\(\int \frac {78+24 x+6 e^5 x \log (x)+(-12+24 x+e^5 (-3+6 x)+e^5 (-3+6 x) \log (x)) \log ((-1+2 x) \log (2))}{(-169+234 x+192 x^2+32 x^3+e^5 (-26 x+44 x^2+16 x^3) \log (x)+e^{10} (-x^2+2 x^3) \log ^2(x)) \log ^2((-1+2 x) \log (2))} \, dx\) [972]

Optimal result

Integrand size = 116, antiderivative size = 28 \[ \int \frac {78+24 x+6 e^5 x \log (x)+\left (-12+24 x+e^5 (-3+6 x)+e^5 (-3+6 x) \log (x)\right ) \log ((-1+2 x) \log (2))}{\left (-169+234 x+192 x^2+32 x^3+e^5 \left (-26 x+44 x^2+16 x^3\right ) \log (x)+e^{10} \left (-x^2+2 x^3\right ) \log ^2(x)\right ) \log ^2((-1+2 x) \log (2))} \, dx=\frac {3}{\left (-13-x \left (4+e^5 \log (x)\right )\right ) \log ((-1+2 x) \log (2))} \]

[Out]

3/(-13-(exp(5)*ln(x)+4)*x)/ln((-1+2*x)*ln(2))

Rubi [F]

\[ \int \frac {78+24 x+6 e^5 x \log (x)+\left (-12+24 x+e^5 (-3+6 x)+e^5 (-3+6 x) \log (x)\right ) \log ((-1+2 x) \log (2))}{\left (-169+234 x+192 x^2+32 x^3+e^5 \left (-26 x+44 x^2+16 x^3\right ) \log (x)+e^{10} \left (-x^2+2 x^3\right ) \log ^2(x)\right ) \log ^2((-1+2 x) \log (2))} \, dx=\int \frac {78+24 x+6 e^5 x \log (x)+\left (-12+24 x+e^5 (-3+6 x)+e^5 (-3+6 x) \log (x)\right ) \log ((-1+2 x) \log (2))}{\left (-169+234 x+192 x^2+32 x^3+e^5 \left (-26 x+44 x^2+16 x^3\right ) \log (x)+e^{10} \left (-x^2+2 x^3\right ) \log ^2(x)\right ) \log ^2((-1+2 x) \log (2))} \, dx \]

[In]

Int[(78 + 24*x + 6*E^5*x*Log[x] + (-12 + 24*x + E^5*(-3 + 6*x) + E^5*(-3 + 6*x)*Log[x])*Log[(-1 + 2*x)*Log[2]]
)/((-169 + 234*x + 192*x^2 + 32*x^3 + E^5*(-26*x + 44*x^2 + 16*x^3)*Log[x] + E^10*(-x^2 + 2*x^3)*Log[x]^2)*Log
[(-1 + 2*x)*Log[2]]^2),x]

[Out]

12*Defer[Int][1/((13 + 4*x + E^5*x*Log[x])^2*Log[-Log[2] + 2*x*Log[2]]^2), x] + 90*Defer[Int][1/((-1 + 2*x)*(1
3 + 4*x + E^5*x*Log[x])^2*Log[-Log[2] + 2*x*Log[2]]^2), x] + 3*E^5*Defer[Int][Log[x]/((13 + 4*x + E^5*x*Log[x]
)^2*Log[-Log[2] + 2*x*Log[2]]^2), x] + 3*E^5*Defer[Int][Log[x]/((-1 + 2*x)*(13 + 4*x + E^5*x*Log[x])^2*Log[-Lo
g[2] + 2*x*Log[2]]^2), x] + 3*(4 + E^5)*Defer[Int][1/((13 + 4*x + E^5*x*Log[x])^2*Log[-Log[2] + 2*x*Log[2]]),
x] + 3*(4 + E^5)*Defer[Int][1/((1 - 2*x)*(13 + 4*x + E^5*x*Log[x])^2*Log[-Log[2] + 2*x*Log[2]]), x] + 3*(4 + E
^5)*Defer[Int][1/((-1 + 2*x)*(13 + 4*x + E^5*x*Log[x])^2*Log[-Log[2] + 2*x*Log[2]]), x] + 3*E^5*Defer[Int][Log
[x]/((13 + 4*x + E^5*x*Log[x])^2*Log[-Log[2] + 2*x*Log[2]]), x] + 3*E^5*Defer[Int][Log[x]/((1 - 2*x)*(13 + 4*x
+ E^5*x*Log[x])^2*Log[-Log[2] + 2*x*Log[2]]), x] + 3*E^5*Defer[Int][Log[x]/((-1 + 2*x)*(13 + 4*x + E^5*x*Log[
x])^2*Log[-Log[2] + 2*x*Log[2]]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-78-24 x-6 e^5 x \log (x)-\left (-12+24 x+e^5 (-3+6 x)+e^5 (-3+6 x) \log (x)\right ) \log ((-1+2 x) \log (2))}{(1-2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))} \, dx \\ & = \int \left (\frac {78}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))}+\frac {24 x}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))}+\frac {6 e^5 x \log (x)}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))}+\frac {12 \left (1+\frac {e^5}{4}\right )}{(1-2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))}+\frac {24 \left (1+\frac {e^5}{4}\right ) x}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))}+\frac {3 e^5 \log (x)}{(1-2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))}+\frac {6 e^5 x \log (x)}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))}\right ) \, dx \\ & = 24 \int \frac {x}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))} \, dx+78 \int \frac {1}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))} \, dx+\left (3 e^5\right ) \int \frac {\log (x)}{(1-2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx+\left (6 e^5\right ) \int \frac {x \log (x)}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))} \, dx+\left (6 e^5\right ) \int \frac {x \log (x)}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx+\left (3 \left (4+e^5\right )\right ) \int \frac {1}{(1-2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx+\left (6 \left (4+e^5\right )\right ) \int \frac {x}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx \\ & = 24 \int \left (\frac {1}{2 \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))}+\frac {1}{2 (-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))}\right ) \, dx+78 \int \frac {1}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))} \, dx+\left (3 e^5\right ) \int \frac {\log (x)}{(1-2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx+\left (6 e^5\right ) \int \left (\frac {\log (x)}{2 \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))}+\frac {\log (x)}{2 (-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))}\right ) \, dx+\left (6 e^5\right ) \int \left (\frac {\log (x)}{2 \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))}+\frac {\log (x)}{2 (-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))}\right ) \, dx+\left (3 \left (4+e^5\right )\right ) \int \frac {1}{(1-2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx+\left (6 \left (4+e^5\right )\right ) \int \left (\frac {1}{2 \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))}+\frac {1}{2 (-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))}\right ) \, dx \\ & = 12 \int \frac {1}{\left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))} \, dx+12 \int \frac {1}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))} \, dx+78 \int \frac {1}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))} \, dx+\left (3 e^5\right ) \int \frac {\log (x)}{\left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))} \, dx+\left (3 e^5\right ) \int \frac {\log (x)}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log ^2(-\log (2)+2 x \log (2))} \, dx+\left (3 e^5\right ) \int \frac {\log (x)}{\left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx+\left (3 e^5\right ) \int \frac {\log (x)}{(1-2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx+\left (3 e^5\right ) \int \frac {\log (x)}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx+\left (3 \left (4+e^5\right )\right ) \int \frac {1}{\left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx+\left (3 \left (4+e^5\right )\right ) \int \frac {1}{(1-2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx+\left (3 \left (4+e^5\right )\right ) \int \frac {1}{(-1+2 x) \left (13+4 x+e^5 x \log (x)\right )^2 \log (-\log (2)+2 x \log (2))} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {78+24 x+6 e^5 x \log (x)+\left (-12+24 x+e^5 (-3+6 x)+e^5 (-3+6 x) \log (x)\right ) \log ((-1+2 x) \log (2))}{\left (-169+234 x+192 x^2+32 x^3+e^5 \left (-26 x+44 x^2+16 x^3\right ) \log (x)+e^{10} \left (-x^2+2 x^3\right ) \log ^2(x)\right ) \log ^2((-1+2 x) \log (2))} \, dx=\frac {3}{\left (-13-4 x-e^5 x \log (x)\right ) \log ((-1+2 x) \log (2))} \]

[In]

Integrate[(78 + 24*x + 6*E^5*x*Log[x] + (-12 + 24*x + E^5*(-3 + 6*x) + E^5*(-3 + 6*x)*Log[x])*Log[(-1 + 2*x)*L
og[2]])/((-169 + 234*x + 192*x^2 + 32*x^3 + E^5*(-26*x + 44*x^2 + 16*x^3)*Log[x] + E^10*(-x^2 + 2*x^3)*Log[x]^
2)*Log[(-1 + 2*x)*Log[2]]^2),x]

[Out]

3/((-13 - 4*x - E^5*x*Log[x])*Log[(-1 + 2*x)*Log[2]])

Maple [A] (verified)

Time = 2.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96

method	result	size
risch	\(-\frac {3}{\left (x \,{\mathrm e}^{5} \ln \left (x \right )+4 x +13\right ) \ln \left (\left (-1+2 x \right ) \ln \left (2\right )\right )}\)	\(27\)
parallelrisch	\(-\frac {3}{\left (x \,{\mathrm e}^{5} \ln \left (x \right )+4 x +13\right ) \ln \left (\left (-1+2 x \right ) \ln \left (2\right )\right )}\)	\(27\)
default	\(-\frac {3}{\left (\ln \left (x \right ) {\mathrm e}^{5+\ln \left (x \right )}+4 x +13\right ) \left (\ln \left (\ln \left (2\right )\right )+\ln \left (-1+2 x \right )\right )}\)	\(30\)

[In]

int((((-3+6*x)*exp(5)*ln(x)+(-3+6*x)*exp(5)+24*x-12)*ln((-1+2*x)*ln(2))+6*x*exp(5)*ln(x)+24*x+78)/((2*x^3-x^2)
*exp(5)^2*ln(x)^2+(16*x^3+44*x^2-26*x)*exp(5)*ln(x)+32*x^3+192*x^2+234*x-169)/ln((-1+2*x)*ln(2))^2,x,method=_R
ETURNVERBOSE)

[Out]

-3/(x*exp(5)*ln(x)+4*x+13)/ln((-1+2*x)*ln(2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {78+24 x+6 e^5 x \log (x)+\left (-12+24 x+e^5 (-3+6 x)+e^5 (-3+6 x) \log (x)\right ) \log ((-1+2 x) \log (2))}{\left (-169+234 x+192 x^2+32 x^3+e^5 \left (-26 x+44 x^2+16 x^3\right ) \log (x)+e^{10} \left (-x^2+2 x^3\right ) \log ^2(x)\right ) \log ^2((-1+2 x) \log (2))} \, dx=-\frac {3}{{\left (x e^{5} \log \left (x\right ) + 4 \, x + 13\right )} \log \left ({\left (2 \, x - 1\right )} \log \left (2\right )\right )} \]

[In]

integrate((((-3+6*x)*exp(5)*log(x)+(-3+6*x)*exp(5)+24*x-12)*log((-1+2*x)*log(2))+6*x*exp(5)*log(x)+24*x+78)/((
2*x^3-x^2)*exp(5)^2*log(x)^2+(16*x^3+44*x^2-26*x)*exp(5)*log(x)+32*x^3+192*x^2+234*x-169)/log((-1+2*x)*log(2))
^2,x, algorithm="fricas")

[Out]

-3/((x*e^5*log(x) + 4*x + 13)*log((2*x - 1)*log(2)))

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {78+24 x+6 e^5 x \log (x)+\left (-12+24 x+e^5 (-3+6 x)+e^5 (-3+6 x) \log (x)\right ) \log ((-1+2 x) \log (2))}{\left (-169+234 x+192 x^2+32 x^3+e^5 \left (-26 x+44 x^2+16 x^3\right ) \log (x)+e^{10} \left (-x^2+2 x^3\right ) \log ^2(x)\right ) \log ^2((-1+2 x) \log (2))} \, dx=- \frac {3}{\left (x e^{5} \log {\left (x \right )} + 4 x + 13\right ) \log {\left (\left (2 x - 1\right ) \log {\left (2 \right )} \right )}} \]

[In]

integrate((((-3+6*x)*exp(5)*ln(x)+(-3+6*x)*exp(5)+24*x-12)*ln((-1+2*x)*ln(2))+6*x*exp(5)*ln(x)+24*x+78)/((2*x*
*3-x**2)*exp(5)**2*ln(x)**2+(16*x**3+44*x**2-26*x)*exp(5)*ln(x)+32*x**3+192*x**2+234*x-169)/ln((-1+2*x)*ln(2))
**2,x)

[Out]

-3/((x*exp(5)*log(x) + 4*x + 13)*log((2*x - 1)*log(2)))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {78+24 x+6 e^5 x \log (x)+\left (-12+24 x+e^5 (-3+6 x)+e^5 (-3+6 x) \log (x)\right ) \log ((-1+2 x) \log (2))}{\left (-169+234 x+192 x^2+32 x^3+e^5 \left (-26 x+44 x^2+16 x^3\right ) \log (x)+e^{10} \left (-x^2+2 x^3\right ) \log ^2(x)\right ) \log ^2((-1+2 x) \log (2))} \, dx=-\frac {3}{x e^{5} \log \left (x\right ) \log \left (\log \left (2\right )\right ) + {\left (x e^{5} \log \left (x\right ) + 4 \, x + 13\right )} \log \left (2 \, x - 1\right ) + 4 \, x \log \left (\log \left (2\right )\right ) + 13 \, \log \left (\log \left (2\right )\right )} \]

[In]

integrate((((-3+6*x)*exp(5)*log(x)+(-3+6*x)*exp(5)+24*x-12)*log((-1+2*x)*log(2))+6*x*exp(5)*log(x)+24*x+78)/((
2*x^3-x^2)*exp(5)^2*log(x)^2+(16*x^3+44*x^2-26*x)*exp(5)*log(x)+32*x^3+192*x^2+234*x-169)/log((-1+2*x)*log(2))
^2,x, algorithm="maxima")

[Out]

-3/(x*e^5*log(x)*log(log(2)) + (x*e^5*log(x) + 4*x + 13)*log(2*x - 1) + 4*x*log(log(2)) + 13*log(log(2)))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {78+24 x+6 e^5 x \log (x)+\left (-12+24 x+e^5 (-3+6 x)+e^5 (-3+6 x) \log (x)\right ) \log ((-1+2 x) \log (2))}{\left (-169+234 x+192 x^2+32 x^3+e^5 \left (-26 x+44 x^2+16 x^3\right ) \log (x)+e^{10} \left (-x^2+2 x^3\right ) \log ^2(x)\right ) \log ^2((-1+2 x) \log (2))} \, dx=-\frac {3}{x e^{5} \log \left (2 \, x \log \left (2\right ) - \log \left (2\right )\right ) \log \left (x\right ) + 4 \, x \log \left (2 \, x \log \left (2\right ) - \log \left (2\right )\right ) + 13 \, \log \left (2 \, x \log \left (2\right ) - \log \left (2\right )\right )} \]

[In]

integrate((((-3+6*x)*exp(5)*log(x)+(-3+6*x)*exp(5)+24*x-12)*log((-1+2*x)*log(2))+6*x*exp(5)*log(x)+24*x+78)/((
2*x^3-x^2)*exp(5)^2*log(x)^2+(16*x^3+44*x^2-26*x)*exp(5)*log(x)+32*x^3+192*x^2+234*x-169)/log((-1+2*x)*log(2))
^2,x, algorithm="giac")

[Out]

-3/(x*e^5*log(2*x*log(2) - log(2))*log(x) + 4*x*log(2*x*log(2) - log(2)) + 13*log(2*x*log(2) - log(2)))

Mupad [F(-1)]

Timed out. \[ \int \frac {78+24 x+6 e^5 x \log (x)+\left (-12+24 x+e^5 (-3+6 x)+e^5 (-3+6 x) \log (x)\right ) \log ((-1+2 x) \log (2))}{\left (-169+234 x+192 x^2+32 x^3+e^5 \left (-26 x+44 x^2+16 x^3\right ) \log (x)+e^{10} \left (-x^2+2 x^3\right ) \log ^2(x)\right ) \log ^2((-1+2 x) \log (2))} \, dx=\int \frac {24\,x+\ln \left (\ln \left (2\right )\,\left (2\,x-1\right )\right )\,\left (24\,x+{\mathrm {e}}^5\,\left (6\,x-3\right )+{\mathrm {e}}^5\,\ln \left (x\right )\,\left (6\,x-3\right )-12\right )+6\,x\,{\mathrm {e}}^5\,\ln \left (x\right )+78}{{\ln \left (\ln \left (2\right )\,\left (2\,x-1\right )\right )}^2\,\left (234\,x+192\,x^2+32\,x^3+{\mathrm {e}}^5\,\ln \left (x\right )\,\left (16\,x^3+44\,x^2-26\,x\right )-{\mathrm {e}}^{10}\,{\ln \left (x\right )}^2\,\left (x^2-2\,x^3\right )-169\right )} \,d x \]

[In]

int((24*x + log(log(2)*(2*x - 1))*(24*x + exp(5)*(6*x - 3) + exp(5)*log(x)*(6*x - 3) - 12) + 6*x*exp(5)*log(x)
+ 78)/(log(log(2)*(2*x - 1))^2*(234*x + 192*x^2 + 32*x^3 + exp(5)*log(x)*(44*x^2 - 26*x + 16*x^3) - exp(10)*l
og(x)^2*(x^2 - 2*x^3) - 169)),x)

[Out]

int((24*x + log(log(2)*(2*x - 1))*(24*x + exp(5)*(6*x - 3) + exp(5)*log(x)*(6*x - 3) - 12) + 6*x*exp(5)*log(x)
+ 78)/(log(log(2)*(2*x - 1))^2*(234*x + 192*x^2 + 32*x^3 + exp(5)*log(x)*(44*x^2 - 26*x + 16*x^3) - exp(10)*l
og(x)^2*(x^2 - 2*x^3) - 169)), x)