3.60.91 \(\int \frac {x+(2 x-3 x^2-6 x^5) \log (x)+2 x \log (x) \log (\log (x))}{\log (x)} \, dx\)

Optimal. Leaf size=19 \[ 5+x \left (x-x \left (x+x^4-\log (\log (x))\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.17, antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {6688, 14, 2309, 2178, 2522} \begin {gather*} -x^6-x^3+x^2+x^2 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + (2*x - 3*x^2 - 6*x^5)*Log[x] + 2*x*Log[x]*Log[Log[x]])/Log[x],x]

[Out]

x^2 - x^3 - x^6 + x^2*Log[Log[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 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 2522

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

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int x \left (2-3 x-6 x^4+\frac {1}{\log (x)}+2 \log (\log (x))\right ) \, dx\\ &=\int \left (-\frac {x \left (-1-2 \log (x)+3 x \log (x)+6 x^4 \log (x)\right )}{\log (x)}+2 x \log (\log (x))\right ) \, dx\\ &=2 \int x \log (\log (x)) \, dx-\int \frac {x \left (-1-2 \log (x)+3 x \log (x)+6 x^4 \log (x)\right )}{\log (x)} \, dx\\ &=x^2 \log (\log (x))-\int x \left (-2+3 x+6 x^4-\frac {1}{\log (x)}\right ) \, dx-\int \frac {x}{\log (x)} \, dx\\ &=x^2 \log (\log (x))-\int \left (x \left (-2+3 x+6 x^4\right )-\frac {x}{\log (x)}\right ) \, dx-\operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\text {Ei}(2 \log (x))+x^2 \log (\log (x))-\int x \left (-2+3 x+6 x^4\right ) \, dx+\int \frac {x}{\log (x)} \, dx\\ &=-\text {Ei}(2 \log (x))+x^2 \log (\log (x))-\int \left (-2 x+3 x^2+6 x^5\right ) \, dx+\operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=x^2-x^3-x^6+x^2 \log (\log (x))\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 21, normalized size = 1.11 \begin {gather*} x^2-x^3-x^6+x^2 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + (2*x - 3*x^2 - 6*x^5)*Log[x] + 2*x*Log[x]*Log[Log[x]])/Log[x],x]

[Out]

x^2 - x^3 - x^6 + x^2*Log[Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 21, normalized size = 1.11 \begin {gather*} -x^{6} - x^{3} + x^{2} \log \left (\log \relax (x)\right ) + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(x)*log(log(x))+(-6*x^5-3*x^2+2*x)*log(x)+x)/log(x),x, algorithm="fricas")

[Out]

-x^6 - x^3 + x^2*log(log(x)) + x^2

________________________________________________________________________________________

giac [A]  time = 0.17, size = 21, normalized size = 1.11 \begin {gather*} -x^{6} - x^{3} + x^{2} \log \left (\log \relax (x)\right ) + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(x)*log(log(x))+(-6*x^5-3*x^2+2*x)*log(x)+x)/log(x),x, algorithm="giac")

[Out]

-x^6 - x^3 + x^2*log(log(x)) + x^2

________________________________________________________________________________________

maple [A]  time = 0.03, size = 22, normalized size = 1.16

method	result	size
risch	\(-x^{6}-x^{3}+x^{2} \ln \left (\ln \relax (x )\right )+x^{2}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*ln(x)*ln(ln(x))+(-6*x^5-3*x^2+2*x)*ln(x)+x)/ln(x),x,method=_RETURNVERBOSE)

[Out]

-x^6-x^3+x^2*ln(ln(x))+x^2

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 21, normalized size = 1.11 \begin {gather*} -x^{6} - x^{3} + x^{2} \log \left (\log \relax (x)\right ) + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(x)*log(log(x))+(-6*x^5-3*x^2+2*x)*log(x)+x)/log(x),x, algorithm="maxima")

[Out]

-x^6 - x^3 + x^2*log(log(x)) + x^2

________________________________________________________________________________________

mupad [B]  time = 4.31, size = 21, normalized size = 1.11 \begin {gather*} x^2\,\ln \left (\ln \relax (x)\right )+x^2-x^3-x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - log(x)*(3*x^2 - 2*x + 6*x^5) + 2*x*log(log(x))*log(x))/log(x),x)

[Out]

x^2*log(log(x)) + x^2 - x^3 - x^6

________________________________________________________________________________________

sympy [A]  time = 0.30, size = 17, normalized size = 0.89 \begin {gather*} - x^{6} - x^{3} + x^{2} \log {\left (\log {\relax (x )} \right )} + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*ln(x)*ln(ln(x))+(-6*x**5-3*x**2+2*x)*ln(x)+x)/ln(x),x)

[Out]

-x**6 - x**3 + x**2*log(log(x)) + x**2

________________________________________________________________________________________