^(1/4)*(a*x+1)^(7/4)/a ^3-1/3*x*(-a*x+1)^(1/4)*(a*x+1)^(7/4)/a^2-17/16*arctan(-1+(-a*x+1)^(1/4)*2 ^(1/2)/(a*x+1)^(1/4))/a^3*2^(1/2)-17/16*arctan(1+(-a*x+1)^(1/4)*2^(1/2)/(a *x+1)^(1/4))/a^3*2^(1/2)+17/32*ln(1-(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+( -a*x+1)^(1/2)/(a*x+1)^(1/2))/a^3*2^(1/2)-17/32*ln(1+(-a*x+1)^(1/4)*2^(1/2) /(a*x+1)^(1/4)+(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a^3*2^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.24 \[ \int e^{\frac {3}{2} \text {arctanh}(a x)} x^2 \, dx=-\frac {\sqrt [4]{1-a x} \left ((1+a x)^{3/4} \left (3+7 a x+4 a^2 x^2\right )+34\ 2^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{4},\frac {5}{4},\frac {1}{2} (1-a x)\right )\right )}{12 a^3} \]
-1/12*((1 - a*x)^(1/4)*((1 + a*x)^(3/4)*(3 + 7*a*x + 4*a^2*x^2) + 34*2^(3/ 4)*Hypergeometric2F1[-3/4, 1/4, 5/4, (1 - a*x)/2]))/a^3
Time = 0.46 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.96, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6676, 101, 27, 90, 60, 73, 770, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int x^2 e^{\frac {3}{2} \text {arctanh}(a x)} \, dx\) |
| \(\Big \downarrow \) 6676 |
| \(\displaystyle \int \frac {x^2 (a x+1)^{3/4}}{(1-a x)^{3/4}}dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {\int -\frac {(a x+1)^{3/4} (3 a x+2)}{2 (1-a x)^{3/4}}dx}{3 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a x+1)^{3/4} (3 a x+2)}{(1-a x)^{3/4}}dx}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {\frac {17}{4} \int \frac {(a x+1)^{3/4}}{(1-a x)^{3/4}}dx-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {17}{4} \left (\frac {3}{2} \int \frac {1}{(1-a x)^{3/4} \sqrt [4]{a x+1}}dx-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {17}{4} \left (-\frac {6 \int \frac {1}{\sqrt [4]{a x+1}}d\sqrt [4]{1-a x}}{a}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {\frac {17}{4} \left (-\frac {6 \int \frac {1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{a}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {17}{4} \left (-\frac {6 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {\sqrt {1-a x}+1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {17}{4} \left (-\frac {6 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )\right )}{a}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {17}{4} \left (-\frac {6 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {17}{4} \left (-\frac {6 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {17}{4} \left (-\frac {6 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {17}{4} \left (-\frac {6 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {17}{4} \left (-\frac {6 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {17}{4} \left (-\frac {6 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )-\frac {3 \sqrt [4]{1-a x} (a x+1)^{7/4}}{2 a}}{6 a^2}-\frac {x \sqrt [4]{1-a x} (a x+1)^{7/4}}{3 a^2}\) |
-1/3*(x*(1 - a*x)^(1/4)*(1 + a*x)^(7/4))/a^2 + ((-3*(1 - a*x)^(1/4)*(1 + a *x)^(7/4))/(2*a) + (17*(-(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/a) - (6*((-(Ar cTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2])/2 + (-1/2*Log[1 + Sqrt [1 - a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2] + Log[1 + S qrt[1 - a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]))/2)) /a))/4)/(6*a^2)
3.1.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
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]
\[\int {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}^{\frac {3}{2}} x^{2}d x\]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.89 \[ \int e^{\frac {3}{2} \text {arctanh}(a x)} x^2 \, dx=\frac {51 \, a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} \log \left (a^{9} \left (-\frac {1}{a^{12}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 51 i \, a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} \log \left (i \, a^{9} \left (-\frac {1}{a^{12}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 51 i \, a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} \log \left (-i \, a^{9} \left (-\frac {1}{a^{12}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 51 \, a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} \log \left (-a^{9} \left (-\frac {1}{a^{12}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 2 \, {\left (8 \, a^{2} x^{2} + 14 \, a x + 23\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, a^{3}} \]
1/48*(51*a^3*(-1/a^12)^(1/4)*log(a^9*(-1/a^12)^(3/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 51*I*a^3*(-1/a^12)^(1/4)*log(I*a^9*(-1/a^12)^(3/4) + s qrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 51*I*a^3*(-1/a^12)^(1/4)*log(-I*a^9* (-1/a^12)^(3/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 51*a^3*(-1/a^12)^ (1/4)*log(-a^9*(-1/a^12)^(3/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 2* (8*a^2*x^2 + 14*a*x + 23)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^3
\[ \int e^{\frac {3}{2} \text {arctanh}(a x)} x^2 \, dx=\int x^{2} \left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}\, dx \]
\[ \int e^{\frac {3}{2} \text {arctanh}(a x)} x^2 \, dx=\int { x^{2} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {3}{2}} \,d x } \]
Exception generated. \[ \int e^{\frac {3}{2} \text {arctanh}(a x)} x^2 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int e^{\frac {3}{2} \text {arctanh}(a x)} x^2 \, dx=\int x^2\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{3/2} \,d x \]