3.62.90 \(\int \frac {x-2 x \log (x)+(9+e^x (-3-3 x)+e^{x^2} (-3-6 x^2)) \log ^2(x)+3 \log ^3(x)}{3 \log ^2(x)} \, dx\)

Optimal. Leaf size=31 \[ 1+2 x+x \left (-e^x-e^{x^2}-\frac {x}{3 \log (x)}+\log (x)\right ) \]

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Rubi [A]  time = 0.35, antiderivative size = 38, normalized size of antiderivative = 1.23, number of steps used = 18, number of rules used = 12, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {12, 6742, 2226, 2204, 2212, 6688, 2176, 2194, 2306, 2309, 2178, 2295} \begin {gather*} -e^{x^2} x-\frac {x^2}{3 \log (x)}+2 x+e^x-e^x (x+1)+x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x - 2*x*Log[x] + (9 + E^x*(-3 - 3*x) + E^x^2*(-3 - 6*x^2))*Log[x]^2 + 3*Log[x]^3)/(3*Log[x]^2),x]

[Out]

E^x + 2*x - E^x^2*x - E^x*(1 + x) - x^2/(3*Log[x]) + x*Log[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 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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 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 2212

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

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, 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} &=\frac {1}{3} \int \frac {x-2 x \log (x)+\left (9+e^x (-3-3 x)+e^{x^2} \left (-3-6 x^2\right )\right ) \log ^2(x)+3 \log ^3(x)}{\log ^2(x)} \, dx\\ &=\frac {1}{3} \int \left (-3 e^{x^2} \left (1+2 x^2\right )+\frac {x-2 x \log (x)+9 \log ^2(x)-3 e^x \log ^2(x)-3 e^x x \log ^2(x)+3 \log ^3(x)}{\log ^2(x)}\right ) \, dx\\ &=\frac {1}{3} \int \frac {x-2 x \log (x)+9 \log ^2(x)-3 e^x \log ^2(x)-3 e^x x \log ^2(x)+3 \log ^3(x)}{\log ^2(x)} \, dx-\int e^{x^2} \left (1+2 x^2\right ) \, dx\\ &=\frac {1}{3} \int \left (9-3 e^x (1+x)+\frac {x}{\log ^2(x)}-\frac {2 x}{\log (x)}+3 \log (x)\right ) \, dx-\int \left (e^{x^2}+2 e^{x^2} x^2\right ) \, dx\\ &=3 x+\frac {1}{3} \int \frac {x}{\log ^2(x)} \, dx-\frac {2}{3} \int \frac {x}{\log (x)} \, dx-2 \int e^{x^2} x^2 \, dx-\int e^{x^2} \, dx-\int e^x (1+x) \, dx+\int \log (x) \, dx\\ &=2 x-e^{x^2} x-e^x (1+x)-\frac {1}{2} \sqrt {\pi } \text {erfi}(x)-\frac {x^2}{3 \log (x)}+x \log (x)+\frac {2}{3} \int \frac {x}{\log (x)} \, dx-\frac {2}{3} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )+\int e^x \, dx+\int e^{x^2} \, dx\\ &=e^x+2 x-e^{x^2} x-e^x (1+x)-\frac {2}{3} \text {Ei}(2 \log (x))-\frac {x^2}{3 \log (x)}+x \log (x)+\frac {2}{3} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=e^x+2 x-e^{x^2} x-e^x (1+x)-\frac {x^2}{3 \log (x)}+x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 33, normalized size = 1.06 \begin {gather*} 2 x-e^x x-e^{x^2} x-\frac {x^2}{3 \log (x)}+x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x - 2*x*Log[x] + (9 + E^x*(-3 - 3*x) + E^x^2*(-3 - 6*x^2))*Log[x]^2 + 3*Log[x]^3)/(3*Log[x]^2),x]

[Out]

2*x - E^x*x - E^x^2*x - x^2/(3*Log[x]) + x*Log[x]

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fricas [A]  time = 0.58, size = 37, normalized size = 1.19 \begin {gather*} \frac {3 \, x \log \relax (x)^{2} - x^{2} - 3 \, {\left (x e^{\left (x^{2}\right )} + x e^{x} - 2 \, x\right )} \log \relax (x)}{3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*log(x)^3+((-6*x^2-3)*exp(x^2)+(-3*x-3)*exp(x)+9)*log(x)^2-2*x*log(x)+x)/log(x)^2,x, algorithm
="fricas")

[Out]

1/3*(3*x*log(x)^2 - x^2 - 3*(x*e^(x^2) + x*e^x - 2*x)*log(x))/log(x)

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giac [A]  time = 0.21, size = 38, normalized size = 1.23 \begin {gather*} -\frac {3 \, x e^{\left (x^{2}\right )} \log \relax (x) + 3 \, x e^{x} \log \relax (x) - 3 \, x \log \relax (x)^{2} + x^{2} - 6 \, x \log \relax (x)}{3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*log(x)^3+((-6*x^2-3)*exp(x^2)+(-3*x-3)*exp(x)+9)*log(x)^2-2*x*log(x)+x)/log(x)^2,x, algorithm
="giac")

[Out]

-1/3*(3*x*e^(x^2)*log(x) + 3*x*e^x*log(x) - 3*x*log(x)^2 + x^2 - 6*x*log(x))/log(x)

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maple [A]  time = 0.03, size = 30, normalized size = 0.97

method	result	size
default	\(2 x -\frac {x^{2}}{3 \ln \relax (x )}-{\mathrm e}^{x} x -{\mathrm e}^{x^{2}} x +x \ln \relax (x )\)	\(30\)
risch	\(2 x -\frac {x^{2}}{3 \ln \relax (x )}-{\mathrm e}^{x} x -{\mathrm e}^{x^{2}} x +x \ln \relax (x )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(3*ln(x)^3+((-6*x^2-3)*exp(x^2)+(-3*x-3)*exp(x)+9)*ln(x)^2-2*x*ln(x)+x)/ln(x)^2,x,method=_RETURNVERBOS
E)

[Out]

2*x-1/3*x^2/ln(x)-exp(x)*x-exp(x^2)*x+x*ln(x)

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maxima [C]  time = 0.38, size = 41, normalized size = 1.32 \begin {gather*} -x e^{\left (x^{2}\right )} - {\left (x - 1\right )} e^{x} + x \log \relax (x) + 2 \, x - \frac {2}{3} \, {\rm Ei}\left (2 \, \log \relax (x)\right ) - e^{x} + \frac {2}{3} \, \Gamma \left (-1, -2 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*log(x)^3+((-6*x^2-3)*exp(x^2)+(-3*x-3)*exp(x)+9)*log(x)^2-2*x*log(x)+x)/log(x)^2,x, algorithm
="maxima")

[Out]

-x*e^(x^2) - (x - 1)*e^x + x*log(x) + 2*x - 2/3*Ei(2*log(x)) - e^x + 2/3*gamma(-1, -2*log(x))

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mupad [B]  time = 4.33, size = 29, normalized size = 0.94 \begin {gather*} 2\,x-x\,{\mathrm {e}}^{x^2}-\frac {x^2}{3\,\ln \relax (x)}-x\,{\mathrm {e}}^x+x\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x/3 + log(x)^3 - (log(x)^2*(exp(x^2)*(6*x^2 + 3) + exp(x)*(3*x + 3) - 9))/3 - (2*x*log(x))/3)/log(x)^2,x)

[Out]

2*x - x*exp(x^2) - x^2/(3*log(x)) - x*exp(x) + x*log(x)

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sympy [A]  time = 0.43, size = 27, normalized size = 0.87 \begin {gather*} - \frac {x^{2}}{3 \log {\relax (x )}} - x e^{x} - x e^{x^{2}} + x \log {\relax (x )} + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*ln(x)**3+((-6*x**2-3)*exp(x**2)+(-3*x-3)*exp(x)+9)*ln(x)**2-2*x*ln(x)+x)/ln(x)**2,x)

[Out]

-x**2/(3*log(x)) - x*exp(x) - x*exp(x**2) + x*log(x) + 2*x

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