\(\int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx\) [3823]

Optimal result

Integrand size = 77, antiderivative size = 23 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=-5+\frac {7 x}{3}-\log (x)+\log (x+3 \log (4) \log (-3+\log (x))) \]

[Out]

7/3*x-ln(x)+ln(6*ln(2)*ln(ln(x)-3)+x)-5

Rubi [F]

\[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx \]

[In]

Int[(-21*x^2 + 9*Log[4] + 7*x^2*Log[x] + ((27 - 63*x)*Log[4] + (-9 + 21*x)*Log[4]*Log[x])*Log[-3 + Log[x]])/(-
9*x^2 + 3*x^2*Log[x] + (-27*x*Log[4] + 9*x*Log[4]*Log[x])*Log[-3 + Log[x]]),x]

[Out]

(7*x*Log[4])/Log[64] - (3*Log[4]*Log[x])/Log[64] - (Log[18014398509481984]*Defer[Int][1/((-3 + Log[x])*(x + Lo
g[64]*Log[-3 + Log[x]])), x])/(3*Log[64]) + (Log[262144]*Defer[Int][1/(x*(-3 + Log[x])*(x + Log[64]*Log[-3 + L
og[x]])), x])/3 + (Log[262144]*Defer[Int][Log[x]/((-3 + Log[x])*(x + Log[64]*Log[-3 + Log[x]])), x])/(3*Log[64
])

Rubi steps \begin{align*} \text {integral}& = \int \frac {21 x^2-9 \log (4)-7 x^2 \log (x)-((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{3 x (3-\log (x)) (x+\log (64) \log (-3+\log (x)))} \, dx \\ & = \frac {1}{3} \int \frac {21 x^2-9 \log (4)-7 x^2 \log (x)-((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{x (3-\log (x)) (x+\log (64) \log (-3+\log (x)))} \, dx \\ & = \frac {1}{3} \int \left (\frac {3 (-3+7 x) \log (4)}{x \log (64)}+\frac {\log (64) \log (262144)-x \log (18014398509481984)+x \log (262144) \log (x)}{x \log (64) (-3+\log (x)) (x+\log (64) \log (-3+\log (x)))}\right ) \, dx \\ & = \frac {\int \frac {\log (64) \log (262144)-x \log (18014398509481984)+x \log (262144) \log (x)}{x (-3+\log (x)) (x+\log (64) \log (-3+\log (x)))} \, dx}{3 \log (64)}+\frac {\log (4) \int \frac {-3+7 x}{x} \, dx}{\log (64)} \\ & = \frac {\int \left (\frac {\log (64) \log (262144)}{x (-3+\log (x)) (x+\log (64) \log (-3+\log (x)))}-\frac {\log (18014398509481984)}{(-3+\log (x)) (x+\log (64) \log (-3+\log (x)))}+\frac {\log (262144) \log (x)}{(-3+\log (x)) (x+\log (64) \log (-3+\log (x)))}\right ) \, dx}{3 \log (64)}+\frac {\log (4) \int \left (7-\frac {3}{x}\right ) \, dx}{\log (64)} \\ & = \frac {7 x \log (4)}{\log (64)}-\frac {3 \log (4) \log (x)}{\log (64)}+\frac {1}{3} \log (262144) \int \frac {1}{x (-3+\log (x)) (x+\log (64) \log (-3+\log (x)))} \, dx+\frac {\log (262144) \int \frac {\log (x)}{(-3+\log (x)) (x+\log (64) \log (-3+\log (x)))} \, dx}{3 \log (64)}-\frac {\log (18014398509481984) \int \frac {1}{(-3+\log (x)) (x+\log (64) \log (-3+\log (x)))} \, dx}{3 \log (64)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\frac {7 x \log (4)}{\log (64)}+\frac {1}{3} \left (-\frac {9 \log (4) \log (x)}{\log (64)}+\frac {9 \log (4) \log (x+\log (64) \log (-3+\log (x)))}{\log (64)}\right ) \]

[In]

Integrate[(-21*x^2 + 9*Log[4] + 7*x^2*Log[x] + ((27 - 63*x)*Log[4] + (-9 + 21*x)*Log[4]*Log[x])*Log[-3 + Log[x
]])/(-9*x^2 + 3*x^2*Log[x] + (-27*x*Log[4] + 9*x*Log[4]*Log[x])*Log[-3 + Log[x]]),x]

[Out]

(7*x*Log[4])/Log[64] + ((-9*Log[4]*Log[x])/Log[64] + (9*Log[4]*Log[x + Log[64]*Log[-3 + Log[x]]])/Log[64])/3

Maple [A] (verified)

Time = 4.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

method	result	size
default	\(-\ln \left (x \right )+\frac {7 x}{3}+\ln \left (6 \ln \left (2\right ) \ln \left (\ln \left (x \right )-3\right )+x \right )\)	\(21\)
norman	\(-\ln \left (x \right )+\frac {7 x}{3}+\ln \left (6 \ln \left (2\right ) \ln \left (\ln \left (x \right )-3\right )+x \right )\)	\(21\)
parallelrisch	\(\ln \left (6 \ln \left (2\right ) \ln \left (\ln \left (x \right )-3\right )+x \right )+\frac {7 x}{3}-\ln \left (x \right )-6\)	\(22\)
risch	\(\frac {7 x}{3}-\ln \left (x \right )+\ln \left (\ln \left (\ln \left (x \right )-3\right )+\frac {x}{6 \ln \left (2\right )}\right )\)	\(23\)

[In]

int(((2*(21*x-9)*ln(2)*ln(x)+2*(-63*x+27)*ln(2))*ln(ln(x)-3)+7*x^2*ln(x)+18*ln(2)-21*x^2)/((18*x*ln(2)*ln(x)-5
4*x*ln(2))*ln(ln(x)-3)+3*x^2*ln(x)-9*x^2),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+7/3*x+ln(6*ln(2)*ln(ln(x)-3)+x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\frac {7}{3} \, x + \log \left (6 \, \log \left (2\right ) \log \left (\log \left (x\right ) - 3\right ) + x\right ) - \log \left (x\right ) \]

[In]

integrate(((2*(21*x-9)*log(2)*log(x)+2*(-63*x+27)*log(2))*log(log(x)-3)+7*x^2*log(x)+18*log(2)-21*x^2)/((18*x*
log(2)*log(x)-54*x*log(2))*log(log(x)-3)+3*x^2*log(x)-9*x^2),x, algorithm="fricas")

[Out]

7/3*x + log(6*log(2)*log(log(x) - 3) + x) - log(x)

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\frac {7 x}{3} - \log {\left (x \right )} + \log {\left (\frac {x}{6 \log {\left (2 \right )}} + \log {\left (\log {\left (x \right )} - 3 \right )} \right )} \]

[In]

integrate(((2*(21*x-9)*ln(2)*ln(x)+2*(-63*x+27)*ln(2))*ln(ln(x)-3)+7*x**2*ln(x)+18*ln(2)-21*x**2)/((18*x*ln(2)
*ln(x)-54*x*ln(2))*ln(ln(x)-3)+3*x**2*ln(x)-9*x**2),x)

[Out]

7*x/3 - log(x) + log(x/(6*log(2)) + log(log(x) - 3))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\frac {7}{3} \, x - \log \left (x\right ) + \log \left (\frac {6 \, \log \left (2\right ) \log \left (\log \left (x\right ) - 3\right ) + x}{6 \, \log \left (2\right )}\right ) \]

[In]

integrate(((2*(21*x-9)*log(2)*log(x)+2*(-63*x+27)*log(2))*log(log(x)-3)+7*x^2*log(x)+18*log(2)-21*x^2)/((18*x*
log(2)*log(x)-54*x*log(2))*log(log(x)-3)+3*x^2*log(x)-9*x^2),x, algorithm="maxima")

[Out]

7/3*x - log(x) + log(1/6*(6*log(2)*log(log(x) - 3) + x)/log(2))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\frac {7}{3} \, x + \log \left (6 \, \log \left (2\right ) \log \left (\log \left (x\right ) - 3\right ) + x\right ) - \log \left (x\right ) \]

[In]

integrate(((2*(21*x-9)*log(2)*log(x)+2*(-63*x+27)*log(2))*log(log(x)-3)+7*x^2*log(x)+18*log(2)-21*x^2)/((18*x*
log(2)*log(x)-54*x*log(2))*log(log(x)-3)+3*x^2*log(x)-9*x^2),x, algorithm="giac")

[Out]

7/3*x + log(6*log(2)*log(log(x) - 3) + x) - log(x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\int -\frac {18\,\ln \left (2\right )+7\,x^2\,\ln \left (x\right )-\ln \left (\ln \left (x\right )-3\right )\,\left (2\,\ln \left (2\right )\,\left (63\,x-27\right )-2\,\ln \left (2\right )\,\ln \left (x\right )\,\left (21\,x-9\right )\right )-21\,x^2}{9\,x^2-3\,x^2\,\ln \left (x\right )+\ln \left (\ln \left (x\right )-3\right )\,\left (54\,x\,\ln \left (2\right )-18\,x\,\ln \left (2\right )\,\ln \left (x\right )\right )} \,d x \]

[In]

int(-(18*log(2) + 7*x^2*log(x) - log(log(x) - 3)*(2*log(2)*(63*x - 27) - 2*log(2)*log(x)*(21*x - 9)) - 21*x^2)
/(9*x^2 - 3*x^2*log(x) + log(log(x) - 3)*(54*x*log(2) - 18*x*log(2)*log(x))),x)

[Out]

int(-(18*log(2) + 7*x^2*log(x) - log(log(x) - 3)*(2*log(2)*(63*x - 27) - 2*log(2)*log(x)*(21*x - 9)) - 21*x^2)
/(9*x^2 - 3*x^2*log(x) + log(log(x) - 3)*(54*x*log(2) - 18*x*log(2)*log(x))), x)