3.28.77 \(\int \frac {-2 x+x^2+(-24-6 x) \log (2)+(-24-2 x+x^2+(-24-6 x) \log (2)) \log (\frac {4+x}{x})+(-24-6 x+(-24-6 x) \log (\frac {4+x}{x})) \log (x+x \log (\frac {4+x}{x}))}{24+6 x+(24+6 x) \log (\frac {4+x}{x})} \, dx\)

Optimal. Leaf size=27 \[ x \left (\frac {x}{12}-\log (2)-\log \left (x+x \log \left (\frac {4+x}{x}\right )\right )\right ) \]

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Rubi [F]  time = 0.82, 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 {-2 x+x^2+(-24-6 x) \log (2)+\left (-24-2 x+x^2+(-24-6 x) \log (2)\right ) \log \left (\frac {4+x}{x}\right )+\left (-24-6 x+(-24-6 x) \log \left (\frac {4+x}{x}\right )\right ) \log \left (x+x \log \left (\frac {4+x}{x}\right )\right )}{24+6 x+(24+6 x) \log \left (\frac {4+x}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x + x^2 + (-24 - 6*x)*Log[2] + (-24 - 2*x + x^2 + (-24 - 6*x)*Log[2])*Log[(4 + x)/x] + (-24 - 6*x + (-
24 - 6*x)*Log[(4 + x)/x])*Log[x + x*Log[(4 + x)/x]])/(24 + 6*x + (24 + 6*x)*Log[(4 + x)/x]),x]

[Out]

x^2/12 - (x*(3 + Log[8]))/3 + 4*Defer[Int][1/((4 + x)*(1 + Log[1 + 4/x])), x] - Defer[Int][Log[x + x*Log[(4 +
x)/x]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x+x^2+(-24-6 x) \log (2)+\left (-24-2 x+x^2+(-24-6 x) \log (2)\right ) \log \left (\frac {4+x}{x}\right )+\left (-24-6 x+(-24-6 x) \log \left (\frac {4+x}{x}\right )\right ) \log \left (x+x \log \left (\frac {4+x}{x}\right )\right )}{6 (4+x) \left (1+\log \left (\frac {4+x}{x}\right )\right )} \, dx\\ &=\frac {1}{6} \int \frac {-2 x+x^2+(-24-6 x) \log (2)+\left (-24-2 x+x^2+(-24-6 x) \log (2)\right ) \log \left (\frac {4+x}{x}\right )+\left (-24-6 x+(-24-6 x) \log \left (\frac {4+x}{x}\right )\right ) \log \left (x+x \log \left (\frac {4+x}{x}\right )\right )}{(4+x) \left (1+\log \left (\frac {4+x}{x}\right )\right )} \, dx\\ &=\frac {1}{6} \int \left (\frac {x^2-24 \log (2)-2 x (1+\log (8))+x^2 \log \left (\frac {4+x}{x}\right )-24 (1+\log (2)) \log \left (\frac {4+x}{x}\right )-2 x (1+\log (8)) \log \left (\frac {4+x}{x}\right )}{(4+x) \left (1+\log \left (\frac {4+x}{x}\right )\right )}-6 \log \left (x+x \log \left (\frac {4+x}{x}\right )\right )\right ) \, dx\\ &=\frac {1}{6} \int \frac {x^2-24 \log (2)-2 x (1+\log (8))+x^2 \log \left (\frac {4+x}{x}\right )-24 (1+\log (2)) \log \left (\frac {4+x}{x}\right )-2 x (1+\log (8)) \log \left (\frac {4+x}{x}\right )}{(4+x) \left (1+\log \left (\frac {4+x}{x}\right )\right )} \, dx-\int \log \left (x+x \log \left (\frac {4+x}{x}\right )\right ) \, dx\\ &=\frac {1}{6} \int \frac {x^2-24 \log (2)-2 x (1+\log (8))+\left (x^2-24 (1+\log (2))-2 x (1+\log (8))\right ) \log \left (\frac {4+x}{x}\right )}{(4+x) \left (1+\log \left (\frac {4+x}{x}\right )\right )} \, dx-\int \log \left (x+x \log \left (\frac {4+x}{x}\right )\right ) \, dx\\ &=\frac {1}{6} \int \left (\frac {x^2-24 (1+\log (2))-2 x (1+\log (8))}{4+x}+\frac {24}{(4+x) \left (1+\log \left (\frac {4+x}{x}\right )\right )}\right ) \, dx-\int \log \left (x+x \log \left (\frac {4+x}{x}\right )\right ) \, dx\\ &=\frac {1}{6} \int \frac {x^2-24 (1+\log (2))-2 x (1+\log (8))}{4+x} \, dx+4 \int \frac {1}{(4+x) \left (1+\log \left (\frac {4+x}{x}\right )\right )} \, dx-\int \log \left (x+x \log \left (\frac {4+x}{x}\right )\right ) \, dx\\ &=\frac {1}{6} \int (x-2 (3+\log (8))) \, dx+4 \int \frac {1}{(4+x) \left (1+\log \left (1+\frac {4}{x}\right )\right )} \, dx-\int \log \left (x+x \log \left (\frac {4+x}{x}\right )\right ) \, dx\\ &=\frac {x^2}{12}-\frac {1}{3} x (3+\log (8))+4 \int \frac {1}{(4+x) \left (1+\log \left (1+\frac {4}{x}\right )\right )} \, dx-\int \log \left (x+x \log \left (\frac {4+x}{x}\right )\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 33, normalized size = 1.22 \begin {gather*} \frac {1}{6} \left (\frac {x^2}{2}-6 x \log (2)-6 x \log \left (x \left (1+\log \left (\frac {4+x}{x}\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x + x^2 + (-24 - 6*x)*Log[2] + (-24 - 2*x + x^2 + (-24 - 6*x)*Log[2])*Log[(4 + x)/x] + (-24 - 6*
x + (-24 - 6*x)*Log[(4 + x)/x])*Log[x + x*Log[(4 + x)/x]])/(24 + 6*x + (24 + 6*x)*Log[(4 + x)/x]),x]

[Out]

(x^2/2 - 6*x*Log[2] - 6*x*Log[x*(1 + Log[(4 + x)/x])])/6

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fricas [A]  time = 0.50, size = 27, normalized size = 1.00 \begin {gather*} \frac {1}{12} \, x^{2} - x \log \relax (2) - x \log \left (x \log \left (\frac {x + 4}{x}\right ) + x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x-24)*log((4+x)/x)-6*x-24)*log(log((4+x)/x)*x+x)+((-6*x-24)*log(2)+x^2-2*x-24)*log((4+x)/x)+(-
6*x-24)*log(2)+x^2-2*x)/((24+6*x)*log((4+x)/x)+24+6*x),x, algorithm="fricas")

[Out]

1/12*x^2 - x*log(2) - x*log(x*log((x + 4)/x) + x)

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giac [A]  time = 0.39, size = 30, normalized size = 1.11 \begin {gather*} \frac {1}{12} \, x^{2} - x \log \relax (2) - x \log \relax (x) - x \log \left (\log \left (\frac {x + 4}{x}\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x-24)*log((4+x)/x)-6*x-24)*log(log((4+x)/x)*x+x)+((-6*x-24)*log(2)+x^2-2*x-24)*log((4+x)/x)+(-
6*x-24)*log(2)+x^2-2*x)/((24+6*x)*log((4+x)/x)+24+6*x),x, algorithm="giac")

[Out]

1/12*x^2 - x*log(2) - x*log(x) - x*log(log((x + 4)/x) + 1)

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maple [A]  time = 0.12, size = 28, normalized size = 1.04

method	result	size
norman	\(\frac {x^{2}}{12}-x \ln \relax (2)-\ln \left (\ln \left (\frac {4+x}{x}\right ) x +x \right ) x\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-6*x-24)*ln((4+x)/x)-6*x-24)*ln(ln((4+x)/x)*x+x)+((-6*x-24)*ln(2)+x^2-2*x-24)*ln((4+x)/x)+(-6*x-24)*ln(
2)+x^2-2*x)/((24+6*x)*ln((4+x)/x)+24+6*x),x,method=_RETURNVERBOSE)

[Out]

1/12*x^2-x*ln(2)-ln(ln((4+x)/x)*x+x)*x

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maxima [A]  time = 0.69, size = 30, normalized size = 1.11 \begin {gather*} \frac {1}{12} \, x^{2} - x \log \relax (2) - x \log \relax (x) - x \log \left (\log \left (x + 4\right ) - \log \relax (x) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x-24)*log((4+x)/x)-6*x-24)*log(log((4+x)/x)*x+x)+((-6*x-24)*log(2)+x^2-2*x-24)*log((4+x)/x)+(-
6*x-24)*log(2)+x^2-2*x)/((24+6*x)*log((4+x)/x)+24+6*x),x, algorithm="maxima")

[Out]

1/12*x^2 - x*log(2) - x*log(x) - x*log(log(x + 4) - log(x) + 1)

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mupad [B]  time = 2.41, size = 27, normalized size = 1.00 \begin {gather*} \frac {x^2}{12}-x\,\ln \left (x+x\,\ln \left (\frac {x+4}{x}\right )\right )-x\,\ln \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + log(2)*(6*x + 24) + log(x + x*log((x + 4)/x))*(6*x + log((x + 4)/x)*(6*x + 24) + 24) + log((x + 4)
/x)*(2*x + log(2)*(6*x + 24) - x^2 + 24) - x^2)/(6*x + log((x + 4)/x)*(6*x + 24) + 24),x)

[Out]

x^2/12 - x*log(x + x*log((x + 4)/x)) - x*log(2)

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sympy [B]  time = 0.73, size = 48, normalized size = 1.78 \begin {gather*} \frac {x^{2}}{12} - x \log {\relax (2 )} + \left (- x - \frac {2}{3}\right ) \log {\left (x \log {\left (\frac {x + 4}{x} \right )} + x \right )} + \frac {2 \log {\relax (x )}}{3} + \frac {2 \log {\left (\log {\left (\frac {x + 4}{x} \right )} + 1 \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x-24)*ln((4+x)/x)-6*x-24)*ln(ln((4+x)/x)*x+x)+((-6*x-24)*ln(2)+x**2-2*x-24)*ln((4+x)/x)+(-6*x-
24)*ln(2)+x**2-2*x)/((24+6*x)*ln((4+x)/x)+24+6*x),x)

[Out]

x**2/12 - x*log(2) + (-x - 2/3)*log(x*log((x + 4)/x) + x) + 2*log(x)/3 + 2*log(log((x + 4)/x) + 1)/3

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