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

Optimal. Leaf size=20 \[ 5+\frac {1}{2} \left (\frac {5}{2}-x\right ) x+\frac {x}{\log (x)} \]

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Rubi [A]  time = 0.05, antiderivative size = 19, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 6742, 2297, 2298} \begin {gather*} -\frac {x^2}{2}+\frac {5 x}{4}+\frac {x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*x)/4 - x^2/2 + x/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 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, 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} &=\frac {1}{4} \int \frac {-4+4 \log (x)+(5-4 x) \log ^2(x)}{\log ^2(x)} \, dx\\ &=\frac {1}{4} \int \left (5-4 x-\frac {4}{\log ^2(x)}+\frac {4}{\log (x)}\right ) \, dx\\ &=\frac {5 x}{4}-\frac {x^2}{2}-\int \frac {1}{\log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx\\ &=\frac {5 x}{4}-\frac {x^2}{2}+\frac {x}{\log (x)}+\text {li}(x)-\int \frac {1}{\log (x)} \, dx\\ &=\frac {5 x}{4}-\frac {x^2}{2}+\frac {x}{\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.95 \begin {gather*} \frac {5 x}{4}-\frac {x^2}{2}+\frac {x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*x)/4 - x^2/2 + x/Log[x]

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fricas [A]  time = 1.03, size = 22, normalized size = 1.10 \begin {gather*} -\frac {{\left (2 \, x^{2} - 5 \, x\right )} \log \relax (x) - 4 \, x}{4 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((2*x^2 - 5*x)*log(x) - 4*x)/log(x)

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giac [A]  time = 0.22, size = 15, normalized size = 0.75 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {5}{4} \, x + \frac {x}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^2 + 5/4*x + x/log(x)

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maple [A]  time = 0.02, size = 16, normalized size = 0.80

method	result	size
default	\(-\frac {x^{2}}{2}+\frac {5 x}{4}+\frac {x}{\ln \relax (x )}\)	\(16\)
risch	\(-\frac {x^{2}}{2}+\frac {5 x}{4}+\frac {x}{\ln \relax (x )}\)	\(16\)
norman	\(\frac {x +\frac {5 x \ln \relax (x )}{4}-\frac {x^{2} \ln \relax (x )}{2}}{\ln \relax (x )}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^2+5/4*x+x/ln(x)

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maxima [C]  time = 0.38, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {5}{4} \, x + {\rm Ei}\left (\log \relax (x)\right ) - \Gamma \left (-1, -\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^2 + 5/4*x + Ei(log(x)) - gamma(-1, -log(x))

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mupad [B]  time = 5.46, size = 15, normalized size = 0.75 \begin {gather*} \frac {x}{\ln \relax (x)}-\frac {x\,\left (2\,x-5\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((log(x)^2*(4*x - 5))/4 - log(x) + 1)/log(x)^2,x)

[Out]

x/log(x) - (x*(2*x - 5))/4

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sympy [A]  time = 0.09, size = 14, normalized size = 0.70 \begin {gather*} - \frac {x^{2}}{2} + \frac {5 x}{4} + \frac {x}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-4*x+5)*ln(x)**2+4*ln(x)-4)/ln(x)**2,x)

[Out]

-x**2/2 + 5*x/4 + x/log(x)

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