\(\int \frac {282-48 x+2 x^2+(-24 x+26 x^2-2 x^3+(24 x-26 x^2+2 x^3) \log (\frac {1-2 x+x^2}{x^2})) \log (-1+\log (\frac {1-2 x+x^2}{x^2}))}{x-x^2+(-x+x^2) \log (\frac {1-2 x+x^2}{x^2})} \, dx\) [10092]

Optimal result

Integrand size = 100, antiderivative size = 21 \[ \int \frac {282-48 x+2 x^2+\left (-24 x+26 x^2-2 x^3+\left (24 x-26 x^2+2 x^3\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )\right ) \log \left (-1+\log \left (\frac {1-2 x+x^2}{x^2}\right )\right )}{x-x^2+\left (-x+x^2\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=\left (-3+(12-x)^2\right ) \log \left (-1+\log \left (\left (-1+\frac {1}{x}\right )^2\right )\right ) \]

[Out]

ln(ln((1/x-1)^2)-1)*((12-x)^2-3)

Rubi [F]

\[ \int \frac {282-48 x+2 x^2+\left (-24 x+26 x^2-2 x^3+\left (24 x-26 x^2+2 x^3\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )\right ) \log \left (-1+\log \left (\frac {1-2 x+x^2}{x^2}\right )\right )}{x-x^2+\left (-x+x^2\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=\int \frac {282-48 x+2 x^2+\left (-24 x+26 x^2-2 x^3+\left (24 x-26 x^2+2 x^3\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )\right ) \log \left (-1+\log \left (\frac {1-2 x+x^2}{x^2}\right )\right )}{x-x^2+\left (-x+x^2\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )} \, dx \]

[In]

Int[(282 - 48*x + 2*x^2 + (-24*x + 26*x^2 - 2*x^3 + (24*x - 26*x^2 + 2*x^3)*Log[(1 - 2*x + x^2)/x^2])*Log[-1 +
Log[(1 - 2*x + x^2)/x^2]])/(x - x^2 + (-x + x^2)*Log[(1 - 2*x + x^2)/x^2]),x]

[Out]

2*Defer[Int][(-1 + Log[(-1 + x)^2/x^2])^(-1), x] + 236*Defer[Int][1/((-1 + x)*(-1 + Log[(-1 + x)^2/x^2])), x]
- 282*Defer[Int][1/(x*(-1 + Log[(-1 + x)^2/x^2])), x] - 24*Defer[Int][Log[-1 + Log[(-1 + x)^2/x^2]], x] + 2*De
fer[Int][x*Log[-1 + Log[(-1 + x)^2/x^2]], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (141-24 x+x^2+x \left (12-13 x+x^2\right ) \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right ) \log \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )\right )}{(1-x) x \left (1-\log \left (\frac {(-1+x)^2}{x^2}\right )\right )} \, dx \\ & = 2 \int \frac {141-24 x+x^2+x \left (12-13 x+x^2\right ) \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right ) \log \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )}{(1-x) x \left (1-\log \left (\frac {(-1+x)^2}{x^2}\right )\right )} \, dx \\ & = 2 \int \left (\frac {141-24 x+x^2}{(-1+x) x \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )}+(-12+x) \log \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )\right ) \, dx \\ & = 2 \int \frac {141-24 x+x^2}{(-1+x) x \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )} \, dx+2 \int (-12+x) \log \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right ) \, dx \\ & = 2 \int \left (\frac {1}{-1+\log \left (\frac {(-1+x)^2}{x^2}\right )}+\frac {118}{(-1+x) \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )}-\frac {141}{x \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )}\right ) \, dx+2 \int \left (-12 \log \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )+x \log \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )\right ) \, dx \\ & = 2 \int \frac {1}{-1+\log \left (\frac {(-1+x)^2}{x^2}\right )} \, dx+2 \int x \log \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right ) \, dx-24 \int \log \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right ) \, dx+236 \int \frac {1}{(-1+x) \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )} \, dx-282 \int \frac {1}{x \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(21)=42\).

Time = 0.22 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.05 \[ \int \frac {282-48 x+2 x^2+\left (-24 x+26 x^2-2 x^3+\left (24 x-26 x^2+2 x^3\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )\right ) \log \left (-1+\log \left (\frac {1-2 x+x^2}{x^2}\right )\right )}{x-x^2+\left (-x+x^2\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=2 \left (\frac {141}{2} \log \left (1-\log \left (\frac {(-1+x)^2}{x^2}\right )\right )+\frac {1}{2} (-24+x) x \log \left (-1+\log \left (\frac {(-1+x)^2}{x^2}\right )\right )\right ) \]

[In]

Integrate[(282 - 48*x + 2*x^2 + (-24*x + 26*x^2 - 2*x^3 + (24*x - 26*x^2 + 2*x^3)*Log[(1 - 2*x + x^2)/x^2])*Lo
g[-1 + Log[(1 - 2*x + x^2)/x^2]])/(x - x^2 + (-x + x^2)*Log[(1 - 2*x + x^2)/x^2]),x]

[Out]

2*((141*Log[1 - Log[(-1 + x)^2/x^2]])/2 + ((-24 + x)*x*Log[-1 + Log[(-1 + x)^2/x^2]])/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(21)=42\).

Time = 1.65 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.81

method	result	size
parallelrisch	\(\ln \left (\ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right )-1\right ) x^{2}-24 \ln \left (\ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right )-1\right ) x +141 \ln \left (\ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right )-1\right )\)	\(59\)

[In]

int((((2*x^3-26*x^2+24*x)*ln((x^2-2*x+1)/x^2)-2*x^3+26*x^2-24*x)*ln(ln((x^2-2*x+1)/x^2)-1)+2*x^2-48*x+282)/((x
^2-x)*ln((x^2-2*x+1)/x^2)-x^2+x),x,method=_RETURNVERBOSE)

[Out]

ln(ln((x^2-2*x+1)/x^2)-1)*x^2-24*ln(ln((x^2-2*x+1)/x^2)-1)*x+141*ln(ln((x^2-2*x+1)/x^2)-1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {282-48 x+2 x^2+\left (-24 x+26 x^2-2 x^3+\left (24 x-26 x^2+2 x^3\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )\right ) \log \left (-1+\log \left (\frac {1-2 x+x^2}{x^2}\right )\right )}{x-x^2+\left (-x+x^2\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )} \, dx={\left (x^{2} - 24 \, x + 141\right )} \log \left (\log \left (\frac {x^{2} - 2 \, x + 1}{x^{2}}\right ) - 1\right ) \]

[In]

integrate((((2*x^3-26*x^2+24*x)*log((x^2-2*x+1)/x^2)-2*x^3+26*x^2-24*x)*log(log((x^2-2*x+1)/x^2)-1)+2*x^2-48*x
+282)/((x^2-x)*log((x^2-2*x+1)/x^2)-x^2+x),x, algorithm="fricas")

[Out]

(x^2 - 24*x + 141)*log(log((x^2 - 2*x + 1)/x^2) - 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (17) = 34\).

Time = 0.37 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {282-48 x+2 x^2+\left (-24 x+26 x^2-2 x^3+\left (24 x-26 x^2+2 x^3\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )\right ) \log \left (-1+\log \left (\frac {1-2 x+x^2}{x^2}\right )\right )}{x-x^2+\left (-x+x^2\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=\left (x^{2} - 24 x + \frac {23}{6}\right ) \log {\left (\log {\left (\frac {x^{2} - 2 x + 1}{x^{2}} \right )} - 1 \right )} + \frac {823 \log {\left (\log {\left (\frac {x^{2} - 2 x + 1}{x^{2}} \right )} - 1 \right )}}{6} \]

[In]

integrate((((2*x**3-26*x**2+24*x)*ln((x**2-2*x+1)/x**2)-2*x**3+26*x**2-24*x)*ln(ln((x**2-2*x+1)/x**2)-1)+2*x**
2-48*x+282)/((x**2-x)*ln((x**2-2*x+1)/x**2)-x**2+x),x)

[Out]

(x**2 - 24*x + 23/6)*log(log((x**2 - 2*x + 1)/x**2) - 1) + 823*log(log((x**2 - 2*x + 1)/x**2) - 1)/6

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int \frac {282-48 x+2 x^2+\left (-24 x+26 x^2-2 x^3+\left (24 x-26 x^2+2 x^3\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )\right ) \log \left (-1+\log \left (\frac {1-2 x+x^2}{x^2}\right )\right )}{x-x^2+\left (-x+x^2\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )} \, dx={\left (x^{2} - 24 \, x\right )} \log \left (2 \, \log \left (x - 1\right ) - 2 \, \log \left (x\right ) - 1\right ) + 141 \, \log \left (\log \left (x - 1\right ) - \log \left (x\right ) - \frac {1}{2}\right ) \]

[In]

integrate((((2*x^3-26*x^2+24*x)*log((x^2-2*x+1)/x^2)-2*x^3+26*x^2-24*x)*log(log((x^2-2*x+1)/x^2)-1)+2*x^2-48*x
+282)/((x^2-x)*log((x^2-2*x+1)/x^2)-x^2+x),x, algorithm="maxima")

[Out]

(x^2 - 24*x)*log(2*log(x - 1) - 2*log(x) - 1) + 141*log(log(x - 1) - log(x) - 1/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (19) = 38\).

Time = 0.62 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.14 \[ \int \frac {282-48 x+2 x^2+\left (-24 x+26 x^2-2 x^3+\left (24 x-26 x^2+2 x^3\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )\right ) \log \left (-1+\log \left (\frac {1-2 x+x^2}{x^2}\right )\right )}{x-x^2+\left (-x+x^2\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )} \, dx={\left (x^{2} - 24 \, x\right )} \log \left (\log \left (\frac {x^{2} - 2 \, x + 1}{x^{2}}\right ) - 1\right ) + 141 \, \log \left (\log \left (x^{2} - 2 \, x + 1\right ) - \log \left (x^{2}\right ) - 1\right ) \]

[In]

integrate((((2*x^3-26*x^2+24*x)*log((x^2-2*x+1)/x^2)-2*x^3+26*x^2-24*x)*log(log((x^2-2*x+1)/x^2)-1)+2*x^2-48*x
+282)/((x^2-x)*log((x^2-2*x+1)/x^2)-x^2+x),x, algorithm="giac")

[Out]

(x^2 - 24*x)*log(log((x^2 - 2*x + 1)/x^2) - 1) + 141*log(log(x^2 - 2*x + 1) - log(x^2) - 1)

Mupad [B] (verification not implemented)

Time = 15.85 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {282-48 x+2 x^2+\left (-24 x+26 x^2-2 x^3+\left (24 x-26 x^2+2 x^3\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )\right ) \log \left (-1+\log \left (\frac {1-2 x+x^2}{x^2}\right )\right )}{x-x^2+\left (-x+x^2\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=\ln \left (\ln \left (\frac {x^2-2\,x+1}{x^2}\right )-1\right )\,\left (x^2-24\,x+141\right ) \]

[In]

int((48*x + log(log((x^2 - 2*x + 1)/x^2) - 1)*(24*x - log((x^2 - 2*x + 1)/x^2)*(24*x - 26*x^2 + 2*x^3) - 26*x^
2 + 2*x^3) - 2*x^2 - 282)/(log((x^2 - 2*x + 1)/x^2)*(x - x^2) - x + x^2),x)

[Out]

log(log((x^2 - 2*x + 1)/x^2) - 1)*(x^2 - 24*x + 141)