3.86.50 \(\int \frac {3 \log ^2(x)+e^{\frac {-x^2+(6 x+13 x^2+18 x^3+12 x^4+3 x^5) \log (x)}{3 \log (x)}} (x-2 x \log (x)+(6+26 x+54 x^2+48 x^3+15 x^4) \log ^2(x))}{3 \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ e^{x+x \left ((1+x)^4+\frac {1}{3} \left (x-\frac {x}{\log (x)}\right )\right )}+x \]

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Rubi [A]  time = 1.36, antiderivative size = 36, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 4, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {12, 6742, 6688, 6706} \begin {gather*} \exp \left (\frac {1}{3} x \left (3 x^4+12 x^3+18 x^2+13 x-\frac {x}{\log (x)}+6\right )\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3*Log[x]^2 + E^((-x^2 + (6*x + 13*x^2 + 18*x^3 + 12*x^4 + 3*x^5)*Log[x])/(3*Log[x]))*(x - 2*x*Log[x] + (6
+ 26*x + 54*x^2 + 48*x^3 + 15*x^4)*Log[x]^2))/(3*Log[x]^2),x]

[Out]

E^((x*(6 + 13*x + 18*x^2 + 12*x^3 + 3*x^4 - x/Log[x]))/3) + 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 6688

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, 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 {3 \log ^2(x)+\exp \left (\frac {-x^2+\left (6 x+13 x^2+18 x^3+12 x^4+3 x^5\right ) \log (x)}{3 \log (x)}\right ) \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx\\ &=\frac {1}{3} \int \left (3+\frac {\exp \left (\frac {1}{3} x \left (6+13 x+18 x^2+12 x^3+3 x^4-\frac {x}{\log (x)}\right )\right ) \left (x-2 x \log (x)+6 \log ^2(x)+26 x \log ^2(x)+54 x^2 \log ^2(x)+48 x^3 \log ^2(x)+15 x^4 \log ^2(x)\right )}{\log ^2(x)}\right ) \, dx\\ &=x+\frac {1}{3} \int \frac {\exp \left (\frac {1}{3} x \left (6+13 x+18 x^2+12 x^3+3 x^4-\frac {x}{\log (x)}\right )\right ) \left (x-2 x \log (x)+6 \log ^2(x)+26 x \log ^2(x)+54 x^2 \log ^2(x)+48 x^3 \log ^2(x)+15 x^4 \log ^2(x)\right )}{\log ^2(x)} \, dx\\ &=x+\frac {1}{3} \int \frac {\exp \left (\frac {1}{3} x \left (6+13 x+18 x^2+12 x^3+3 x^4-\frac {x}{\log (x)}\right )\right ) \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx\\ &=\exp \left (\frac {1}{3} x \left (6+13 x+18 x^2+12 x^3+3 x^4-\frac {x}{\log (x)}\right )\right )+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.45, size = 39, normalized size = 1.44 \begin {gather*} e^{2 x+\frac {13 x^2}{3}+6 x^3+4 x^4+x^5-\frac {x^2}{3 \log (x)}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*Log[x]^2 + E^((-x^2 + (6*x + 13*x^2 + 18*x^3 + 12*x^4 + 3*x^5)*Log[x])/(3*Log[x]))*(x - 2*x*Log[x
] + (6 + 26*x + 54*x^2 + 48*x^3 + 15*x^4)*Log[x]^2))/(3*Log[x]^2),x]

[Out]

E^(2*x + (13*x^2)/3 + 6*x^3 + 4*x^4 + x^5 - x^2/(3*Log[x])) + x

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fricas [A]  time = 0.61, size = 41, normalized size = 1.52 \begin {gather*} x + e^{\left (-\frac {x^{2} - {\left (3 \, x^{5} + 12 \, x^{4} + 18 \, x^{3} + 13 \, x^{2} + 6 \, x\right )} \log \relax (x)}{3 \, \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((15*x^4+48*x^3+54*x^2+26*x+6)*log(x)^2-2*x*log(x)+x)*exp(1/3*((3*x^5+12*x^4+18*x^3+13*x^2+6*x)
*log(x)-x^2)/log(x))+3*log(x)^2)/log(x)^2,x, algorithm="fricas")

[Out]

x + e^(-1/3*(x^2 - (3*x^5 + 12*x^4 + 18*x^3 + 13*x^2 + 6*x)*log(x))/log(x))

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giac [A]  time = 0.22, size = 48, normalized size = 1.78 \begin {gather*} x + e^{\left (\frac {3 \, x^{5} \log \relax (x) + 12 \, x^{4} \log \relax (x) + 18 \, x^{3} \log \relax (x) + 13 \, x^{2} \log \relax (x) - x^{2} + 6 \, x \log \relax (x)}{3 \, \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((15*x^4+48*x^3+54*x^2+26*x+6)*log(x)^2-2*x*log(x)+x)*exp(1/3*((3*x^5+12*x^4+18*x^3+13*x^2+6*x)
*log(x)-x^2)/log(x))+3*log(x)^2)/log(x)^2,x, algorithm="giac")

[Out]

x + e^(1/3*(3*x^5*log(x) + 12*x^4*log(x) + 18*x^3*log(x) + 13*x^2*log(x) - x^2 + 6*x*log(x))/log(x))

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maple [A]  time = 0.05, size = 45, normalized size = 1.67

method	result	size
risch	\(x +{\mathrm e}^{\frac {x \left (3 x^{4} \ln \relax (x )+12 x^{3} \ln \relax (x )+18 x^{2} \ln \relax (x )+13 x \ln \relax (x )+6 \ln \relax (x )-x \right )}{3 \ln \relax (x )}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(((15*x^4+48*x^3+54*x^2+26*x+6)*ln(x)^2-2*x*ln(x)+x)*exp(1/3*((3*x^5+12*x^4+18*x^3+13*x^2+6*x)*ln(x)-x
^2)/ln(x))+3*ln(x)^2)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

x+exp(1/3*x*(3*x^4*ln(x)+12*x^3*ln(x)+18*x^2*ln(x)+13*x*ln(x)+6*ln(x)-x)/ln(x))

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maxima [A]  time = 0.56, size = 34, normalized size = 1.26 \begin {gather*} x + e^{\left (x^{5} + 4 \, x^{4} + 6 \, x^{3} + \frac {13}{3} \, x^{2} + 2 \, x - \frac {x^{2}}{3 \, \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((15*x^4+48*x^3+54*x^2+26*x+6)*log(x)^2-2*x*log(x)+x)*exp(1/3*((3*x^5+12*x^4+18*x^3+13*x^2+6*x)
*log(x)-x^2)/log(x))+3*log(x)^2)/log(x)^2,x, algorithm="maxima")

[Out]

x + e^(x^5 + 4*x^4 + 6*x^3 + 13/3*x^2 + 2*x - 1/3*x^2/log(x))

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mupad [B]  time = 5.30, size = 39, normalized size = 1.44 \begin {gather*} x+{\left ({\mathrm {e}}^{x^2}\right )}^{13/3}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{x^5}\,{\mathrm {e}}^{4\,x^4}\,{\mathrm {e}}^{6\,x^3}\,{\mathrm {e}}^{-\frac {x^2}{3\,\ln \relax (x)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2 + (exp(((log(x)*(6*x + 13*x^2 + 18*x^3 + 12*x^4 + 3*x^5))/3 - x^2/3)/log(x))*(x + log(x)^2*(26*x
+ 54*x^2 + 48*x^3 + 15*x^4 + 6) - 2*x*log(x)))/3)/log(x)^2,x)

[Out]

x + exp(x^2)^(13/3)*exp(2*x)*exp(x^5)*exp(4*x^4)*exp(6*x^3)*exp(-x^2/(3*log(x)))

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sympy [A]  time = 0.49, size = 39, normalized size = 1.44 \begin {gather*} x + e^{\frac {- \frac {x^{2}}{3} + \frac {\left (3 x^{5} + 12 x^{4} + 18 x^{3} + 13 x^{2} + 6 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*(((15*x**4+48*x**3+54*x**2+26*x+6)*ln(x)**2-2*x*ln(x)+x)*exp(1/3*((3*x**5+12*x**4+18*x**3+13*x**
2+6*x)*ln(x)-x**2)/ln(x))+3*ln(x)**2)/ln(x)**2,x)

[Out]

x + exp((-x**2/3 + (3*x**5 + 12*x**4 + 18*x**3 + 13*x**2 + 6*x)*log(x)/3)/log(x))

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