3.51.63 \(\int \frac {2+(-2 x^2+324 x^4) \log (x)}{x \log (x)} \, dx\)

Optimal. Leaf size=16 \[ -x^2+81 x^4+\log \left (\log ^2(x)\right ) \]

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Rubi [A]  time = 0.20, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6741, 12, 6688, 2302, 29} \begin {gather*} 81 x^4-x^2+2 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + (-2*x^2 + 324*x^4)*Log[x])/(x*Log[x]),x]

[Out]

-x^2 + 81*x^4 + 2*Log[Log[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 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]]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (1-x^2 \log (x)+162 x^4 \log (x)\right )}{x \log (x)} \, dx\\ &=2 \int \frac {1-x^2 \log (x)+162 x^4 \log (x)}{x \log (x)} \, dx\\ &=2 \int \left (-x+162 x^3+\frac {1}{x \log (x)}\right ) \, dx\\ &=-x^2+81 x^4+2 \int \frac {1}{x \log (x)} \, dx\\ &=-x^2+81 x^4+2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-x^2+81 x^4+2 \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -x^2+81 x^4+2 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + (-2*x^2 + 324*x^4)*Log[x])/(x*Log[x]),x]

[Out]

-x^2 + 81*x^4 + 2*Log[Log[x]]

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fricas [A]  time = 0.60, size = 16, normalized size = 1.00 \begin {gather*} 81 \, x^{4} - x^{2} + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((324*x^4-2*x^2)*log(x)+2)/x/log(x),x, algorithm="fricas")

[Out]

81*x^4 - x^2 + 2*log(log(x))

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giac [A]  time = 0.14, size = 16, normalized size = 1.00 \begin {gather*} 81 \, x^{4} - x^{2} + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((324*x^4-2*x^2)*log(x)+2)/x/log(x),x, algorithm="giac")

[Out]

81*x^4 - x^2 + 2*log(log(x))

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maple [A]  time = 0.02, size = 17, normalized size = 1.06

method	result	size
default	\(81 x^{4}-x^{2}+2 \ln \left (\ln \relax (x )\right )\)	\(17\)
norman	\(81 x^{4}-x^{2}+2 \ln \left (\ln \relax (x )\right )\)	\(17\)
risch	\(81 x^{4}-x^{2}+2 \ln \left (\ln \relax (x )\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((324*x^4-2*x^2)*ln(x)+2)/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

81*x^4-x^2+2*ln(ln(x))

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maxima [A]  time = 0.34, size = 16, normalized size = 1.00 \begin {gather*} 81 \, x^{4} - x^{2} + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((324*x^4-2*x^2)*log(x)+2)/x/log(x),x, algorithm="maxima")

[Out]

81*x^4 - x^2 + 2*log(log(x))

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mupad [B]  time = 3.40, size = 16, normalized size = 1.00 \begin {gather*} 2\,\ln \left (\ln \relax (x)\right )-x^2+81\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(2*x^2 - 324*x^4) - 2)/(x*log(x)),x)

[Out]

2*log(log(x)) - x^2 + 81*x^4

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sympy [A]  time = 0.09, size = 14, normalized size = 0.88 \begin {gather*} 81 x^{4} - x^{2} + 2 \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((324*x**4-2*x**2)*ln(x)+2)/x/ln(x),x)

[Out]

81*x**4 - x**2 + 2*log(log(x))

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