3.70.95 \(\int \frac {e^{x^2} (x+e^{-2+2 x} x+x^2+(x+2 x^3+2 x^4+e^{-2+2 x} (2 x+2 x^3)) \log (x)+(1+e^{-2+2 x}+x+(-1-x+2 x^2+2 x^3+e^{-2+2 x} (-1+2 x^2)) \log (x)) \log (1+e^{-2+2 x}+x))}{x^2+e^{-2+2 x} x^2+x^3} \, dx\)

Optimal. Leaf size=24 \[ \frac {e^{x^2} \log (x) \left (x+\log \left (1+e^{-2+2 x}+x\right )\right )}{x} \]

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Rubi [A]  time = 14.17, antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps used = 91, number of rules used = 18, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.142, Rules used = {6742, 2554, 14, 6688, 2210, 2199, 2194, 2178, 12, 6483, 6475, 2288, 2214, 2204, 6351, 6360, 2557, 6715} \begin {gather*} e^{x^2} \log (x)+\frac {e^{x^2} \log \left (x+e^{2 x-2}+1\right ) \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x^2*(x + E^(-2 + 2*x)*x + x^2 + (x + 2*x^3 + 2*x^4 + E^(-2 + 2*x)*(2*x + 2*x^3))*Log[x] + (1 + E^(-2 +
2*x) + x + (-1 - x + 2*x^2 + 2*x^3 + E^(-2 + 2*x)*(-1 + 2*x^2))*Log[x])*Log[1 + E^(-2 + 2*x) + x]))/(x^2 + E^(
-2 + 2*x)*x^2 + x^3),x]

[Out]

E^x^2*Log[x] + (E^x^2*Log[x]*Log[1 + E^(-2 + 2*x) + x])/x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 2557

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[(z*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(z*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]

Rule 6351

Int[Erfi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*Erfi[a + b*x])/b, x] - Simp[E^(a + b*x)^2/(b*Sqrt[P
i]), x] /; FreeQ[{a, b}, x]

Rule 6360

Int[Erfi[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[(2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, b^2*x^2])/Sqrt[P
i], x] /; FreeQ[b, x]

Rule 6475

Int[ExpIntegralE[1, (b_.)*(x_)]/(x_), x_Symbol] :> Simp[b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)], x
] + (-Simp[EulerGamma*Log[x], x] - Simp[(1*Log[b*x]^2)/2, x]) /; FreeQ[b, x]

Rule 6483

Int[ExpIntegralEi[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[Log[x]*(ExpIntegralEi[b*x] + ExpIntegralE[1, -(b*x)]), x
] - Int[ExpIntegralE[1, -(b*x)]/x, x] /; FreeQ[b, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {e^{2+x^2} (1+2 x) \log (x)}{x \left (e^2+e^{2 x}+e^2 x\right )}+\frac {e^{x^2} \left (x+2 x \log (x)+2 x^3 \log (x)+\log \left (1+e^{-2+2 x}+x\right )-\log (x) \log \left (1+e^{-2+2 x}+x\right )+2 x^2 \log (x) \log \left (1+e^{-2+2 x}+x\right )\right )}{x^2}\right ) \, dx\\ &=-\int \frac {e^{2+x^2} (1+2 x) \log (x)}{x \left (e^2+e^{2 x}+e^2 x\right )} \, dx+\int \frac {e^{x^2} \left (x+2 x \log (x)+2 x^3 \log (x)+\log \left (1+e^{-2+2 x}+x\right )-\log (x) \log \left (1+e^{-2+2 x}+x\right )+2 x^2 \log (x) \log \left (1+e^{-2+2 x}+x\right )\right )}{x^2} \, dx\\ &=-\left (\log (x) \int \frac {e^{2+x^2}}{x \left (e^2+e^{2 x}+e^2 x\right )} \, dx\right )-(2 \log (x)) \int \frac {e^{2+x^2}}{e^2+e^{2 x}+e^2 x} \, dx+\int \frac {e^{x^2} \left (x+\log \left (1+e^{-2+2 x}+x\right )+\log (x) \left (2 \left (x+x^3\right )+\left (-1+2 x^2\right ) \log \left (1+e^{-2+2 x}+x\right )\right )\right )}{x^2} \, dx+\int \frac {2 \int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx+\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx}{x} \, dx\\ &=-\left (\log (x) \int \frac {e^{2+x^2}}{x \left (e^2+e^{2 x}+e^2 x\right )} \, dx\right )-(2 \log (x)) \int \frac {e^{2+x^2}}{e^2+e^{2 x}+e^2 x} \, dx+\int \left (\frac {e^{x^2} \left (1+2 \log (x)+2 x^2 \log (x)\right )}{x}+\frac {e^{x^2} \left (1-\log (x)+2 x^2 \log (x)\right ) \log \left (1+e^{-2+2 x}+x\right )}{x^2}\right ) \, dx+\int \left (\frac {2 \int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx}{x}+\frac {\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx}{x}\right ) \, dx\\ &=2 \int \frac {\int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx-\log (x) \int \frac {e^{2+x^2}}{x \left (e^2+e^{2 x}+e^2 x\right )} \, dx-(2 \log (x)) \int \frac {e^{2+x^2}}{e^2+e^{2 x}+e^2 x} \, dx+\int \frac {e^{x^2} \left (1+2 \log (x)+2 x^2 \log (x)\right )}{x} \, dx+\int \frac {e^{x^2} \left (1-\log (x)+2 x^2 \log (x)\right ) \log \left (1+e^{-2+2 x}+x\right )}{x^2} \, dx+\int \frac {\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx}{x} \, dx\\ &=2 \int \frac {\int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx-\log (x) \int \frac {e^{2+x^2}}{x \left (e^2+e^{2 x}+e^2 x\right )} \, dx-(2 \log (x)) \int \frac {e^{2+x^2}}{e^2+e^{2 x}+e^2 x} \, dx+\int \left (\frac {e^{x^2}}{x}+\frac {2 e^{x^2} \left (1+x^2\right ) \log (x)}{x}\right ) \, dx+\int \left (\frac {e^{x^2} \log \left (1+e^{-2+2 x}+x\right )}{x^2}+2 e^{x^2} \log (x) \log \left (1+e^{-2+2 x}+x\right )-\frac {e^{x^2} \log (x) \log \left (1+e^{-2+2 x}+x\right )}{x^2}\right ) \, dx+\int \frac {\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx}{x} \, dx\\ &=2 \int \frac {e^{x^2} \left (1+x^2\right ) \log (x)}{x} \, dx+2 \int e^{x^2} \log (x) \log \left (1+e^{-2+2 x}+x\right ) \, dx+2 \int \frac {\int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx-\log (x) \int \frac {e^{2+x^2}}{x \left (e^2+e^{2 x}+e^2 x\right )} \, dx-(2 \log (x)) \int \frac {e^{2+x^2}}{e^2+e^{2 x}+e^2 x} \, dx+\int \frac {e^{x^2}}{x} \, dx+\int \frac {e^{x^2} \log \left (1+e^{-2+2 x}+x\right )}{x^2} \, dx-\int \frac {e^{x^2} \log (x) \log \left (1+e^{-2+2 x}+x\right )}{x^2} \, dx+\int \frac {\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx}{x} \, dx\\ &=\frac {\text {Ei}\left (x^2\right )}{2}+e^{x^2} \log (x)+\text {Ei}\left (x^2\right ) \log (x)-\frac {e^{x^2} \log \left (1+e^{-2+2 x}+x\right )}{x}+\sqrt {\pi } \text {erfi}(x) \log \left (1+e^{-2+2 x}+x\right )+\frac {e^{x^2} \log (x) \log \left (1+e^{-2+2 x}+x\right )}{x}-2 \int \frac {e^{x^2}+\text {Ei}\left (x^2\right )}{2 x} \, dx-2 \int \frac {\left (1+2 e^{-2+2 x}\right ) \sqrt {\pi } \text {erfi}(x) \log (x)}{2 \left (1+e^{-2+2 x}+x\right )} \, dx-2 \int \frac {\sqrt {\pi } \text {erfi}(x) \log \left (1+e^{-2+2 x}+x\right )}{2 x} \, dx+2 \int \frac {\int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx-\log (x) \int \frac {e^{2+x^2}}{x \left (e^2+e^{2 x}+e^2 x\right )} \, dx-(2 \log (x)) \int \frac {e^{2+x^2}}{e^2+e^{2 x}+e^2 x} \, dx-\int \frac {\left (1+2 e^{-2+2 x}\right ) \left (-\frac {e^{x^2}}{x}+\sqrt {\pi } \text {erfi}(x)\right )}{1+e^{-2+2 x}+x} \, dx+\int \frac {\left (1+2 e^{-2+2 x}\right ) \left (-\frac {e^{x^2}}{x}+\sqrt {\pi } \text {erfi}(x)\right ) \log (x)}{1+e^{-2+2 x}+x} \, dx+\int \frac {\left (-e^{x^2}+\sqrt {\pi } x \text {erfi}(x)\right ) \log \left (1+e^{-2+2 x}+x\right )}{x^2} \, dx+\int \frac {\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx}{x} \, dx\\ &=\frac {\text {Ei}\left (x^2\right )}{2}-e^{x^2} \log (x)+2 \sqrt {\pi } x \text {erfi}(x) \log (x)+2 x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right ) \log \left (1+e^{-2+2 x}+x\right )+\frac {e^{x^2} \log (x) \log \left (1+e^{-2+2 x}+x\right )}{x}+2 \int \frac {\int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx-\sqrt {\pi } \int \frac {\left (1+2 e^{-2+2 x}\right ) \text {erfi}(x) \log (x)}{1+e^{-2+2 x}+x} \, dx-\sqrt {\pi } \int \frac {\text {erfi}(x) \log \left (1+e^{-2+2 x}+x\right )}{x} \, dx-\left (e^2 \sqrt {\pi } \log (x)\right ) \int \frac {\text {erfi}(x)}{e^2+e^{2 x}+e^2 x} \, dx-\left (2 e^2 \sqrt {\pi } \log (x)\right ) \int \frac {x \text {erfi}(x)}{e^2+e^{2 x}+e^2 x} \, dx-\int \left (-\frac {e^{x^2} \left (e^2+2 e^{2 x}\right )}{x \left (e^2+e^{2 x}+e^2 x\right )}+\frac {\left (e^2+2 e^{2 x}\right ) \sqrt {\pi } \text {erfi}(x)}{e^2+e^{2 x}+e^2 x}\right ) \, dx-\int \frac {e^{x^2}+\text {Ei}\left (x^2\right )}{x} \, dx-\int \frac {\left (1+2 e^{-2+2 x}\right ) \left (\frac {e^{x^2}}{x}-\sqrt {\pi } \text {erfi}(x)+2 x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )\right )}{1+e^{-2+2 x}+x} \, dx+\int \frac {\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx}{x} \, dx-\int \frac {-2 e^{x^2}+2 \sqrt {\pi } x \text {erfi}(x)-\text {Ei}\left (x^2\right )+2 \int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx+\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx-e^2 \sqrt {\pi } \int \frac {\text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx-2 e^2 \sqrt {\pi } \int \frac {x \text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx\\ &=\frac {\text {Ei}\left (x^2\right )}{2}+e^{x^2} \log (x)+\frac {e^{x^2} \log (x) \log \left (1+e^{-2+2 x}+x\right )}{x}+2 \int \frac {\int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx-\sqrt {\pi } \int \frac {\left (e^2+2 e^{2 x}\right ) \text {erfi}(x)}{e^2+e^{2 x}+e^2 x} \, dx+\sqrt {\pi } \int \frac {2 \left (1+2 e^{-2+2 x}\right ) x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )}{\sqrt {\pi } \left (1+e^{-2+2 x}+x\right )} \, dx+\sqrt {\pi } \int \frac {-\frac {2 e^{x^2}}{\sqrt {\pi }}+2 x \text {erfi}(x)-e^2 \int \frac {\text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx-2 e^2 \int \frac {x \text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx+\int \frac {e^{x^2} \left (e^2+2 e^{2 x}\right )}{x \left (e^2+e^{2 x}+e^2 x\right )} \, dx-\int \left (\frac {e^{x^2}}{x}+\frac {\text {Ei}\left (x^2\right )}{x}\right ) \, dx-\int \left (\frac {e^{x^2} \left (e^2+2 e^{2 x}\right )}{x \left (e^2+e^{2 x}+e^2 x\right )}+\frac {\left (e^2+2 e^{2 x}\right ) \left (-\sqrt {\pi } \text {erfi}(x)+2 x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )\right )}{e^2+e^{2 x}+e^2 x}\right ) \, dx+\int \frac {\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx}{x} \, dx-\int \left (-\frac {2 e^{x^2}}{x}+\frac {2 \sqrt {\pi } x \text {erfi}(x)-\text {Ei}\left (x^2\right )+2 \int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx+\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx-e^2 \sqrt {\pi } \int \frac {\text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx-2 e^2 \sqrt {\pi } \int \frac {x \text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x}\right ) \, dx\\ &=\frac {\text {Ei}\left (x^2\right )}{2}+e^{x^2} \log (x)+\frac {e^{x^2} \log (x) \log \left (1+e^{-2+2 x}+x\right )}{x}+2 \int \frac {e^{x^2}}{x} \, dx+2 \int \frac {\left (1+2 e^{-2+2 x}\right ) x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )}{1+e^{-2+2 x}+x} \, dx+2 \int \frac {\int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx-\sqrt {\pi } \int \left (2 \text {erfi}(x)-\frac {e^2 (1+2 x) \text {erfi}(x)}{e^2+e^{2 x}+e^2 x}\right ) \, dx+\sqrt {\pi } \int \left (-\frac {2 e^{x^2}}{\sqrt {\pi } x}+\frac {2 x \text {erfi}(x)-e^2 \int \frac {\text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx-2 e^2 \int \frac {x \text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x}\right ) \, dx-\int \frac {e^{x^2}}{x} \, dx-\int \frac {e^{x^2} \left (e^2+2 e^{2 x}\right )}{x \left (e^2+e^{2 x}+e^2 x\right )} \, dx+\int \left (\frac {2 e^{x^2}}{x}-\frac {e^{2+x^2} (1+2 x)}{x \left (e^2+e^{2 x}+e^2 x\right )}\right ) \, dx-\int \frac {\text {Ei}\left (x^2\right )}{x} \, dx-\int \frac {\left (e^2+2 e^{2 x}\right ) \left (-\sqrt {\pi } \text {erfi}(x)+2 x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )\right )}{e^2+e^{2 x}+e^2 x} \, dx+\int \frac {\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx}{x} \, dx-\int \frac {2 \sqrt {\pi } x \text {erfi}(x)-\text {Ei}\left (x^2\right )+2 \int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx+\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx-e^2 \sqrt {\pi } \int \frac {\text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx-2 e^2 \sqrt {\pi } \int \frac {x \text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx\\ &=\text {Ei}\left (x^2\right )+e^{x^2} \log (x)+\frac {e^{x^2} \log (x) \log \left (1+e^{-2+2 x}+x\right )}{x}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\text {Ei}(x)}{x} \, dx,x,x^2\right )+2 \int \left (2 x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )-\frac {e^2 x (1+2 x) \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )}{e^2+e^{2 x}+e^2 x}\right ) \, dx+2 \int \frac {\int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx+\sqrt {\pi } \int \frac {2 x \text {erfi}(x)-e^2 \int \frac {\text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx-2 e^2 \int \frac {x \text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx-\left (2 \sqrt {\pi }\right ) \int \text {erfi}(x) \, dx+\left (e^2 \sqrt {\pi }\right ) \int \frac {(1+2 x) \text {erfi}(x)}{e^2+e^{2 x}+e^2 x} \, dx-\int \frac {e^{2+x^2} (1+2 x)}{x \left (e^2+e^{2 x}+e^2 x\right )} \, dx-\int \left (\frac {2 e^{x^2}}{x}-\frac {e^{2+x^2} (1+2 x)}{x \left (e^2+e^{2 x}+e^2 x\right )}\right ) \, dx-\int \left (-2 \left (\sqrt {\pi } \text {erfi}(x)-2 x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )\right )-\frac {e^2 (1+2 x) \left (-\sqrt {\pi } \text {erfi}(x)+2 x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )\right )}{e^2+e^{2 x}+e^2 x}\right ) \, dx+\int \frac {\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx}{x} \, dx-\int \left (\frac {2 \sqrt {\pi } x \text {erfi}(x)-\text {Ei}\left (x^2\right )+2 \int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx+\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx-e^2 \sqrt {\pi } \int \frac {\text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x}-\frac {2 e^2 \sqrt {\pi } \int \frac {x \text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x}\right ) \, dx\\ &=2 e^{x^2}-2 \sqrt {\pi } x \text {erfi}(x)+\text {Ei}\left (x^2\right )+e^{x^2} \log (x)-\frac {1}{2} \left (E_1\left (-x^2\right )+\text {Ei}\left (x^2\right )\right ) \log \left (x^2\right )+\frac {e^{x^2} \log (x) \log \left (1+e^{-2+2 x}+x\right )}{x}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {E_1(-x)}{x} \, dx,x,x^2\right )-2 \int \frac {e^{x^2}}{x} \, dx+2 \int \left (\sqrt {\pi } \text {erfi}(x)-2 x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )\right ) \, dx+2 \int \frac {\int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx+4 \int x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right ) \, dx+e^2 \int \frac {(1+2 x) \left (-\sqrt {\pi } \text {erfi}(x)+2 x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )\right )}{e^2+e^{2 x}+e^2 x} \, dx-\left (2 e^2\right ) \int \frac {x (1+2 x) \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )}{e^2+e^{2 x}+e^2 x} \, dx+\sqrt {\pi } \int \left (\frac {2 x \text {erfi}(x)-e^2 \int \frac {\text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x}-\frac {2 e^2 \int \frac {x \text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x}\right ) \, dx+\left (e^2 \sqrt {\pi }\right ) \int \left (\frac {\text {erfi}(x)}{e^2+e^{2 x}+e^2 x}+\frac {2 x \text {erfi}(x)}{e^2+e^{2 x}+e^2 x}\right ) \, dx+\left (2 e^2 \sqrt {\pi }\right ) \int \frac {\int \frac {x \text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx+\int \frac {e^{2+x^2} (1+2 x)}{x \left (e^2+e^{2 x}+e^2 x\right )} \, dx-\int \left (\frac {2 e^{2+x^2}}{e^2+e^{2 x}+e^2 x}+\frac {e^{2+x^2}}{x \left (e^2+e^{2 x}+e^2 x\right )}\right ) \, dx+\int \frac {\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx}{x} \, dx-\int \frac {2 \sqrt {\pi } x \text {erfi}(x)-\text {Ei}\left (x^2\right )+2 \int \frac {e^{2+x^2}}{e^{2 x}+e^2 (1+x)} \, dx+\int \frac {e^{2+x^2}}{x \left (e^{2 x}+e^2 (1+x)\right )} \, dx-e^2 \sqrt {\pi } \int \frac {\text {erfi}(x)}{e^{2 x}+e^2 (1+x)} \, dx}{x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 24, normalized size = 1.00 \begin {gather*} \frac {e^{x^2} \log (x) \left (x+\log \left (1+e^{-2+2 x}+x\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x^2*(x + E^(-2 + 2*x)*x + x^2 + (x + 2*x^3 + 2*x^4 + E^(-2 + 2*x)*(2*x + 2*x^3))*Log[x] + (1 + E^
(-2 + 2*x) + x + (-1 - x + 2*x^2 + 2*x^3 + E^(-2 + 2*x)*(-1 + 2*x^2))*Log[x])*Log[1 + E^(-2 + 2*x) + x]))/(x^2
+ E^(-2 + 2*x)*x^2 + x^3),x]

[Out]

(E^x^2*Log[x]*(x + Log[1 + E^(-2 + 2*x) + x]))/x

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fricas [A]  time = 0.50, size = 24, normalized size = 1.00 \begin {gather*} \frac {{\left (x + \log \left (x + e^{\left (2 \, x - 2\right )} + 1\right )\right )} e^{\left (x^{2} + \log \left (\log \relax (x)\right )\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x^2-1)*exp(x-1)^2+2*x^3+2*x^2-x-1)*log(x)+exp(x-1)^2+x+1)*log(exp(x-1)^2+x+1)+((2*x^3+2*x)*exp
(x-1)^2+2*x^4+2*x^3+x)*log(x)+x*exp(x-1)^2+x^2+x)*exp(log(log(x))+x^2)/(x^2*exp(x-1)^2+x^3+x^2)/log(x),x, algo
rithm="fricas")

[Out]

(x + log(x + e^(2*x - 2) + 1))*e^(x^2 + log(log(x)))/x

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giac [A]  time = 0.22, size = 40, normalized size = 1.67 \begin {gather*} \frac {x e^{\left (x^{2}\right )} \log \relax (x) + e^{\left (x^{2}\right )} \log \left (x e^{2} + e^{2} + e^{\left (2 \, x\right )}\right ) \log \relax (x) - 2 \, e^{\left (x^{2}\right )} \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x^2-1)*exp(x-1)^2+2*x^3+2*x^2-x-1)*log(x)+exp(x-1)^2+x+1)*log(exp(x-1)^2+x+1)+((2*x^3+2*x)*exp
(x-1)^2+2*x^4+2*x^3+x)*log(x)+x*exp(x-1)^2+x^2+x)*exp(log(log(x))+x^2)/(x^2*exp(x-1)^2+x^3+x^2)/log(x),x, algo
rithm="giac")

[Out]

(x*e^(x^2)*log(x) + e^(x^2)*log(x*e^2 + e^2 + e^(2*x))*log(x) - 2*e^(x^2)*log(x))/x

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maple [A]  time = 0.05, size = 23, normalized size = 0.96

method	result	size
risch	\(\frac {\left (\ln \left ({\mathrm e}^{2 x -2}+x +1\right )+x \right ) {\mathrm e}^{x^{2}} \ln \relax (x )}{x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((2*x^2-1)*exp(x-1)^2+2*x^3+2*x^2-x-1)*ln(x)+exp(x-1)^2+x+1)*ln(exp(x-1)^2+x+1)+((2*x^3+2*x)*exp(x-1)^2+
2*x^4+2*x^3+x)*ln(x)+x*exp(x-1)^2+x^2+x)*exp(ln(ln(x))+x^2)/(x^2*exp(x-1)^2+x^3+x^2)/ln(x),x,method=_RETURNVER
BOSE)

[Out]

(ln(exp(2*x-2)+x+1)+x)*exp(x^2)*ln(x)/x

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maxima [A]  time = 0.46, size = 34, normalized size = 1.42 \begin {gather*} \frac {{\left (x - 2\right )} e^{\left (x^{2}\right )} \log \relax (x) + e^{\left (x^{2}\right )} \log \left (x e^{2} + e^{2} + e^{\left (2 \, x\right )}\right ) \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x^2-1)*exp(x-1)^2+2*x^3+2*x^2-x-1)*log(x)+exp(x-1)^2+x+1)*log(exp(x-1)^2+x+1)+((2*x^3+2*x)*exp
(x-1)^2+2*x^4+2*x^3+x)*log(x)+x*exp(x-1)^2+x^2+x)*exp(log(log(x))+x^2)/(x^2*exp(x-1)^2+x^3+x^2)/log(x),x, algo
rithm="maxima")

[Out]

((x - 2)*e^(x^2)*log(x) + e^(x^2)*log(x*e^2 + e^2 + e^(2*x))*log(x))/x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{\ln \left (\ln \relax (x)\right )+x^2}\,\left (x+\ln \left (x+{\mathrm {e}}^{2\,x-2}+1\right )\,\left (x+{\mathrm {e}}^{2\,x-2}+\ln \relax (x)\,\left ({\mathrm {e}}^{2\,x-2}\,\left (2\,x^2-1\right )-x+2\,x^2+2\,x^3-1\right )+1\right )+x\,{\mathrm {e}}^{2\,x-2}+x^2+\ln \relax (x)\,\left (x+{\mathrm {e}}^{2\,x-2}\,\left (2\,x^3+2\,x\right )+2\,x^3+2\,x^4\right )\right )}{\ln \relax (x)\,\left (x^2\,{\mathrm {e}}^{2\,x-2}+x^2+x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(log(x)) + x^2)*(x + log(x + exp(2*x - 2) + 1)*(x + exp(2*x - 2) + log(x)*(exp(2*x - 2)*(2*x^2 - 1
) - x + 2*x^2 + 2*x^3 - 1) + 1) + x*exp(2*x - 2) + x^2 + log(x)*(x + exp(2*x - 2)*(2*x + 2*x^3) + 2*x^3 + 2*x^
4)))/(log(x)*(x^2*exp(2*x - 2) + x^2 + x^3)),x)

[Out]

int((exp(log(log(x)) + x^2)*(x + log(x + exp(2*x - 2) + 1)*(x + exp(2*x - 2) + log(x)*(exp(2*x - 2)*(2*x^2 - 1
) - x + 2*x^2 + 2*x^3 - 1) + 1) + x*exp(2*x - 2) + x^2 + log(x)*(x + exp(2*x - 2)*(2*x + 2*x^3) + 2*x^3 + 2*x^
4)))/(log(x)*(x^2*exp(2*x - 2) + x^2 + x^3)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x**2-1)*exp(x-1)**2+2*x**3+2*x**2-x-1)*ln(x)+exp(x-1)**2+x+1)*ln(exp(x-1)**2+x+1)+((2*x**3+2*x
)*exp(x-1)**2+2*x**4+2*x**3+x)*ln(x)+x*exp(x-1)**2+x**2+x)*exp(ln(ln(x))+x**2)/(x**2*exp(x-1)**2+x**3+x**2)/ln
(x),x)

[Out]

Timed out

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