3.12.81 \(\int \frac {1+(20 x+18 x^2+3 x^3) \log (x)}{x \log (x)} \, dx\)

Optimal. Leaf size=13 \[ -4+x (4+x) (5+x)+\log (\log (x)) \]

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Rubi [A]  time = 0.10, antiderivative size = 15, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6688, 2302, 29} \begin {gather*} x^3+9 x^2+20 x+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + (20*x + 18*x^2 + 3*x^3)*Log[x])/(x*Log[x]),x]

[Out]

20*x + 9*x^2 + x^3 + Log[Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, 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 (20+18 x+3 x^2+\frac {1}{x \log (x)}\right ) \, dx\\ &=20 x+9 x^2+x^3+\int \frac {1}{x \log (x)} \, dx\\ &=20 x+9 x^2+x^3+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=20 x+9 x^2+x^3+\log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.15 \begin {gather*} 20 x+9 x^2+x^3+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + (20*x + 18*x^2 + 3*x^3)*Log[x])/(x*Log[x]),x]

[Out]

20*x + 9*x^2 + x^3 + Log[Log[x]]

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fricas [A]  time = 0.76, size = 15, normalized size = 1.15 \begin {gather*} x^{3} + 9 \, x^{2} + 20 \, x + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+18*x^2+20*x)*log(x)+1)/x/log(x),x, algorithm="fricas")

[Out]

x^3 + 9*x^2 + 20*x + log(log(x))

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giac [A]  time = 0.35, size = 15, normalized size = 1.15 \begin {gather*} x^{3} + 9 \, x^{2} + 20 \, x + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+18*x^2+20*x)*log(x)+1)/x/log(x),x, algorithm="giac")

[Out]

x^3 + 9*x^2 + 20*x + log(log(x))

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maple [A]  time = 0.01, size = 16, normalized size = 1.23

method	result	size
default	\(x^{3}+9 x^{2}+20 x +\ln \left (\ln \relax (x )\right )\)	\(16\)
norman	\(x^{3}+9 x^{2}+20 x +\ln \left (\ln \relax (x )\right )\)	\(16\)
risch	\(x^{3}+9 x^{2}+20 x +\ln \left (\ln \relax (x )\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^3+18*x^2+20*x)*ln(x)+1)/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

x^3+9*x^2+20*x+ln(ln(x))

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maxima [A]  time = 0.61, size = 15, normalized size = 1.15 \begin {gather*} x^{3} + 9 \, x^{2} + 20 \, x + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+18*x^2+20*x)*log(x)+1)/x/log(x),x, algorithm="maxima")

[Out]

x^3 + 9*x^2 + 20*x + log(log(x))

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mupad [B]  time = 0.84, size = 15, normalized size = 1.15 \begin {gather*} 20\,x+\ln \left (\ln \relax (x)\right )+9\,x^2+x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(20*x + 18*x^2 + 3*x^3) + 1)/(x*log(x)),x)

[Out]

20*x + log(log(x)) + 9*x^2 + x^3

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sympy [A]  time = 0.10, size = 15, normalized size = 1.15 \begin {gather*} x^{3} + 9 x^{2} + 20 x + \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**3+18*x**2+20*x)*ln(x)+1)/x/ln(x),x)

[Out]

x**3 + 9*x**2 + 20*x + log(log(x))

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