3.98.59 \(\int \frac {1+(2 x+12 x^2+16 x^3+5 x^4) \log (x)+\log (x) \log (\log (x))}{\log (x)} \, dx\)

Optimal. Leaf size=18 \[ -6+x \left (x+x^2 (2+x)^2+\log (\log (x))\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 7, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6688, 14, 2298, 2520} \begin {gather*} x^5+4 x^4+4 x^3+x^2+x \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + (2*x + 12*x^2 + 16*x^3 + 5*x^4)*Log[x] + Log[x]*Log[Log[x]])/Log[x],x]

[Out]

x^2 + 4*x^3 + 4*x^4 + x^5 + x*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 2298

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

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 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 \left (x \left (2+12 x+16 x^2+5 x^3\right )+\frac {1}{\log (x)}+\log (\log (x))\right ) \, dx\\ &=\int x \left (2+12 x+16 x^2+5 x^3\right ) \, dx+\int \frac {1}{\log (x)} \, dx+\int \log (\log (x)) \, dx\\ &=x \log (\log (x))+\text {li}(x)+\int \left (2 x+12 x^2+16 x^3+5 x^4\right ) \, dx-\int \frac {1}{\log (x)} \, dx\\ &=x^2+4 x^3+4 x^4+x^5+x \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 22, normalized size = 1.22 \begin {gather*} x^2+4 x^3+4 x^4+x^5+x \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + (2*x + 12*x^2 + 16*x^3 + 5*x^4)*Log[x] + Log[x]*Log[Log[x]])/Log[x],x]

[Out]

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

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fricas [A]  time = 0.84, size = 22, normalized size = 1.22 \begin {gather*} x^{5} + 4 \, x^{4} + 4 \, x^{3} + x^{2} + x \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x))+(5*x^4+16*x^3+12*x^2+2*x)*log(x)+1)/log(x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 22, normalized size = 1.22 \begin {gather*} x^{5} + 4 \, x^{4} + 4 \, x^{3} + x^{2} + x \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x))+(5*x^4+16*x^3+12*x^2+2*x)*log(x)+1)/log(x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.02, size = 23, normalized size = 1.28

method	result	size
default	\(x^{2}+4 x^{3}+4 x^{4}+x^{5}+x \ln \left (\ln \relax (x )\right )\)	\(23\)
risch	\(x^{2}+4 x^{3}+4 x^{4}+x^{5}+x \ln \left (\ln \relax (x )\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)*ln(ln(x))+(5*x^4+16*x^3+12*x^2+2*x)*ln(x)+1)/ln(x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 22, normalized size = 1.22 \begin {gather*} x^{5} + 4 \, x^{4} + 4 \, x^{3} + x^{2} + x \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x))+(5*x^4+16*x^3+12*x^2+2*x)*log(x)+1)/log(x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.93, size = 22, normalized size = 1.22 \begin {gather*} x\,\ln \left (\ln \relax (x)\right )+x^2+4\,x^3+4\,x^4+x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(2*x + 12*x^2 + 16*x^3 + 5*x^4) + log(log(x))*log(x) + 1)/log(x),x)

[Out]

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

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sympy [A]  time = 0.33, size = 22, normalized size = 1.22 \begin {gather*} x^{5} + 4 x^{4} + 4 x^{3} + x^{2} + x \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x)*ln(ln(x))+(5*x**4+16*x**3+12*x**2+2*x)*ln(x)+1)/ln(x),x)

[Out]

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

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