3.50.45 \(\int \frac {-2-2 x-3 x^2+(-2-4 x) \log (x)-\log ^2(x)+(2+x^2+2 x \log (x)+\log ^2(x)) \log (\frac {1}{2} (2 x+x^3+2 x^2 \log (x)+x \log ^2(x)))}{(2+x^2+2 x \log (x)+\log ^2(x)) \log ^2(\frac {1}{2} (2 x+x^3+2 x^2 \log (x)+x \log ^2(x)))} \, dx\)

Optimal. Leaf size=20 \[ 4+\frac {x}{\log \left (x+\frac {1}{2} x (x+\log (x))^2\right )} \]

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Rubi [F]  time = 1.87, 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 {-2-2 x-3 x^2+(-2-4 x) \log (x)-\log ^2(x)+\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log \left (\frac {1}{2} \left (2 x+x^3+2 x^2 \log (x)+x \log ^2(x)\right )\right )}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} \left (2 x+x^3+2 x^2 \log (x)+x \log ^2(x)\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-2*Defer[Int][1/((2 + x^2 + 2*x*Log[x] + Log[x]^2)*Log[(x*(2 + x^2 + 2*x*Log[x] + Log[x]^2))/2]^2), x] - 2*Def
er[Int][x/((2 + x^2 + 2*x*Log[x] + Log[x]^2)*Log[(x*(2 + x^2 + 2*x*Log[x] + Log[x]^2))/2]^2), x] - 3*Defer[Int
][x^2/((2 + x^2 + 2*x*Log[x] + Log[x]^2)*Log[(x*(2 + x^2 + 2*x*Log[x] + Log[x]^2))/2]^2), x] - 2*Defer[Int][Lo
g[x]/((2 + x^2 + 2*x*Log[x] + Log[x]^2)*Log[(x*(2 + x^2 + 2*x*Log[x] + Log[x]^2))/2]^2), x] - 4*Defer[Int][(x*
Log[x])/((2 + x^2 + 2*x*Log[x] + Log[x]^2)*Log[(x*(2 + x^2 + 2*x*Log[x] + Log[x]^2))/2]^2), x] - Defer[Int][Lo
g[x]^2/((2 + x^2 + 2*x*Log[x] + Log[x]^2)*Log[(x*(2 + x^2 + 2*x*Log[x] + Log[x]^2))/2]^2), x] + Defer[Int][Log
[(x*(2 + x^2 + 2*x*Log[x] + Log[x]^2))/2]^(-1), x]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 24, normalized size = 1.20 \begin {gather*} \frac {x}{\log \left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/Log[(x*(2 + x^2 + 2*x*Log[x] + Log[x]^2))/2]

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fricas [A]  time = 0.87, size = 25, normalized size = 1.25 \begin {gather*} \frac {x}{\log \left (\frac {1}{2} \, x^{3} + x^{2} \log \relax (x) + \frac {1}{2} \, x \log \relax (x)^{2} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)^2+2*x*log(x)+x^2+2)*log(1/2*x*log(x)^2+x^2*log(x)+1/2*x^3+x)-log(x)^2+(-4*x-2)*log(x)-3*x^2
-2*x-2)/(log(x)^2+2*x*log(x)+x^2+2)/log(1/2*x*log(x)^2+x^2*log(x)+1/2*x^3+x)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.27, size = 29, normalized size = 1.45 \begin {gather*} -\frac {x}{\log \relax (2) - \log \left (x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2} + 2\right ) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)^2+2*x*log(x)+x^2+2)*log(1/2*x*log(x)^2+x^2*log(x)+1/2*x^3+x)-log(x)^2+(-4*x-2)*log(x)-3*x^2
-2*x-2)/(log(x)^2+2*x*log(x)+x^2+2)/log(1/2*x*log(x)^2+x^2*log(x)+1/2*x^3+x)^2,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.21, size = 174, normalized size = 8.70

method	result	size
risch	\(\frac {2 i x}{\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right )^{2}+\pi \mathrm {csgn}\left (i x \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right )^{3}-2 i \ln \relax (2)+2 i \ln \relax (x )+2 i \ln \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )}\)	\(174\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((ln(x)^2+2*x*ln(x)+x^2+2)*ln(1/2*x*ln(x)^2+x^2*ln(x)+1/2*x^3+x)-ln(x)^2+(-4*x-2)*ln(x)-3*x^2-2*x-2)/(ln(x
)^2+2*x*ln(x)+x^2+2)/ln(1/2*x*ln(x)^2+x^2*ln(x)+1/2*x^3+x)^2,x,method=_RETURNVERBOSE)

[Out]

2*I*x/(Pi*csgn(I*x)*csgn(I*(ln(x)^2+2*x*ln(x)+x^2+2))*csgn(I*x*(ln(x)^2+2*x*ln(x)+x^2+2))-Pi*csgn(I*x)*csgn(I*
x*(ln(x)^2+2*x*ln(x)+x^2+2))^2-Pi*csgn(I*(ln(x)^2+2*x*ln(x)+x^2+2))*csgn(I*x*(ln(x)^2+2*x*ln(x)+x^2+2))^2+Pi*c
sgn(I*x*(ln(x)^2+2*x*ln(x)+x^2+2))^3-2*I*ln(2)+2*I*ln(x)+2*I*ln(ln(x)^2+2*x*ln(x)+x^2+2))

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maxima [A]  time = 0.48, size = 29, normalized size = 1.45 \begin {gather*} -\frac {x}{\log \relax (2) - \log \left (x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2} + 2\right ) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)^2+2*x*log(x)+x^2+2)*log(1/2*x*log(x)^2+x^2*log(x)+1/2*x^3+x)-log(x)^2+(-4*x-2)*log(x)-3*x^2
-2*x-2)/(log(x)^2+2*x*log(x)+x^2+2)/log(1/2*x*log(x)^2+x^2*log(x)+1/2*x^3+x)^2,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {2\,x+{\ln \relax (x)}^2-\ln \left (\frac {x^3}{2}+x^2\,\ln \relax (x)+\frac {x\,{\ln \relax (x)}^2}{2}+x\right )\,\left (x^2+2\,x\,\ln \relax (x)+{\ln \relax (x)}^2+2\right )+\ln \relax (x)\,\left (4\,x+2\right )+3\,x^2+2}{{\ln \left (\frac {x^3}{2}+x^2\,\ln \relax (x)+\frac {x\,{\ln \relax (x)}^2}{2}+x\right )}^2\,\left (x^2+2\,x\,\ln \relax (x)+{\ln \relax (x)}^2+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.42, size = 24, normalized size = 1.20 \begin {gather*} \frac {x}{\log {\left (\frac {x^{3}}{2} + x^{2} \log {\relax (x )} + \frac {x \log {\relax (x )}^{2}}{2} + x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((ln(x)**2+2*x*ln(x)+x**2+2)*ln(1/2*x*ln(x)**2+x**2*ln(x)+1/2*x**3+x)-ln(x)**2+(-4*x-2)*ln(x)-3*x**2
-2*x-2)/(ln(x)**2+2*x*ln(x)+x**2+2)/ln(1/2*x*ln(x)**2+x**2*ln(x)+1/2*x**3+x)**2,x)

[Out]

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

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