\(\int \frac {8 x-9 \log (2)+(-2 x+2 \log (2)) \log (x)+(x-\log (2)) \log (2 x-2 \log (2))}{-49 x+28 x^2-4 x^3+(49-28 x+4 x^2) \log (2)+(-4 x+4 \log (2)) \log ^2(x)+(-14 x+4 x^2+(14-4 x) \log (2)) \log (2 x-2 \log (2))+(-x+\log (2)) \log ^2(2 x-2 \log (2))+\log (x) (28 x-8 x^2+(-28+8 x) \log (2)+(4 x-4 \log (2)) \log (2 x-2 \log (2)))} \, dx\) [9462]

Optimal result

Integrand size = 161, antiderivative size = 23 \[ \int \frac {8 x-9 \log (2)+(-2 x+2 \log (2)) \log (x)+(x-\log (2)) \log (2 x-2 \log (2))}{-49 x+28 x^2-4 x^3+\left (49-28 x+4 x^2\right ) \log (2)+(-4 x+4 \log (2)) \log ^2(x)+\left (-14 x+4 x^2+(14-4 x) \log (2)\right ) \log (2 x-2 \log (2))+(-x+\log (2)) \log ^2(2 x-2 \log (2))+\log (x) \left (28 x-8 x^2+(-28+8 x) \log (2)+(4 x-4 \log (2)) \log (2 x-2 \log (2))\right )} \, dx=-\frac {x}{7-2 x-2 \log (x)+\log (-2 (-x+\log (2)))} \]

[Out]

-x/(ln(-2*ln(2)+2*x)-2*ln(x)-2*x+7)

Rubi [F]

\[ \int \frac {8 x-9 \log (2)+(-2 x+2 \log (2)) \log (x)+(x-\log (2)) \log (2 x-2 \log (2))}{-49 x+28 x^2-4 x^3+\left (49-28 x+4 x^2\right ) \log (2)+(-4 x+4 \log (2)) \log ^2(x)+\left (-14 x+4 x^2+(14-4 x) \log (2)\right ) \log (2 x-2 \log (2))+(-x+\log (2)) \log ^2(2 x-2 \log (2))+\log (x) \left (28 x-8 x^2+(-28+8 x) \log (2)+(4 x-4 \log (2)) \log (2 x-2 \log (2))\right )} \, dx=\int \frac {8 x-9 \log (2)+(-2 x+2 \log (2)) \log (x)+(x-\log (2)) \log (2 x-2 \log (2))}{-49 x+28 x^2-4 x^3+\left (49-28 x+4 x^2\right ) \log (2)+(-4 x+4 \log (2)) \log ^2(x)+\left (-14 x+4 x^2+(14-4 x) \log (2)\right ) \log (2 x-2 \log (2))+(-x+\log (2)) \log ^2(2 x-2 \log (2))+\log (x) \left (28 x-8 x^2+(-28+8 x) \log (2)+(4 x-4 \log (2)) \log (2 x-2 \log (2))\right )} \, dx \]

[In]

Int[(8*x - 9*Log[2] + (-2*x + 2*Log[2])*Log[x] + (x - Log[2])*Log[2*x - 2*Log[2]])/(-49*x + 28*x^2 - 4*x^3 + (
49 - 28*x + 4*x^2)*Log[2] + (-4*x + 4*Log[2])*Log[x]^2 + (-14*x + 4*x^2 + (14 - 4*x)*Log[2])*Log[2*x - 2*Log[2
]] + (-x + Log[2])*Log[2*x - 2*Log[2]]^2 + Log[x]*(28*x - 8*x^2 + (-28 + 8*x)*Log[2] + (4*x - 4*Log[2])*Log[2*
x - 2*Log[2]])),x]

[Out]

-Defer[Int][(-7 + 2*x + 2*Log[x] - Log[2*x - Log[4]])^(-2), x] - 2*Defer[Int][x/(-7 + 2*x + 2*Log[x] - Log[2*x
- Log[4]])^2, x] + Log[2]*Defer[Int][1/((x - Log[2])*(-7 + 2*x + 2*Log[x] - Log[2*x - Log[4]])^2), x] + Defer
[Int][(-7 + 2*x + 2*Log[x] - Log[2*x - Log[4]])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-8 x+\log (512)+2 (x-\log (2)) \log (x)+(-x+\log (2)) \log (2 x-\log (4))}{(x-\log (2)) (7-2 x-2 \log (x)+\log (2 x-\log (4)))^2} \, dx \\ & = \int \left (\frac {1}{-7+2 x+2 \log (x)-\log (2 x-\log (4))}+\frac {-2 x^2-x (1-\log (4))+\log (4)}{(x-\log (2)) (7-2 x-2 \log (x)+\log (2 x-\log (4)))^2}\right ) \, dx \\ & = \int \frac {1}{-7+2 x+2 \log (x)-\log (2 x-\log (4))} \, dx+\int \frac {-2 x^2-x (1-\log (4))+\log (4)}{(x-\log (2)) (7-2 x-2 \log (x)+\log (2 x-\log (4)))^2} \, dx \\ & = \int \left (-\frac {1}{(-7+2 x+2 \log (x)-\log (2 x-\log (4)))^2}-\frac {2 x}{(-7+2 x+2 \log (x)-\log (2 x-\log (4)))^2}+\frac {\log (2)}{(x-\log (2)) (-7+2 x+2 \log (x)-\log (2 x-\log (4)))^2}\right ) \, dx+\int \frac {1}{-7+2 x+2 \log (x)-\log (2 x-\log (4))} \, dx \\ & = -\left (2 \int \frac {x}{(-7+2 x+2 \log (x)-\log (2 x-\log (4)))^2} \, dx\right )+\log (2) \int \frac {1}{(x-\log (2)) (-7+2 x+2 \log (x)-\log (2 x-\log (4)))^2} \, dx-\int \frac {1}{(-7+2 x+2 \log (x)-\log (2 x-\log (4)))^2} \, dx+\int \frac {1}{-7+2 x+2 \log (x)-\log (2 x-\log (4))} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {8 x-9 \log (2)+(-2 x+2 \log (2)) \log (x)+(x-\log (2)) \log (2 x-2 \log (2))}{-49 x+28 x^2-4 x^3+\left (49-28 x+4 x^2\right ) \log (2)+(-4 x+4 \log (2)) \log ^2(x)+\left (-14 x+4 x^2+(14-4 x) \log (2)\right ) \log (2 x-2 \log (2))+(-x+\log (2)) \log ^2(2 x-2 \log (2))+\log (x) \left (28 x-8 x^2+(-28+8 x) \log (2)+(4 x-4 \log (2)) \log (2 x-2 \log (2))\right )} \, dx=\int \frac {8 x-9 \log (2)+(-2 x+2 \log (2)) \log (x)+(x-\log (2)) \log (2 x-2 \log (2))}{-49 x+28 x^2-4 x^3+\left (49-28 x+4 x^2\right ) \log (2)+(-4 x+4 \log (2)) \log ^2(x)+\left (-14 x+4 x^2+(14-4 x) \log (2)\right ) \log (2 x-2 \log (2))+(-x+\log (2)) \log ^2(2 x-2 \log (2))+\log (x) \left (28 x-8 x^2+(-28+8 x) \log (2)+(4 x-4 \log (2)) \log (2 x-2 \log (2))\right )} \, dx \]

[In]

Integrate[(8*x - 9*Log[2] + (-2*x + 2*Log[2])*Log[x] + (x - Log[2])*Log[2*x - 2*Log[2]])/(-49*x + 28*x^2 - 4*x
^3 + (49 - 28*x + 4*x^2)*Log[2] + (-4*x + 4*Log[2])*Log[x]^2 + (-14*x + 4*x^2 + (14 - 4*x)*Log[2])*Log[2*x - 2
*Log[2]] + (-x + Log[2])*Log[2*x - 2*Log[2]]^2 + Log[x]*(28*x - 8*x^2 + (-28 + 8*x)*Log[2] + (4*x - 4*Log[2])*
Log[2*x - 2*Log[2]])),x]

[Out]

Integrate[(8*x - 9*Log[2] + (-2*x + 2*Log[2])*Log[x] + (x - Log[2])*Log[2*x - 2*Log[2]])/(-49*x + 28*x^2 - 4*x
^3 + (49 - 28*x + 4*x^2)*Log[2] + (-4*x + 4*Log[2])*Log[x]^2 + (-14*x + 4*x^2 + (14 - 4*x)*Log[2])*Log[2*x - 2
*Log[2]] + (-x + Log[2])*Log[2*x - 2*Log[2]]^2 + Log[x]*(28*x - 8*x^2 + (-28 + 8*x)*Log[2] + (4*x - 4*Log[2])*
Log[2*x - 2*Log[2]])), x]

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04

method	result	size
default	\(-\frac {x}{\ln \left (2\right )-2 x +\ln \left (x -\ln \left (2\right )\right )-2 \ln \left (x \right )+7}\)	\(24\)
risch	\(\frac {x}{2 \ln \left (x \right )-\ln \left (-2 \ln \left (2\right )+2 x \right )+2 x -7}\)	\(25\)
parallelrisch	\(\frac {x}{2 \ln \left (x \right )-\ln \left (-2 \ln \left (2\right )+2 x \right )+2 x -7}\)	\(25\)

[In]

int(((2*ln(2)-2*x)*ln(x)+(x-ln(2))*ln(-2*ln(2)+2*x)-9*ln(2)+8*x)/((4*ln(2)-4*x)*ln(x)^2+((-4*ln(2)+4*x)*ln(-2*
ln(2)+2*x)+(8*x-28)*ln(2)-8*x^2+28*x)*ln(x)+(ln(2)-x)*ln(-2*ln(2)+2*x)^2+((-4*x+14)*ln(2)+4*x^2-14*x)*ln(-2*ln
(2)+2*x)+(4*x^2-28*x+49)*ln(2)-4*x^3+28*x^2-49*x),x,method=_RETURNVERBOSE)

[Out]

-x/(ln(2)-2*x+ln(x-ln(2))-2*ln(x)+7)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {8 x-9 \log (2)+(-2 x+2 \log (2)) \log (x)+(x-\log (2)) \log (2 x-2 \log (2))}{-49 x+28 x^2-4 x^3+\left (49-28 x+4 x^2\right ) \log (2)+(-4 x+4 \log (2)) \log ^2(x)+\left (-14 x+4 x^2+(14-4 x) \log (2)\right ) \log (2 x-2 \log (2))+(-x+\log (2)) \log ^2(2 x-2 \log (2))+\log (x) \left (28 x-8 x^2+(-28+8 x) \log (2)+(4 x-4 \log (2)) \log (2 x-2 \log (2))\right )} \, dx=\frac {x}{2 \, x - \log \left (2 \, x - 2 \, \log \left (2\right )\right ) + 2 \, \log \left (x\right ) - 7} \]

[In]

integrate(((2*log(2)-2*x)*log(x)+(x-log(2))*log(-2*log(2)+2*x)-9*log(2)+8*x)/((4*log(2)-4*x)*log(x)^2+((-4*log
(2)+4*x)*log(-2*log(2)+2*x)+(8*x-28)*log(2)-8*x^2+28*x)*log(x)+(log(2)-x)*log(-2*log(2)+2*x)^2+((-4*x+14)*log(
2)+4*x^2-14*x)*log(-2*log(2)+2*x)+(4*x^2-28*x+49)*log(2)-4*x^3+28*x^2-49*x),x, algorithm="fricas")

[Out]

x/(2*x - log(2*x - 2*log(2)) + 2*log(x) - 7)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {8 x-9 \log (2)+(-2 x+2 \log (2)) \log (x)+(x-\log (2)) \log (2 x-2 \log (2))}{-49 x+28 x^2-4 x^3+\left (49-28 x+4 x^2\right ) \log (2)+(-4 x+4 \log (2)) \log ^2(x)+\left (-14 x+4 x^2+(14-4 x) \log (2)\right ) \log (2 x-2 \log (2))+(-x+\log (2)) \log ^2(2 x-2 \log (2))+\log (x) \left (28 x-8 x^2+(-28+8 x) \log (2)+(4 x-4 \log (2)) \log (2 x-2 \log (2))\right )} \, dx=- \frac {x}{- 2 x - 2 \log {\left (x \right )} + \log {\left (2 x - 2 \log {\left (2 \right )} \right )} + 7} \]

[In]

integrate(((2*ln(2)-2*x)*ln(x)+(x-ln(2))*ln(-2*ln(2)+2*x)-9*ln(2)+8*x)/((4*ln(2)-4*x)*ln(x)**2+((-4*ln(2)+4*x)
*ln(-2*ln(2)+2*x)+(8*x-28)*ln(2)-8*x**2+28*x)*ln(x)+(ln(2)-x)*ln(-2*ln(2)+2*x)**2+((-4*x+14)*ln(2)+4*x**2-14*x
)*ln(-2*ln(2)+2*x)+(4*x**2-28*x+49)*ln(2)-4*x**3+28*x**2-49*x),x)

[Out]

-x/(-2*x - 2*log(x) + log(2*x - 2*log(2)) + 7)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {8 x-9 \log (2)+(-2 x+2 \log (2)) \log (x)+(x-\log (2)) \log (2 x-2 \log (2))}{-49 x+28 x^2-4 x^3+\left (49-28 x+4 x^2\right ) \log (2)+(-4 x+4 \log (2)) \log ^2(x)+\left (-14 x+4 x^2+(14-4 x) \log (2)\right ) \log (2 x-2 \log (2))+(-x+\log (2)) \log ^2(2 x-2 \log (2))+\log (x) \left (28 x-8 x^2+(-28+8 x) \log (2)+(4 x-4 \log (2)) \log (2 x-2 \log (2))\right )} \, dx=\frac {x}{2 \, x - \log \left (2\right ) - \log \left (x - \log \left (2\right )\right ) + 2 \, \log \left (x\right ) - 7} \]

[In]

integrate(((2*log(2)-2*x)*log(x)+(x-log(2))*log(-2*log(2)+2*x)-9*log(2)+8*x)/((4*log(2)-4*x)*log(x)^2+((-4*log
(2)+4*x)*log(-2*log(2)+2*x)+(8*x-28)*log(2)-8*x^2+28*x)*log(x)+(log(2)-x)*log(-2*log(2)+2*x)^2+((-4*x+14)*log(
2)+4*x^2-14*x)*log(-2*log(2)+2*x)+(4*x^2-28*x+49)*log(2)-4*x^3+28*x^2-49*x),x, algorithm="maxima")

[Out]

x/(2*x - log(2) - log(x - log(2)) + 2*log(x) - 7)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {8 x-9 \log (2)+(-2 x+2 \log (2)) \log (x)+(x-\log (2)) \log (2 x-2 \log (2))}{-49 x+28 x^2-4 x^3+\left (49-28 x+4 x^2\right ) \log (2)+(-4 x+4 \log (2)) \log ^2(x)+\left (-14 x+4 x^2+(14-4 x) \log (2)\right ) \log (2 x-2 \log (2))+(-x+\log (2)) \log ^2(2 x-2 \log (2))+\log (x) \left (28 x-8 x^2+(-28+8 x) \log (2)+(4 x-4 \log (2)) \log (2 x-2 \log (2))\right )} \, dx=\frac {x}{2 \, x - \log \left (2\right ) - \log \left (x - \log \left (2\right )\right ) + 2 \, \log \left (x\right ) - 7} \]

[In]

integrate(((2*log(2)-2*x)*log(x)+(x-log(2))*log(-2*log(2)+2*x)-9*log(2)+8*x)/((4*log(2)-4*x)*log(x)^2+((-4*log
(2)+4*x)*log(-2*log(2)+2*x)+(8*x-28)*log(2)-8*x^2+28*x)*log(x)+(log(2)-x)*log(-2*log(2)+2*x)^2+((-4*x+14)*log(
2)+4*x^2-14*x)*log(-2*log(2)+2*x)+(4*x^2-28*x+49)*log(2)-4*x^3+28*x^2-49*x),x, algorithm="giac")

[Out]

x/(2*x - log(2) - log(x - log(2)) + 2*log(x) - 7)

Mupad [F(-1)]

Timed out. \[ \int \frac {8 x-9 \log (2)+(-2 x+2 \log (2)) \log (x)+(x-\log (2)) \log (2 x-2 \log (2))}{-49 x+28 x^2-4 x^3+\left (49-28 x+4 x^2\right ) \log (2)+(-4 x+4 \log (2)) \log ^2(x)+\left (-14 x+4 x^2+(14-4 x) \log (2)\right ) \log (2 x-2 \log (2))+(-x+\log (2)) \log ^2(2 x-2 \log (2))+\log (x) \left (28 x-8 x^2+(-28+8 x) \log (2)+(4 x-4 \log (2)) \log (2 x-2 \log (2))\right )} \, dx=\int -\frac {8\,x-9\,\ln \left (2\right )-\ln \left (x\right )\,\left (2\,x-2\,\ln \left (2\right )\right )+\ln \left (2\,x-2\,\ln \left (2\right )\right )\,\left (x-\ln \left (2\right )\right )}{49\,x-\ln \left (2\right )\,\left (4\,x^2-28\,x+49\right )+\ln \left (2\,x-2\,\ln \left (2\right )\right )\,\left (14\,x+\ln \left (2\right )\,\left (4\,x-14\right )-4\,x^2\right )-\ln \left (x\right )\,\left (28\,x+\ln \left (2\right )\,\left (8\,x-28\right )+\ln \left (2\,x-2\,\ln \left (2\right )\right )\,\left (4\,x-4\,\ln \left (2\right )\right )-8\,x^2\right )+{\ln \left (x\right )}^2\,\left (4\,x-4\,\ln \left (2\right )\right )+{\ln \left (2\,x-2\,\ln \left (2\right )\right )}^2\,\left (x-\ln \left (2\right )\right )-28\,x^2+4\,x^3} \,d x \]

[In]

int(-(8*x - 9*log(2) - log(x)*(2*x - 2*log(2)) + log(2*x - 2*log(2))*(x - log(2)))/(49*x - log(2)*(4*x^2 - 28*
x + 49) + log(2*x - 2*log(2))*(14*x + log(2)*(4*x - 14) - 4*x^2) - log(x)*(28*x + log(2)*(8*x - 28) + log(2*x
- 2*log(2))*(4*x - 4*log(2)) - 8*x^2) + log(x)^2*(4*x - 4*log(2)) + log(2*x - 2*log(2))^2*(x - log(2)) - 28*x^
2 + 4*x^3),x)

[Out]

int(-(8*x - 9*log(2) - log(x)*(2*x - 2*log(2)) + log(2*x - 2*log(2))*(x - log(2)))/(49*x - log(2)*(4*x^2 - 28*
x + 49) + log(2*x - 2*log(2))*(14*x + log(2)*(4*x - 14) - 4*x^2) - log(x)*(28*x + log(2)*(8*x - 28) + log(2*x
- 2*log(2))*(4*x - 4*log(2)) - 8*x^2) + log(x)^2*(4*x - 4*log(2)) + log(2*x - 2*log(2))^2*(x - log(2)) - 28*x^
2 + 4*x^3), x)