\(\int \frac {-2 x+2 \log (x)+(x-\log (x)) \log (2 x^2)+(1-x) \log (2 x^2) \log (\frac {x}{\log (2 x^2)})}{(x^3-2 x^2 \log (x)+x \log ^2(x)) \log (2 x^2)} \, dx\) [222]

Optimal result

Integrand size = 72, antiderivative size = 22 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=1+\frac {\log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{x-\log (x)} \]

[Out]

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

Rubi [F]

\[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx \]

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\frac {\log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{x-\log (x)} \]

[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])

Maple [A] (verified)

Time = 2.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
parallelrisch	\(\frac {\ln \left (\frac {x}{\ln \left (2 x^{2}\right )}\right )}{x -\ln \left (x \right )}\)	\(21\)
risch	\(\text {Expression too large to display}\)	\(878\)

[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]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\frac {\log \left (\frac {x}{\log \left (2\right ) + 2 \, \log \left (x\right )}\right )}{x - \log \left (x\right )} \]

[In]

integrate(((1-x)*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))

Sympy [F(-2)]

Exception generated. \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((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)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\frac {x - \log \left (\log \left (2\right ) + 2 \, \log \left (x\right )\right )}{x - \log \left (x\right )} \]

[In]

integrate(((1-x)*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))

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\frac {x}{x - \log \left (x\right )} - \frac {\log \left (\log \left (2 \, x^{2}\right )\right )}{x - \log \left (x\right )} \]

[In]

integrate(((1-x)*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))

Mupad [B] (verification not implemented)

Time = 8.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\frac {\ln \left (\frac {x}{\ln \left (2\,x^2\right )}\right )}{x-\ln \left (x\right )} \]

[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))