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

Optimal. Leaf size=22 \[ 1+\frac {\log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{x-\log (x)} \]

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Rubi [F]  time = 1.21, 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+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.27, size = 20, normalized size = 0.91 \begin {gather*} \frac {\log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{x-\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x + 2*Log[x] + (x - Log[x])*Log[2*x^2] + (1 - x)*Log[2*x^2]*Log[x/Log[2*x^2]])/((x^3 - 2*x^2*Log
[x] + x*Log[x]^2)*Log[2*x^2]),x]

[Out]

Log[x/Log[2*x^2]]/(x - Log[x])

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fricas [A]  time = 0.54, size = 21, normalized size = 0.95 \begin {gather*} \frac {\log \left (\frac {x}{\log \relax (2) + 2 \, \log \relax (x)}\right )}{x - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.51, size = 28, normalized size = 1.27 \begin {gather*} \frac {x}{x - \log \relax (x)} - \frac {\log \left (\log \left (2 \, x^{2}\right )\right )}{x - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.56, size = 878, normalized size = 39.91

method	result	size
risch	\(-\frac {\ln \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )\right )}{x -\ln \relax (x )}+\frac {-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i x}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {x}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right )-i \pi \mathrm {csgn}\left (\frac {x}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i x}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i x}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {x}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {x}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (2)+4 i \ln \relax (x )}\right )^{3}+i \pi +2 \ln \relax (2)+2 x}{2 x -2 \ln \relax (x )}\)	\(878\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(x-ln(x))*ln(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2)+4*I*ln(x))+
1/2*(-I*Pi*csgn(I*x)*csgn(I/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2
)+4*I*ln(x)))*csgn(I*x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2)+4*I
*ln(x)))+I*Pi*csgn(I*x)*csgn(I*x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I
*ln(2)+4*I*ln(x)))^2-I*Pi*csgn(I*x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2
*I*ln(2)+4*I*ln(x)))*csgn(x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2
)+4*I*ln(x)))-I*Pi*csgn(x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2)+
4*I*ln(x)))^2+I*Pi*csgn(I/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2)+
4*I*ln(x)))*csgn(I*x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2)+4*I*l
n(x)))^2-I*Pi*csgn(I*x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2)+4*I
*ln(x)))^3+I*Pi*csgn(I*x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2)+4
*I*ln(x)))*csgn(x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2)+4*I*ln(x
)))^2+I*Pi*csgn(x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+2*I*ln(2)+4*I*ln(x
)))^3+I*Pi+2*ln(2)+2*x)/(x-ln(x))

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maxima [A]  time = 0.70, size = 21, normalized size = 0.95 \begin {gather*} \frac {x - \log \left (\log \relax (2) + 2 \, \log \relax (x)\right )}{x - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.74, size = 20, normalized size = 0.91 \begin {gather*} \frac {\ln \left (\frac {x}{\ln \left (2\,x^2\right )}\right )}{x-\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x/log(2*x^2))/(x - log(x))

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sympy [A]  time = 0.51, size = 15, normalized size = 0.68 \begin {gather*} \frac {\log {\left (\frac {x}{2 \log {\relax (x )} + \log {\relax (2 )}} \right )}}{x - \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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