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

Optimal. Leaf size=12 \[ \log (x)+x (-2+\log (1+\log (x))) \]

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Rubi [F]  time = 0.28, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1-x+(1-2 x) \log (x)+(x+x \log (x)) \log (1+\log (x))}{x+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-2*x + ExpIntegralEi[1 + Log[x]]/E + Log[x] + Defer[Int][Log[1 + Log[x]], x]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 13, normalized size = 1.08 \begin {gather*} -2 x+\log (x)+x \log (1+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*x + Log[x] + x*Log[1 + Log[x]]

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fricas [A]  time = 0.51, size = 13, normalized size = 1.08 \begin {gather*} x \log \left (\log \relax (x) + 1\right ) - 2 \, x + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(log(x) + 1) - 2*x + log(x)

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giac [A]  time = 0.12, size = 13, normalized size = 1.08 \begin {gather*} x \log \left (\log \relax (x) + 1\right ) - 2 \, x + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(log(x) + 1) - 2*x + log(x)

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maple [A]  time = 0.04, size = 14, normalized size = 1.17

method	result	size
norman	\(\ln \relax (x )+\ln \left (\ln \relax (x )+1\right ) x -2 x\)	\(14\)
risch	\(\ln \relax (x )+\ln \left (\ln \relax (x )+1\right ) x -2 x\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)+ln(ln(x)+1)*x-2*x

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maxima [A]  time = 0.36, size = 20, normalized size = 1.67 \begin {gather*} {\left (x - 1\right )} \log \left (\log \relax (x) + 1\right ) - 2 \, x + \log \relax (x) + \log \left (\log \relax (x) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x - 1)*log(log(x) + 1) - 2*x + log(x) + log(log(x) + 1)

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mupad [B]  time = 5.59, size = 13, normalized size = 1.08 \begin {gather*} \ln \relax (x)-2\,x+x\,\ln \left (\ln \relax (x)+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - 2*x + x*log(log(x) + 1)

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sympy [A]  time = 0.32, size = 14, normalized size = 1.17 \begin {gather*} x \log {\left (\log {\relax (x )} + 1 \right )} - 2 x + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(log(x) + 1) - 2*x + log(x)

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