3.35.43 \(\int \frac {6 x+3 x^2+(5-6 x-6 x^2-6 x^3-4 x^4) \log (x)+(6 x+6 x^2) \log (x) \log (\log (x))}{x \log (x)} \, dx\)

Optimal. Leaf size=23 \[ 6+5 \log (x)+x (2+x) \left (-3-x^2+3 \log (\log (x))\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.22, antiderivative size = 37, normalized size of antiderivative = 1.61, number of steps used = 14, number of rules used = 8, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {6688, 2330, 2298, 2309, 2178, 6742, 2520, 2522} \begin {gather*} -x^4-2 x^3-3 x^2+3 x^2 \log (\log (x))-6 x+6 x \log (\log (x))+5 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-6*x - 3*x^2 - 2*x^3 - x^4 + 5*Log[x] + 6*x*Log[Log[x]] + 3*x^2*Log[Log[x]]

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 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

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 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2520

Int[Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)], x_Symbol] :> Simp[x*Log[c*Log[d*x^n]^p], x] - Dist[n*p, Int[1/Log[
d*x^n], x], x] /; FreeQ[{c, d, n, p}, x]

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]]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.17, size = 21, normalized size = 0.91 \begin {gather*} 5 \log (x)-x (2+x) \left (3+x^2-3 \log (\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 35, normalized size = 1.52 \begin {gather*} -x^{4} - 2 \, x^{3} - 3 \, x^{2} + 3 \, {\left (x^{2} + 2 \, x\right )} \log \left (\log \relax (x)\right ) - 6 \, x + 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+6*x)*log(x)*log(log(x))+(-4*x^4-6*x^3-6*x^2-6*x+5)*log(x)+3*x^2+6*x)/x/log(x),x, algorithm="
fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.22, size = 35, normalized size = 1.52 \begin {gather*} -x^{4} - 2 \, x^{3} - 3 \, x^{2} + 3 \, {\left (x^{2} + 2 \, x\right )} \log \left (\log \relax (x)\right ) - 6 \, x + 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+6*x)*log(x)*log(log(x))+(-4*x^4-6*x^3-6*x^2-6*x+5)*log(x)+3*x^2+6*x)/x/log(x),x, algorithm="
giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.07, size = 37, normalized size = 1.61

method	result	size
risch	\(\left (3 x^{2}+6 x \right ) \ln \left (\ln \relax (x )\right )-x^{4}-2 x^{3}-3 x^{2}-6 x +5 \ln \relax (x )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^2+6*x)*ln(x)*ln(ln(x))+(-4*x^4-6*x^3-6*x^2-6*x+5)*ln(x)+3*x^2+6*x)/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

(3*x^2+6*x)*ln(ln(x))-x^4-2*x^3-3*x^2-6*x+5*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 37, normalized size = 1.61 \begin {gather*} -x^{4} - 2 \, x^{3} + 3 \, x^{2} \log \left (\log \relax (x)\right ) - 3 \, x^{2} + 6 \, x \log \left (\log \relax (x)\right ) - 6 \, x + 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+6*x)*log(x)*log(log(x))+(-4*x^4-6*x^3-6*x^2-6*x+5)*log(x)+3*x^2+6*x)/x/log(x),x, algorithm="
maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 2.12, size = 36, normalized size = 1.57 \begin {gather*} 5\,\ln \relax (x)-6\,x+\ln \left (\ln \relax (x)\right )\,\left (3\,x^2+6\,x\right )-3\,x^2-2\,x^3-x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x - log(x)*(6*x + 6*x^2 + 6*x^3 + 4*x^4 - 5) + 3*x^2 + log(log(x))*log(x)*(6*x + 6*x^2))/(x*log(x)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.34, size = 34, normalized size = 1.48 \begin {gather*} - x^{4} - 2 x^{3} - 3 x^{2} - 6 x + \left (3 x^{2} + 6 x\right ) \log {\left (\log {\relax (x )} \right )} + 5 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**2+6*x)*ln(x)*ln(ln(x))+(-4*x**4-6*x**3-6*x**2-6*x+5)*ln(x)+3*x**2+6*x)/x/ln(x),x)

[Out]

-x**4 - 2*x**3 - 3*x**2 - 6*x + (3*x**2 + 6*x)*log(log(x)) + 5*log(x)

________________________________________________________________________________________