\(\int \frac {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+(-50+50 x-300 x \log ^2(x)) \log (x^2)+50 x \log ^2(x) \log ^2(x^2)}{(-9 x+9 x^2) \log ^2(x)+(6 x-6 x^2) \log ^2(x) \log (x^2)+(-x+x^2) \log ^2(x) \log ^2(x^2)} \, dx\) [3676]

Optimal result

Integrand size = 104, antiderivative size = 32 \[ \int \frac {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+\left (-50+50 x-300 x \log ^2(x)\right ) \log \left (x^2\right )+50 x \log ^2(x) \log ^2\left (x^2\right )}{\left (-9 x+9 x^2\right ) \log ^2(x)+\left (6 x-6 x^2\right ) \log ^2(x) \log \left (x^2\right )+\left (-x+x^2\right ) \log ^2(x) \log ^2\left (x^2\right )} \, dx=10 \left (-3+5 \left (\log \left (\frac {1-x}{3}\right )+\frac {1}{\log (x) \left (3-\log \left (x^2\right )\right )}\right )\right ) \]

[Out]

50*ln(1/3-1/3*x)+50/(3-ln(x^2))/ln(x)-30

Rubi [F]

\[ \int \frac {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+\left (-50+50 x-300 x \log ^2(x)\right ) \log \left (x^2\right )+50 x \log ^2(x) \log ^2\left (x^2\right )}{\left (-9 x+9 x^2\right ) \log ^2(x)+\left (6 x-6 x^2\right ) \log ^2(x) \log \left (x^2\right )+\left (-x+x^2\right ) \log ^2(x) \log ^2\left (x^2\right )} \, dx=\int \frac {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+\left (-50+50 x-300 x \log ^2(x)\right ) \log \left (x^2\right )+50 x \log ^2(x) \log ^2\left (x^2\right )}{\left (-9 x+9 x^2\right ) \log ^2(x)+\left (6 x-6 x^2\right ) \log ^2(x) \log \left (x^2\right )+\left (-x+x^2\right ) \log ^2(x) \log ^2\left (x^2\right )} \, dx \]

[In]

Int[(150 - 150*x + (-100 + 100*x)*Log[x] + 450*x*Log[x]^2 + (-50 + 50*x - 300*x*Log[x]^2)*Log[x^2] + 50*x*Log[
x]^2*Log[x^2]^2)/((-9*x + 9*x^2)*Log[x]^2 + (6*x - 6*x^2)*Log[x]^2*Log[x^2] + (-x + x^2)*Log[x]^2*Log[x^2]^2),
x]

[Out]

50*Log[1 - x] + 100*Defer[Int][1/(x*Log[x]*(-3 + Log[x^2])^2), x] + 50*Defer[Int][1/(x*Log[x]^2*(-3 + Log[x^2]
)), x]

Rubi steps \begin{align*} \text {integral}& = \int 50 \left (\frac {1}{-1+x}+\frac {2}{x \log (x) \left (-3+\log \left (x^2\right )\right )^2}+\frac {1}{x \log ^2(x) \left (-3+\log \left (x^2\right )\right )}\right ) \, dx \\ & = 50 \int \left (\frac {1}{-1+x}+\frac {2}{x \log (x) \left (-3+\log \left (x^2\right )\right )^2}+\frac {1}{x \log ^2(x) \left (-3+\log \left (x^2\right )\right )}\right ) \, dx \\ & = 50 \log (1-x)+50 \int \frac {1}{x \log ^2(x) \left (-3+\log \left (x^2\right )\right )} \, dx+100 \int \frac {1}{x \log (x) \left (-3+\log \left (x^2\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int \frac {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+\left (-50+50 x-300 x \log ^2(x)\right ) \log \left (x^2\right )+50 x \log ^2(x) \log ^2\left (x^2\right )}{\left (-9 x+9 x^2\right ) \log ^2(x)+\left (6 x-6 x^2\right ) \log ^2(x) \log \left (x^2\right )+\left (-x+x^2\right ) \log ^2(x) \log ^2\left (x^2\right )} \, dx=50 \left (\log (-1+x)+\frac {1}{3 \log (x)-\log (x) \log \left (x^2\right )}\right ) \]

[In]

Integrate[(150 - 150*x + (-100 + 100*x)*Log[x] + 450*x*Log[x]^2 + (-50 + 50*x - 300*x*Log[x]^2)*Log[x^2] + 50*
x*Log[x]^2*Log[x^2]^2)/((-9*x + 9*x^2)*Log[x]^2 + (6*x - 6*x^2)*Log[x]^2*Log[x^2] + (-x + x^2)*Log[x]^2*Log[x^
2]^2),x]

[Out]

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

Maple [A] (verified)

Time = 2.86 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16

method	result	size
parallelrisch	\(\frac {-1200+1200 \ln \left (-1+x \right ) \ln \left (x \right ) \ln \left (x^{2}\right )-3600 \ln \left (x \right ) \ln \left (-1+x \right )}{24 \left (\ln \left (x^{2}\right )-3\right ) \ln \left (x \right )}\)	\(37\)
default	\(50 \ln \left (-1+x \right )-\frac {50}{\left (-2 \ln \left (x \right )+\ln \left (x^{2}\right )-3\right ) \ln \left (x \right )}+\frac {100}{\left (-2 \ln \left (x \right )+\ln \left (x^{2}\right )-3\right ) \left (\ln \left (x^{2}\right )-3\right )}\)	\(48\)
parts	\(50 \ln \left (-1+x \right )-\frac {50}{\left (-2 \ln \left (x \right )+\ln \left (x^{2}\right )-3\right ) \ln \left (x \right )}+\frac {100}{\left (-2 \ln \left (x \right )+\ln \left (x^{2}\right )-3\right ) \left (\ln \left (x^{2}\right )-3\right )}\)	\(48\)
risch	\(50 \ln \left (-1+x \right )-\frac {100 i}{\left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (x \right )-6 i\right ) \ln \left (x \right )}\)	\(69\)

[In]

int((50*x*ln(x)^2*ln(x^2)^2+(-300*x*ln(x)^2+50*x-50)*ln(x^2)+450*x*ln(x)^2+(100*x-100)*ln(x)-150*x+150)/((x^2-
x)*ln(x)^2*ln(x^2)^2+(-6*x^2+6*x)*ln(x)^2*ln(x^2)+(9*x^2-9*x)*ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

1/24*(-1200+1200*ln(-1+x)*ln(x)*ln(x^2)-3600*ln(x)*ln(-1+x))/(ln(x^2)-3)/ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+\left (-50+50 x-300 x \log ^2(x)\right ) \log \left (x^2\right )+50 x \log ^2(x) \log ^2\left (x^2\right )}{\left (-9 x+9 x^2\right ) \log ^2(x)+\left (6 x-6 x^2\right ) \log ^2(x) \log \left (x^2\right )+\left (-x+x^2\right ) \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {50 \, {\left (2 \, \log \left (x - 1\right ) \log \left (x\right )^{2} - 3 \, \log \left (x - 1\right ) \log \left (x\right ) - 1\right )}}{2 \, \log \left (x\right )^{2} - 3 \, \log \left (x\right )} \]

[In]

integrate((50*x*log(x)^2*log(x^2)^2+(-300*x*log(x)^2+50*x-50)*log(x^2)+450*x*log(x)^2+(100*x-100)*log(x)-150*x
+150)/((x^2-x)*log(x)^2*log(x^2)^2+(-6*x^2+6*x)*log(x)^2*log(x^2)+(9*x^2-9*x)*log(x)^2),x, algorithm="fricas")

[Out]

50*(2*log(x - 1)*log(x)^2 - 3*log(x - 1)*log(x) - 1)/(2*log(x)^2 - 3*log(x))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59 \[ \int \frac {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+\left (-50+50 x-300 x \log ^2(x)\right ) \log \left (x^2\right )+50 x \log ^2(x) \log ^2\left (x^2\right )}{\left (-9 x+9 x^2\right ) \log ^2(x)+\left (6 x-6 x^2\right ) \log ^2(x) \log \left (x^2\right )+\left (-x+x^2\right ) \log ^2(x) \log ^2\left (x^2\right )} \, dx=50 \log {\left (x - 1 \right )} - \frac {50}{2 \log {\left (x \right )}^{2} - 3 \log {\left (x \right )}} \]

[In]

integrate((50*x*ln(x)**2*ln(x**2)**2+(-300*x*ln(x)**2+50*x-50)*ln(x**2)+450*x*ln(x)**2+(100*x-100)*ln(x)-150*x
+150)/((x**2-x)*ln(x)**2*ln(x**2)**2+(-6*x**2+6*x)*ln(x)**2*ln(x**2)+(9*x**2-9*x)*ln(x)**2),x)

[Out]

50*log(x - 1) - 50/(2*log(x)**2 - 3*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int \frac {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+\left (-50+50 x-300 x \log ^2(x)\right ) \log \left (x^2\right )+50 x \log ^2(x) \log ^2\left (x^2\right )}{\left (-9 x+9 x^2\right ) \log ^2(x)+\left (6 x-6 x^2\right ) \log ^2(x) \log \left (x^2\right )+\left (-x+x^2\right ) \log ^2(x) \log ^2\left (x^2\right )} \, dx=-\frac {50}{2 \, \log \left (x\right )^{2} - 3 \, \log \left (x\right )} + 50 \, \log \left (x - 1\right ) \]

[In]

integrate((50*x*log(x)^2*log(x^2)^2+(-300*x*log(x)^2+50*x-50)*log(x^2)+450*x*log(x)^2+(100*x-100)*log(x)-150*x
+150)/((x^2-x)*log(x)^2*log(x^2)^2+(-6*x^2+6*x)*log(x)^2*log(x^2)+(9*x^2-9*x)*log(x)^2),x, algorithm="maxima")

[Out]

-50/(2*log(x)^2 - 3*log(x)) + 50*log(x - 1)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+\left (-50+50 x-300 x \log ^2(x)\right ) \log \left (x^2\right )+50 x \log ^2(x) \log ^2\left (x^2\right )}{\left (-9 x+9 x^2\right ) \log ^2(x)+\left (6 x-6 x^2\right ) \log ^2(x) \log \left (x^2\right )+\left (-x+x^2\right ) \log ^2(x) \log ^2\left (x^2\right )} \, dx=-\frac {100}{3 \, {\left (2 \, \log \left (x\right ) - 3\right )}} + \frac {50}{3 \, \log \left (x\right )} + 50 \, \log \left (x - 1\right ) \]

[In]

integrate((50*x*log(x)^2*log(x^2)^2+(-300*x*log(x)^2+50*x-50)*log(x^2)+450*x*log(x)^2+(100*x-100)*log(x)-150*x
+150)/((x^2-x)*log(x)^2*log(x^2)^2+(-6*x^2+6*x)*log(x)^2*log(x^2)+(9*x^2-9*x)*log(x)^2),x, algorithm="giac")

[Out]

-100/3/(2*log(x) - 3) + 50/3/log(x) + 50*log(x - 1)

Mupad [B] (verification not implemented)

Time = 9.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int \frac {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+\left (-50+50 x-300 x \log ^2(x)\right ) \log \left (x^2\right )+50 x \log ^2(x) \log ^2\left (x^2\right )}{\left (-9 x+9 x^2\right ) \log ^2(x)+\left (6 x-6 x^2\right ) \log ^2(x) \log \left (x^2\right )+\left (-x+x^2\right ) \log ^2(x) \log ^2\left (x^2\right )} \, dx=50\,\ln \left (x-1\right )-\frac {50}{\ln \left (x\right )\,\left (\ln \left (x^2\right )-3\right )} \]

[In]

int(-(450*x*log(x)^2 - 150*x + log(x)*(100*x - 100) - log(x^2)*(300*x*log(x)^2 - 50*x + 50) + 50*x*log(x^2)^2*
log(x)^2 + 150)/(log(x)^2*(9*x - 9*x^2) - log(x^2)*log(x)^2*(6*x - 6*x^2) + log(x^2)^2*log(x)^2*(x - x^2)),x)

[Out]

50*log(x - 1) - 50/(log(x)*(log(x^2) - 3))