3.35.71 \(\int \frac {6-2 x-6 x^2+2 e^3 x^2+4 x^3+(-6-x+e^3 x+2 x^2) \log (x)+(4 x^2+2 x \log (x)) \log (\frac {2 x}{2 x+\log (x)})}{4 x^2+2 x \log (x)} \, dx\)

Optimal. Leaf size=26 \[ (-3+x) \left (\frac {1}{2} \left (e^3+x\right )+\log \left (\frac {2 x}{2 x+\log (x)}\right )\right ) \]

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Rubi [F]  time = 1.07, 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 {6-2 x-6 x^2+2 e^3 x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{4 x^2+2 x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6 - 2*x - 6*x^2 + 2*E^3*x^2 + 4*x^3 + (-6 - x + E^3*x + 2*x^2)*Log[x] + (4*x^2 + 2*x*Log[x])*Log[(2*x)/(2
*x + Log[x])])/(4*x^2 + 2*x*Log[x]),x]

[Out]

-x - ((1 - E^3)*x)/2 + x^2/2 - 3*Log[x] + x*Log[(2*x)/(2*x + Log[x])] - Defer[Int][(-2*x - Log[x])^(-1), x] +
5*Defer[Int][(2*x + Log[x])^(-1), x] + 3*Defer[Int][1/(x*(2*x + Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6-2 x+\left (-6+2 e^3\right ) x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{4 x^2+2 x \log (x)} \, dx\\ &=\int \frac {6-2 x+\left (-6+2 e^3\right ) x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{x (4 x+2 \log (x))} \, dx\\ &=\int \frac {6-2 x+\left (-6+2 e^3\right ) x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{2 x (2 x+\log (x))} \, dx\\ &=\frac {1}{2} \int \frac {6-2 x+\left (-6+2 e^3\right ) x^2+4 x^3+\left (-6-x+e^3 x+2 x^2\right ) \log (x)+\left (4 x^2+2 x \log (x)\right ) \log \left (\frac {2 x}{2 x+\log (x)}\right )}{x (2 x+\log (x))} \, dx\\ &=\frac {1}{2} \int \left (\frac {6-2 x-6 \left (1-\frac {e^3}{3}\right ) x^2+4 x^3-6 \log (x)-\left (1-e^3\right ) x \log (x)+2 x^2 \log (x)}{x (2 x+\log (x))}+2 \log \left (\frac {2 x}{2 x+\log (x)}\right )\right ) \, dx\\ &=\frac {1}{2} \int \frac {6-2 x-6 \left (1-\frac {e^3}{3}\right ) x^2+4 x^3-6 \log (x)-\left (1-e^3\right ) x \log (x)+2 x^2 \log (x)}{x (2 x+\log (x))} \, dx+\int \log \left (\frac {2 x}{2 x+\log (x)}\right ) \, dx\\ &=x \log \left (\frac {2 x}{2 x+\log (x)}\right )+\frac {1}{2} \int \left (\frac {-6-\left (1-e^3\right ) x+2 x^2}{x}-\frac {2 (-3+x) (1+2 x)}{x (2 x+\log (x))}\right ) \, dx-\int \frac {-1+\log (x)}{2 x+\log (x)} \, dx\\ &=x \log \left (\frac {2 x}{2 x+\log (x)}\right )+\frac {1}{2} \int \frac {-6-\left (1-e^3\right ) x+2 x^2}{x} \, dx-\int \frac {(-3+x) (1+2 x)}{x (2 x+\log (x))} \, dx-\int \left (1+\frac {-1-2 x}{2 x+\log (x)}\right ) \, dx\\ &=-x+x \log \left (\frac {2 x}{2 x+\log (x)}\right )+\frac {1}{2} \int \left (-1+e^3-\frac {6}{x}+2 x\right ) \, dx-\int \frac {-1-2 x}{2 x+\log (x)} \, dx-\int \left (-\frac {5}{2 x+\log (x)}-\frac {3}{x (2 x+\log (x))}+\frac {2 x}{2 x+\log (x)}\right ) \, dx\\ &=-x-\frac {1}{2} \left (1-e^3\right ) x+\frac {x^2}{2}-3 \log (x)+x \log \left (\frac {2 x}{2 x+\log (x)}\right )-2 \int \frac {x}{2 x+\log (x)} \, dx+3 \int \frac {1}{x (2 x+\log (x))} \, dx+5 \int \frac {1}{2 x+\log (x)} \, dx-\int \left (\frac {1}{-2 x-\log (x)}-\frac {2 x}{2 x+\log (x)}\right ) \, dx\\ &=-x-\frac {1}{2} \left (1-e^3\right ) x+\frac {x^2}{2}-3 \log (x)+x \log \left (\frac {2 x}{2 x+\log (x)}\right )+3 \int \frac {1}{x (2 x+\log (x))} \, dx+5 \int \frac {1}{2 x+\log (x)} \, dx-\int \frac {1}{-2 x-\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 43, normalized size = 1.65 \begin {gather*} \frac {1}{2} \left (\left (-3+e^3\right ) x+x^2-6 \log (x)+2 x \log \left (\frac {2 x}{2 x+\log (x)}\right )+6 \log (2 x+\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 - 2*x - 6*x^2 + 2*E^3*x^2 + 4*x^3 + (-6 - x + E^3*x + 2*x^2)*Log[x] + (4*x^2 + 2*x*Log[x])*Log[(2
*x)/(2*x + Log[x])])/(4*x^2 + 2*x*Log[x]),x]

[Out]

((-3 + E^3)*x + x^2 - 6*Log[x] + 2*x*Log[(2*x)/(2*x + Log[x])] + 6*Log[2*x + Log[x]])/2

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fricas [A]  time = 0.97, size = 30, normalized size = 1.15 \begin {gather*} \frac {1}{2} \, x^{2} + \frac {1}{2} \, x e^{3} + {\left (x - 3\right )} \log \left (\frac {2 \, x}{2 \, x + \log \relax (x)}\right ) - \frac {3}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+4*x^2)*log(2*x/(2*x+log(x)))+(x*exp(3)+2*x^2-x-6)*log(x)+2*x^2*exp(3)+4*x^3-6*x^2-2*x+6
)/(2*x*log(x)+4*x^2),x, algorithm="fricas")

[Out]

1/2*x^2 + 1/2*x*e^3 + (x - 3)*log(2*x/(2*x + log(x))) - 3/2*x

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giac [A]  time = 0.31, size = 45, normalized size = 1.73 \begin {gather*} \frac {1}{2} \, x^{2} + \frac {1}{2} \, x e^{3} + x \log \relax (2) - x \log \left (2 \, x + \log \relax (x)\right ) + x \log \relax (x) - \frac {3}{2} \, x + 3 \, \log \left (2 \, x + \log \relax (x)\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+4*x^2)*log(2*x/(2*x+log(x)))+(x*exp(3)+2*x^2-x-6)*log(x)+2*x^2*exp(3)+4*x^3-6*x^2-2*x+6
)/(2*x*log(x)+4*x^2),x, algorithm="giac")

[Out]

1/2*x^2 + 1/2*x*e^3 + x*log(2) - x*log(2*x + log(x)) + x*log(x) - 3/2*x + 3*log(2*x + log(x)) - 3*log(x)

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maple [C]  time = 0.10, size = 154, normalized size = 5.92

method	result	size
risch	\(-x \ln \left (x +\frac {\ln \relax (x )}{2}\right )+x \ln \relax (x )-\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{x +\frac {\ln \relax (x )}{2}}\right ) \mathrm {csgn}\left (\frac {i x}{x +\frac {\ln \relax (x )}{2}}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{x +\frac {\ln \relax (x )}{2}}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x +\frac {\ln \relax (x )}{2}}\right ) \mathrm {csgn}\left (\frac {i x}{x +\frac {\ln \relax (x )}{2}}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i x}{x +\frac {\ln \relax (x )}{2}}\right )^{3}}{2}+\frac {x \,{\mathrm e}^{3}}{2}+\frac {x^{2}}{2}-\frac {3 x}{2}-3 \ln \relax (x )+3 \ln \left (2 x +\ln \relax (x )\right )\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*ln(x)+4*x^2)*ln(2*x/(2*x+ln(x)))+(x*exp(3)+2*x^2-x-6)*ln(x)+2*x^2*exp(3)+4*x^3-6*x^2-2*x+6)/(2*x*ln(
x)+4*x^2),x,method=_RETURNVERBOSE)

[Out]

-x*ln(x+1/2*ln(x))+x*ln(x)-1/2*I*Pi*x*csgn(I*x)*csgn(I/(x+1/2*ln(x)))*csgn(I*x/(x+1/2*ln(x)))+1/2*I*Pi*x*csgn(
I*x)*csgn(I*x/(x+1/2*ln(x)))^2+1/2*I*Pi*x*csgn(I/(x+1/2*ln(x)))*csgn(I*x/(x+1/2*ln(x)))^2-1/2*I*Pi*x*csgn(I*x/
(x+1/2*ln(x)))^3+1/2*x*exp(3)+1/2*x^2-3/2*x-3*ln(x)+3*ln(2*x+ln(x))

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maxima [A]  time = 0.66, size = 35, normalized size = 1.35 \begin {gather*} \frac {1}{2} \, x^{2} + \frac {1}{2} \, x {\left (e^{3} + 2 \, \log \relax (2) - 3\right )} - {\left (x - 3\right )} \log \left (2 \, x + \log \relax (x)\right ) + {\left (x - 3\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+4*x^2)*log(2*x/(2*x+log(x)))+(x*exp(3)+2*x^2-x-6)*log(x)+2*x^2*exp(3)+4*x^3-6*x^2-2*x+6
)/(2*x*log(x)+4*x^2),x, algorithm="maxima")

[Out]

1/2*x^2 + 1/2*x*(e^3 + 2*log(2) - 3) - (x - 3)*log(2*x + log(x)) + (x - 3)*log(x)

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mupad [B]  time = 2.38, size = 41, normalized size = 1.58 \begin {gather*} 3\,\ln \left (2\,x+\ln \relax (x)\right )-3\,\ln \relax (x)+x\,\ln \left (\frac {2\,x}{2\,x+\ln \relax (x)}\right )+\frac {x^2}{2}+x\,\left (\frac {{\mathrm {e}}^3}{2}-\frac {3}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((2*x)/(2*x + log(x)))*(2*x*log(x) + 4*x^2) - 2*x - log(x)*(x - x*exp(3) - 2*x^2 + 6) + 2*x^2*exp(3) -
6*x^2 + 4*x^3 + 6)/(2*x*log(x) + 4*x^2),x)

[Out]

3*log(2*x + log(x)) - 3*log(x) + x*log((2*x)/(2*x + log(x))) + x^2/2 + x*(exp(3)/2 - 3/2)

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sympy [A]  time = 0.49, size = 41, normalized size = 1.58 \begin {gather*} \frac {x^{2}}{2} + x \log {\left (\frac {2 x}{2 x + \log {\relax (x )}} \right )} + \frac {x \left (-3 + e^{3}\right )}{2} - 3 \log {\relax (x )} + 3 \log {\left (2 x + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*ln(x)+4*x**2)*ln(2*x/(2*x+ln(x)))+(x*exp(3)+2*x**2-x-6)*ln(x)+2*x**2*exp(3)+4*x**3-6*x**2-2*x+
6)/(2*x*ln(x)+4*x**2),x)

[Out]

x**2/2 + x*log(2*x/(2*x + log(x))) + x*(-3 + exp(3))/2 - 3*log(x) + 3*log(2*x + log(x))

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