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

Optimal. Leaf size=11 \[ x-x^2+\log ^4(x) \]

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Rubi [A]  time = 0.08, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6688, 2302, 30} \begin {gather*} -x^2+x+\log ^4(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - x^2 + Log[x]^4

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} x-x^2+\log ^4(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - x^2 + Log[x]^4

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fricas [A]  time = 0.64, size = 11, normalized size = 1.00 \begin {gather*} \log \relax (x)^{4} - x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)^4 - x^2 + x

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giac [A]  time = 0.24, size = 11, normalized size = 1.00 \begin {gather*} \log \relax (x)^{4} - x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)^4 - x^2 + x

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maple [A]  time = 0.02, size = 12, normalized size = 1.09

method	result	size
default	\(\ln \relax (x )^{4}-x^{2}+x\)	\(12\)
norman	\(\ln \relax (x )^{4}-x^{2}+x\)	\(12\)
risch	\(\ln \relax (x )^{4}-x^{2}+x\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 11, normalized size = 1.00 \begin {gather*} \log \relax (x)^{4} - x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)^4 - x^2 + x

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mupad [B]  time = 2.18, size = 11, normalized size = 1.00 \begin {gather*} -x^2+x+{\ln \relax (x)}^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x)^4 - x^2

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sympy [A]  time = 0.09, size = 8, normalized size = 0.73 \begin {gather*} - x^{2} + x + \log {\relax (x )}^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2 + x + log(x)**4

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