3.66.29 \(\int \frac {\log (2)-\log (2) \log (x)+(1-\log (2)) \log ^2(x)}{\log ^2(x)} \, dx\)

Optimal. Leaf size=16 \[ x+\frac {x \log (2) (-1-\log (x))}{\log (x)} \]

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Rubi [A]  time = 0.07, antiderivative size = 18, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6742, 2297, 2298} \begin {gather*} x (1-\log (2))-\frac {x \log (2)}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Log[2] - Log[2]*Log[x] + (1 - Log[2])*Log[x]^2)/Log[x]^2,x]

[Out]

x*(1 - Log[2]) - (x*Log[2])/Log[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} &=\int \left (1-\log (2)+\frac {\log (2)}{\log ^2(x)}-\frac {\log (2)}{\log (x)}\right ) \, dx\\ &=x (1-\log (2))+\log (2) \int \frac {1}{\log ^2(x)} \, dx-\log (2) \int \frac {1}{\log (x)} \, dx\\ &=x (1-\log (2))-\frac {x \log (2)}{\log (x)}-\log (2) \text {li}(x)+\log (2) \int \frac {1}{\log (x)} \, dx\\ &=x (1-\log (2))-\frac {x \log (2)}{\log (x)}\\ \end {aligned} \end {gather*}

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

Antiderivative was successfully verified.

[In]

Integrate[(Log[2] - Log[2]*Log[x] + (1 - Log[2])*Log[x]^2)/Log[x]^2,x]

[Out]

x - x*Log[2] - (x*Log[2])/Log[x]

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fricas [A]  time = 0.63, size = 22, normalized size = 1.38 \begin {gather*} -\frac {x \log \relax (2) + {\left (x \log \relax (2) - x\right )} \log \relax (x)}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*log(2) + (x*log(2) - x)*log(x))/log(x)

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giac [A]  time = 0.13, size = 16, normalized size = 1.00 \begin {gather*} -x \log \relax (2) + x - \frac {x \log \relax (2)}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*log(2) + x - x*log(2)/log(x)

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maple [A]  time = 0.05, size = 17, normalized size = 1.06

method	result	size
risch	\(-x \ln \relax (2)+x -\frac {\ln \relax (2) x}{\ln \relax (x )}\)	\(17\)
norman	\(\frac {x \left (1-\ln \relax (2)\right ) \ln \relax (x )-x \ln \relax (2)}{\ln \relax (x )}\)	\(22\)
default	\(-x \ln \relax (2)+x +\ln \relax (2) \expIntegralEi \left (1, -\ln \relax (x )\right )+\ln \relax (2) \left (-\frac {x}{\ln \relax (x )}-\expIntegralEi \left (1, -\ln \relax (x )\right )\right )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*ln(2)+x-ln(2)/ln(x)*x

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maxima [C]  time = 0.40, size = 23, normalized size = 1.44 \begin {gather*} -x \log \relax (2) - {\rm Ei}\left (\log \relax (x)\right ) \log \relax (2) + \Gamma \left (-1, -\log \relax (x)\right ) \log \relax (2) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*log(2) - Ei(log(x))*log(2) + gamma(-1, -log(x))*log(2) + x

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mupad [B]  time = 4.18, size = 17, normalized size = 1.06 \begin {gather*} -x\,\left (\ln \relax (2)-1\right )-\frac {x\,\ln \relax (2)}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- x*(log(2) - 1) - (x*log(2))/log(x)

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sympy [A]  time = 0.08, size = 14, normalized size = 0.88 \begin {gather*} x \left (1 - \log {\relax (2 )}\right ) - \frac {x \log {\relax (2 )}}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(1 - log(2)) - x*log(2)/log(x)

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