Integrand size = 12, antiderivative size = 255 \[ \int e^{-\frac {1}{2} \text {arctanh}(a x)} x \, dx=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a^2}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 a^2}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt {2} a^2}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt {2} a^2}-\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^2}+\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^2} \]
-1/4*(-a*x+1)^(1/4)*(a*x+1)^(3/4)/a^2-1/2*(-a*x+1)^(5/4)*(a*x+1)^(3/4)/a^2 +1/8*arctan(-1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))/a^2*2^(1/2)+1/8*arcta n(1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))/a^2*2^(1/2)-1/16*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^2*2^(1/2)+1/16* 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^ 2*2^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.22 \[ \int e^{-\frac {1}{2} \text {arctanh}(a x)} x \, dx=\frac {(1-a x)^{5/4} \left (-5 (1+a x)^{3/4}+2^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-a x)\right )\right )}{10 a^2} \]
((1 - a*x)^(5/4)*(-5*(1 + a*x)^(3/4) + 2^(3/4)*Hypergeometric2F1[1/4, 5/4, 9/4, (1 - a*x)/2]))/(10*a^2)
Time = 0.42 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.94, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6676, 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.
| \(\displaystyle \int x e^{-\frac {1}{2} \text {arctanh}(a x)} \, dx\) |
| \(\Big \downarrow \) 6676 |
| \(\displaystyle \int \frac {x \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle -\frac {\int \frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}dx}{4 a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {1}{(1-a x)^{3/4} \sqrt [4]{a x+1}}dx+\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}}{4 a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \int \frac {1}{\sqrt [4]{a x+1}}d\sqrt [4]{1-a x}}{a}}{4 a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \int \frac {1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{a}}{4 a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {\sqrt {1-a x}+1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}}{4 a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )\right )}{a}}{4 a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}}{4 a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}}{4 a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}}{4 a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}}{4 a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}}{4 a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a^2}-\frac {\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}}{4 a}\) |
-1/2*((1 - a*x)^(5/4)*(1 + a*x)^(3/4))/a^2 - (((1 - a*x)^(1/4)*(1 + a*x)^( 3/4))/a - (2*((-(ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqr t[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 + Sqrt[1 - a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4 )]/(2*Sqrt[2]))/2))/a)/(4*a)
3.1.93.3.1 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)^(-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]
\[\int \frac {x}{\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}d x\]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.95 \[ \int e^{-\frac {1}{2} \text {arctanh}(a x)} x \, dx=-\frac {a^{2} \left (-\frac {1}{a^{8}}\right )^{\frac {1}{4}} \log \left (a^{6} \left (-\frac {1}{a^{8}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - i \, a^{2} \left (-\frac {1}{a^{8}}\right )^{\frac {1}{4}} \log \left (i \, a^{6} \left (-\frac {1}{a^{8}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + i \, a^{2} \left (-\frac {1}{a^{8}}\right )^{\frac {1}{4}} \log \left (-i \, a^{6} \left (-\frac {1}{a^{8}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - a^{2} \left (-\frac {1}{a^{8}}\right )^{\frac {1}{4}} \log \left (-a^{6} \left (-\frac {1}{a^{8}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - 3\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, a^{2}} \]
-1/8*(a^2*(-1/a^8)^(1/4)*log(a^6*(-1/a^8)^(3/4) + sqrt(-sqrt(-a^2*x^2 + 1) /(a*x - 1))) - I*a^2*(-1/a^8)^(1/4)*log(I*a^6*(-1/a^8)^(3/4) + sqrt(-sqrt( -a^2*x^2 + 1)/(a*x - 1))) + I*a^2*(-1/a^8)^(1/4)*log(-I*a^6*(-1/a^8)^(3/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - a^2*(-1/a^8)^(1/4)*log(-a^6*(-1/ a^8)^(3/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 2*sqrt(-a^2*x^2 + 1)*( 2*a*x - 3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^2
\[ \int e^{-\frac {1}{2} \text {arctanh}(a x)} x \, dx=\int \frac {x}{\sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}\, dx \]
\[ \int e^{-\frac {1}{2} \text {arctanh}(a x)} x \, dx=\int { \frac {x}{\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}} \,d x } \]
\[ \int e^{-\frac {1}{2} \text {arctanh}(a x)} x \, dx=\int { \frac {x}{\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}} \,d x } \]
Timed out. \[ \int e^{-\frac {1}{2} \text {arctanh}(a x)} x \, dx=\int \frac {x}{\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}} \,d x \]