\(\int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+(-24 x^2+(24 x+24 x^2) \log ^2(2)) \log (x)+(-4 x^2-4 x^3) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx\) [10036]

Optimal result

Integrand size = 94, antiderivative size = 29 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=4+\frac {4 \left (\frac {1}{2}+x-\frac {x}{\log ^2(2)-\frac {1}{3} x \log (x)}\right )}{x^2} \]

[Out]

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

Rubi [F]

\[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=\int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx \]

[In]

Int[(-12*x^2 + 36*x*Log[2]^2 + (-36 - 36*x)*Log[2]^4 + (-24*x^2 + (24*x + 24*x^2)*Log[2]^2)*Log[x] + (-4*x^2 -
4*x^3)*Log[x]^2)/(9*x^3*Log[2]^4 - 6*x^4*Log[2]^2*Log[x] + x^5*Log[x]^2),x]

[Out]

(2*(1 + x)^2)/x^2 - 36*Log[2]^2*Defer[Int][1/(x^2*(-3*Log[2]^2 + x*Log[x])^2), x] - 12*Defer[Int][1/(x*(-3*Log
[2]^2 + x*Log[x])^2), x] - 24*Defer[Int][1/(x^2*(-3*Log[2]^2 + x*Log[x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{x^3 \left (3 \log ^2(2)-x \log (x)\right )^2} \, dx \\ & = \int \left (-\frac {4 (1+x)}{x^3}-\frac {12 \left (x+3 \log ^2(2)\right )}{x^2 \left (-3 \log ^2(2)+x \log (x)\right )^2}-\frac {24}{x^2 \left (-3 \log ^2(2)+x \log (x)\right )}\right ) \, dx \\ & = -\left (4 \int \frac {1+x}{x^3} \, dx\right )-12 \int \frac {x+3 \log ^2(2)}{x^2 \left (-3 \log ^2(2)+x \log (x)\right )^2} \, dx-24 \int \frac {1}{x^2 \left (-3 \log ^2(2)+x \log (x)\right )} \, dx \\ & = \frac {2 (1+x)^2}{x^2}-12 \int \left (\frac {1}{x \left (-3 \log ^2(2)+x \log (x)\right )^2}+\frac {3 \log ^2(2)}{x^2 \left (-3 \log ^2(2)+x \log (x)\right )^2}\right ) \, dx-24 \int \frac {1}{x^2 \left (-3 \log ^2(2)+x \log (x)\right )} \, dx \\ & = \frac {2 (1+x)^2}{x^2}-12 \int \frac {1}{x \left (-3 \log ^2(2)+x \log (x)\right )^2} \, dx-24 \int \frac {1}{x^2 \left (-3 \log ^2(2)+x \log (x)\right )} \, dx-\left (36 \log ^2(2)\right ) \int \frac {1}{x^2 \left (-3 \log ^2(2)+x \log (x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=-\frac {2 \left (-1+x \left (-2-\frac {6}{-3 \log ^2(2)+x \log (x)}\right )\right )}{x^2} \]

[In]

Integrate[(-12*x^2 + 36*x*Log[2]^2 + (-36 - 36*x)*Log[2]^4 + (-24*x^2 + (24*x + 24*x^2)*Log[2]^2)*Log[x] + (-4
*x^2 - 4*x^3)*Log[x]^2)/(9*x^3*Log[2]^4 - 6*x^4*Log[2]^2*Log[x] + x^5*Log[x]^2),x]

[Out]

(-2*(-1 + x*(-2 - 6/(-3*Log[2]^2 + x*Log[x]))))/x^2

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07

method	result	size
risch	\(\frac {4 x +2}{x^{2}}-\frac {12}{x \left (3 \ln \left (2\right )^{2}-x \ln \left (x \right )\right )}\)	\(31\)
default	\(-\frac {4 \left (\left (3-3 \ln \left (2\right )^{2}\right ) x +x^{2} \ln \left (x \right )-\frac {3 \ln \left (2\right )^{2}}{2}+\frac {x \ln \left (x \right )}{2}\right )}{x^{2} \left (3 \ln \left (2\right )^{2}-x \ln \left (x \right )\right )}\)	\(48\)
norman	\(\frac {\left (12 \ln \left (2\right )^{2}-12\right ) x -4 x^{2} \ln \left (x \right )+6 \ln \left (2\right )^{2}-2 x \ln \left (x \right )}{x^{2} \left (3 \ln \left (2\right )^{2}-x \ln \left (x \right )\right )}\)	\(48\)
parallelrisch	\(\frac {12 x \ln \left (2\right )^{2}-4 x^{2} \ln \left (x \right )+6 \ln \left (2\right )^{2}-2 x \ln \left (x \right )-12 x}{x^{2} \left (3 \ln \left (2\right )^{2}-x \ln \left (x \right )\right )}\)	\(48\)

[In]

int(((-4*x^3-4*x^2)*ln(x)^2+((24*x^2+24*x)*ln(2)^2-24*x^2)*ln(x)+(-36*x-36)*ln(2)^4+36*x*ln(2)^2-12*x^2)/(x^5*
ln(x)^2-6*x^4*ln(2)^2*ln(x)+9*x^3*ln(2)^4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=\frac {2 \, {\left (3 \, {\left (2 \, x + 1\right )} \log \left (2\right )^{2} - {\left (2 \, x^{2} + x\right )} \log \left (x\right ) - 6 \, x\right )}}{3 \, x^{2} \log \left (2\right )^{2} - x^{3} \log \left (x\right )} \]

[In]

integrate(((-4*x^3-4*x^2)*log(x)^2+((24*x^2+24*x)*log(2)^2-24*x^2)*log(x)+(-36*x-36)*log(2)^4+36*x*log(2)^2-12
*x^2)/(x^5*log(x)^2-6*x^4*log(2)^2*log(x)+9*x^3*log(2)^4),x, algorithm="fricas")

[Out]

2*(3*(2*x + 1)*log(2)^2 - (2*x^2 + x)*log(x) - 6*x)/(3*x^2*log(2)^2 - x^3*log(x))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=\frac {12}{x^{2} \log {\left (x \right )} - 3 x \log {\left (2 \right )}^{2}} - \frac {- 4 x - 2}{x^{2}} \]

[In]

integrate(((-4*x**3-4*x**2)*ln(x)**2+((24*x**2+24*x)*ln(2)**2-24*x**2)*ln(x)+(-36*x-36)*ln(2)**4+36*x*ln(2)**2
-12*x**2)/(x**5*ln(x)**2-6*x**4*ln(2)**2*ln(x)+9*x**3*ln(2)**4),x)

[Out]

12/(x**2*log(x) - 3*x*log(2)**2) - (-4*x - 2)/x**2

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=\frac {2 \, {\left (6 \, {\left (\log \left (2\right )^{2} - 1\right )} x + 3 \, \log \left (2\right )^{2} - {\left (2 \, x^{2} + x\right )} \log \left (x\right )\right )}}{3 \, x^{2} \log \left (2\right )^{2} - x^{3} \log \left (x\right )} \]

[In]

integrate(((-4*x^3-4*x^2)*log(x)^2+((24*x^2+24*x)*log(2)^2-24*x^2)*log(x)+(-36*x-36)*log(2)^4+36*x*log(2)^2-12
*x^2)/(x^5*log(x)^2-6*x^4*log(2)^2*log(x)+9*x^3*log(2)^4),x, algorithm="maxima")

[Out]

2*(6*(log(2)^2 - 1)*x + 3*log(2)^2 - (2*x^2 + x)*log(x))/(3*x^2*log(2)^2 - x^3*log(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=-\frac {12}{3 \, x \log \left (2\right )^{2} - x^{2} \log \left (x\right )} + \frac {2 \, {\left (2 \, x + 1\right )}}{x^{2}} \]

[In]

integrate(((-4*x^3-4*x^2)*log(x)^2+((24*x^2+24*x)*log(2)^2-24*x^2)*log(x)+(-36*x-36)*log(2)^4+36*x*log(2)^2-12
*x^2)/(x^5*log(x)^2-6*x^4*log(2)^2*log(x)+9*x^3*log(2)^4),x, algorithm="giac")

[Out]

-12/(3*x*log(2)^2 - x^2*log(x)) + 2*(2*x + 1)/x^2

Mupad [B] (verification not implemented)

Time = 18.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=\frac {4\,x^2\,\ln \left (x\right )+x\,\left (2\,\ln \left (x\right )-12\,{\ln \left (2\right )}^2+12\right )-6\,{\ln \left (2\right )}^2}{x^2\,\left (x\,\ln \left (x\right )-3\,{\ln \left (2\right )}^2\right )} \]

[In]

int(-(log(2)^4*(36*x + 36) + log(x)^2*(4*x^2 + 4*x^3) - log(x)*(log(2)^2*(24*x + 24*x^2) - 24*x^2) - 36*x*log(
2)^2 + 12*x^2)/(9*x^3*log(2)^4 + x^5*log(x)^2 - 6*x^4*log(2)^2*log(x)),x)

[Out]

(4*x^2*log(x) + x*(2*log(x) - 12*log(2)^2 + 12) - 6*log(2)^2)/(x^2*(x*log(x) - 3*log(2)^2))