\(\int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))})}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx\) [2849]

Optimal result

Integrand size = 109, antiderivative size = 24 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=x \log \left (\frac {4}{3 x+\frac {\log (x)}{2}+\log (4+\log (2 x))}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=\int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx \]

[In]

Int[(-6 - 24*x + (-1 - 6*x)*Log[2*x] + (24*x + 4*Log[x] + (6*x + Log[x])*Log[2*x] + (8 + 2*Log[2*x])*Log[4 + L
og[2*x]])*Log[8/(6*x + Log[x] + 2*Log[4 + Log[2*x]])])/(24*x + 4*Log[x] + (6*x + Log[x])*Log[2*x] + (8 + 2*Log
[2*x])*Log[4 + Log[2*x]]),x]

[Out]

x*Log[8/(6*x + Log[x] + 2*Log[4 + Log[2*x]])] + Defer[Int][(6*x + Log[x] + 2*Log[4 + Log[2*x]])^(-1), x] + 6*D
efer[Int][x/(6*x + Log[x] + 2*Log[4 + Log[2*x]]), x] - 4*Defer[Int][1/((4 + Log[2*x])*(6*x + Log[x] + 2*Log[4
+ Log[2*x]])), x] - 24*Defer[Int][x/((4 + Log[2*x])*(6*x + Log[x] + 2*Log[4 + Log[2*x]])), x] - Defer[Int][Log
[2*x]/((4 + Log[2*x])*(6*x + Log[x] + 2*Log[4 + Log[2*x]])), x] - 6*Defer[Int][(x*Log[2*x])/((4 + Log[2*x])*(6
*x + Log[x] + 2*Log[4 + Log[2*x]])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))} \, dx \\ & = \int \left (\frac {-6-24 x-\log (2 x)-6 x \log (2 x)}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))}+\log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )\right ) \, dx \\ & = \int \frac {-6-24 x-\log (2 x)-6 x \log (2 x)}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))} \, dx+\int \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right ) \, dx \\ & = x \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )+\int \frac {x \left (6+\frac {1+\frac {2}{4+\log (2 x)}}{x}\right )}{6 x+\log (x)+2 \log (4+\log (2 x))} \, dx+\int \left (-\frac {6}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))}-\frac {24 x}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))}-\frac {\log (2 x)}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))}-\frac {6 x \log (2 x)}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))}\right ) \, dx \\ & = x \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )-6 \int \frac {1}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))} \, dx-6 \int \frac {x \log (2 x)}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))} \, dx-24 \int \frac {x}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))} \, dx-\int \frac {\log (2 x)}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))} \, dx+\int \left (\frac {1}{6 x+\log (x)+2 \log (4+\log (2 x))}+\frac {6 x}{6 x+\log (x)+2 \log (4+\log (2 x))}+\frac {2}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))}\right ) \, dx \\ & = x \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )+2 \int \frac {1}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))} \, dx+6 \int \frac {x}{6 x+\log (x)+2 \log (4+\log (2 x))} \, dx-6 \int \frac {1}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))} \, dx-6 \int \frac {x \log (2 x)}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))} \, dx-24 \int \frac {x}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))} \, dx+\int \frac {1}{6 x+\log (x)+2 \log (4+\log (2 x))} \, dx-\int \frac {\log (2 x)}{(4+\log (2 x)) (6 x+\log (x)+2 \log (4+\log (2 x)))} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=x \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right ) \]

[In]

Integrate[(-6 - 24*x + (-1 - 6*x)*Log[2*x] + (24*x + 4*Log[x] + (6*x + Log[x])*Log[2*x] + (8 + 2*Log[2*x])*Log
[4 + Log[2*x]])*Log[8/(6*x + Log[x] + 2*Log[4 + Log[2*x]])])/(24*x + 4*Log[x] + (6*x + Log[x])*Log[2*x] + (8 +
2*Log[2*x])*Log[4 + Log[2*x]]),x]

[Out]

x*Log[8/(6*x + Log[x] + 2*Log[4 + Log[2*x]])]

Maple [A] (verified)

Time = 14.41 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96

method	result	size
parallelrisch	\(\ln \left (\frac {8}{2 \ln \left (\ln \left (2 x \right )+4\right )+\ln \left (x \right )+6 x}\right ) x\)	\(23\)
risch	\(-x \ln \left (\frac {\ln \left (\ln \left (2\right )+\ln \left (x \right )+4\right )}{3}+\frac {\ln \left (x \right )}{6}+x \right )-x \ln \left (3\right )+2 x \ln \left (2\right )\)	\(31\)

[In]

int((((2*ln(2*x)+8)*ln(ln(2*x)+4)+(ln(x)+6*x)*ln(2*x)+4*ln(x)+24*x)*ln(8/(2*ln(ln(2*x)+4)+ln(x)+6*x))+(-6*x-1)
*ln(2*x)-24*x-6)/((2*ln(2*x)+8)*ln(ln(2*x)+4)+(ln(x)+6*x)*ln(2*x)+4*ln(x)+24*x),x,method=_RETURNVERBOSE)

[Out]

ln(8/(2*ln(ln(2*x)+4)+ln(x)+6*x))*x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=x \log \left (\frac {8}{6 \, x + \log \left (x\right ) + 2 \, \log \left (\log \left (2\right ) + \log \left (x\right ) + 4\right )}\right ) \]

[In]

integrate((((2*log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+24*x)*log(8/(2*log(log(2*x)+4)+log(x
)+6*x))+(-6*x-1)*log(2*x)-24*x-6)/((2*log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+24*x),x, algo
rithm="fricas")

[Out]

x*log(8/(6*x + log(x) + 2*log(log(2) + log(x) + 4)))

Sympy [F(-2)]

Exception generated. \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((((2*ln(2*x)+8)*ln(ln(2*x)+4)+(ln(x)+6*x)*ln(2*x)+4*ln(x)+24*x)*ln(8/(2*ln(ln(2*x)+4)+ln(x)+6*x))+(-
6*x-1)*ln(2*x)-24*x-6)/((2*ln(2*x)+8)*ln(ln(2*x)+4)+(ln(x)+6*x)*ln(2*x)+4*ln(x)+24*x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=3 \, x \log \left (2\right ) - x \log \left (6 \, x + \log \left (x\right ) + 2 \, \log \left (\log \left (2\right ) + \log \left (x\right ) + 4\right )\right ) \]

[In]

integrate((((2*log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+24*x)*log(8/(2*log(log(2*x)+4)+log(x
)+6*x))+(-6*x-1)*log(2*x)-24*x-6)/((2*log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+24*x),x, algo
rithm="maxima")

[Out]

3*x*log(2) - x*log(6*x + log(x) + 2*log(log(2) + log(x) + 4))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=3 \, x \log \left (2\right ) - x \log \left (6 \, x + \log \left (x\right ) + 2 \, \log \left (\log \left (2\right ) + \log \left (x\right ) + 4\right )\right ) \]

[In]

integrate((((2*log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+24*x)*log(8/(2*log(log(2*x)+4)+log(x
)+6*x))+(-6*x-1)*log(2*x)-24*x-6)/((2*log(2*x)+8)*log(log(2*x)+4)+(log(x)+6*x)*log(2*x)+4*log(x)+24*x),x, algo
rithm="giac")

[Out]

3*x*log(2) - x*log(6*x + log(x) + 2*log(log(2) + log(x) + 4))

Mupad [B] (verification not implemented)

Time = 9.61 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-6-24 x+(-1-6 x) \log (2 x)+(24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))) \log \left (\frac {8}{6 x+\log (x)+2 \log (4+\log (2 x))}\right )}{24 x+4 \log (x)+(6 x+\log (x)) \log (2 x)+(8+2 \log (2 x)) \log (4+\log (2 x))} \, dx=x\,\left (\ln \left (\frac {1}{6\,x+2\,\ln \left (\ln \left (2\,x\right )+4\right )+\ln \left (x\right )}\right )+3\,\ln \left (2\right )\right ) \]

[In]

int(-(24*x - log(8/(6*x + 2*log(log(2*x) + 4) + log(x)))*(24*x + 4*log(x) + log(log(2*x) + 4)*(2*log(2*x) + 8)
+ log(2*x)*(6*x + log(x))) + log(2*x)*(6*x + 1) + 6)/(24*x + 4*log(x) + log(log(2*x) + 4)*(2*log(2*x) + 8) +
log(2*x)*(6*x + log(x))),x)

[Out]

x*(log(1/(6*x + 2*log(log(2*x) + 4) + log(x))) + 3*log(2))