\(\int \frac {-72+216 x^2+72 x^5-72 x^7+4 x^9-144 x^{12}+(-72-72 x^3-12 x^7+432 x^{10}) \log (x)+(72 x^3+12 x^5-432 x^8) \log ^2(x)+(-4 x^3+144 x^6) \log ^3(x)}{-x^9+3 x^7 \log (x)-3 x^5 \log ^2(x)+x^3 \log ^3(x)} \, dx\) [4827]

Optimal result

Integrand size = 117, antiderivative size = 27 \[ \int \frac {-72+216 x^2+72 x^5-72 x^7+4 x^9-144 x^{12}+\left (-72-72 x^3-12 x^7+432 x^{10}\right ) \log (x)+\left (72 x^3+12 x^5-432 x^8\right ) \log ^2(x)+\left (-4 x^3+144 x^6\right ) \log ^3(x)}{-x^9+3 x^7 \log (x)-3 x^5 \log ^2(x)+x^3 \log ^3(x)} \, dx=-1-4 x+36 \left (x^2+\frac {1}{x \left (-x^2+\log (x)\right )}\right )^2 \]

[Out]

6*(1/(ln(x)-x^2)/x+x^2)*(6/(ln(x)-x^2)/x+6*x^2)-4*x-1

Rubi [F]

\[ \int \frac {-72+216 x^2+72 x^5-72 x^7+4 x^9-144 x^{12}+\left (-72-72 x^3-12 x^7+432 x^{10}\right ) \log (x)+\left (72 x^3+12 x^5-432 x^8\right ) \log ^2(x)+\left (-4 x^3+144 x^6\right ) \log ^3(x)}{-x^9+3 x^7 \log (x)-3 x^5 \log ^2(x)+x^3 \log ^3(x)} \, dx=\int \frac {-72+216 x^2+72 x^5-72 x^7+4 x^9-144 x^{12}+\left (-72-72 x^3-12 x^7+432 x^{10}\right ) \log (x)+\left (72 x^3+12 x^5-432 x^8\right ) \log ^2(x)+\left (-4 x^3+144 x^6\right ) \log ^3(x)}{-x^9+3 x^7 \log (x)-3 x^5 \log ^2(x)+x^3 \log ^3(x)} \, dx \]

[In]

Int[(-72 + 216*x^2 + 72*x^5 - 72*x^7 + 4*x^9 - 144*x^12 + (-72 - 72*x^3 - 12*x^7 + 432*x^10)*Log[x] + (72*x^3
+ 12*x^5 - 432*x^8)*Log[x]^2 + (-4*x^3 + 144*x^6)*Log[x]^3)/(-x^9 + 3*x^7*Log[x] - 3*x^5*Log[x]^2 + x^3*Log[x]
^3),x]

[Out]

-4*x + 36*x^4 + 72*Defer[Int][1/(x^3*(x^2 - Log[x])^3), x] - 144*Defer[Int][1/(x*(x^2 - Log[x])^3), x] - 72*De
fer[Int][(x^2 - Log[x])^(-2), x] - 72*Defer[Int][1/(x^3*(x^2 - Log[x])^2), x] + 144*Defer[Int][x^2/(x^2 - Log[
x])^2, x] - 72*Defer[Int][(x^2 - Log[x])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {72-216 x^2-72 x^5+72 x^7-4 x^9+144 x^{12}-\left (-72-72 x^3-12 x^7+432 x^{10}\right ) \log (x)-\left (72 x^3+12 x^5-432 x^8\right ) \log ^2(x)-\left (-4 x^3+144 x^6\right ) \log ^3(x)}{x^3 \left (x^2-\log (x)\right )^3} \, dx \\ & = \int \left (4 \left (-1+36 x^3\right )-\frac {72 \left (-1+2 x^2\right )}{x^3 \left (x^2-\log (x)\right )^3}+\frac {72 \left (-1-x^3+2 x^5\right )}{x^3 \left (x^2-\log (x)\right )^2}-\frac {72}{x^2-\log (x)}\right ) \, dx \\ & = 4 \int \left (-1+36 x^3\right ) \, dx-72 \int \frac {-1+2 x^2}{x^3 \left (x^2-\log (x)\right )^3} \, dx+72 \int \frac {-1-x^3+2 x^5}{x^3 \left (x^2-\log (x)\right )^2} \, dx-72 \int \frac {1}{x^2-\log (x)} \, dx \\ & = -4 x+36 x^4-72 \int \left (-\frac {1}{x^3 \left (x^2-\log (x)\right )^3}+\frac {2}{x \left (x^2-\log (x)\right )^3}\right ) \, dx+72 \int \left (-\frac {1}{\left (x^2-\log (x)\right )^2}-\frac {1}{x^3 \left (x^2-\log (x)\right )^2}+\frac {2 x^2}{\left (x^2-\log (x)\right )^2}\right ) \, dx-72 \int \frac {1}{x^2-\log (x)} \, dx \\ & = -4 x+36 x^4+72 \int \frac {1}{x^3 \left (x^2-\log (x)\right )^3} \, dx-72 \int \frac {1}{\left (x^2-\log (x)\right )^2} \, dx-72 \int \frac {1}{x^3 \left (x^2-\log (x)\right )^2} \, dx-72 \int \frac {1}{x^2-\log (x)} \, dx-144 \int \frac {1}{x \left (x^2-\log (x)\right )^3} \, dx+144 \int \frac {x^2}{\left (x^2-\log (x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {-72+216 x^2+72 x^5-72 x^7+4 x^9-144 x^{12}+\left (-72-72 x^3-12 x^7+432 x^{10}\right ) \log (x)+\left (72 x^3+12 x^5-432 x^8\right ) \log ^2(x)+\left (-4 x^3+144 x^6\right ) \log ^3(x)}{-x^9+3 x^7 \log (x)-3 x^5 \log ^2(x)+x^3 \log ^3(x)} \, dx=4 \left (-x+9 x^4+\frac {9}{x^2 \left (-x^2+\log (x)\right )^2}+\frac {18 x}{-x^2+\log (x)}\right ) \]

[In]

Integrate[(-72 + 216*x^2 + 72*x^5 - 72*x^7 + 4*x^9 - 144*x^12 + (-72 - 72*x^3 - 12*x^7 + 432*x^10)*Log[x] + (7
2*x^3 + 12*x^5 - 432*x^8)*Log[x]^2 + (-4*x^3 + 144*x^6)*Log[x]^3)/(-x^9 + 3*x^7*Log[x] - 3*x^5*Log[x]^2 + x^3*
Log[x]^3),x]

[Out]

4*(-x + 9*x^4 + 9/(x^2*(-x^2 + Log[x])^2) + (18*x)/(-x^2 + Log[x]))

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44

method	result	size
risch	\(36 x^{4}-4 x -\frac {36 \left (2 x^{5}-2 x^{3} \ln \left (x \right )-1\right )}{x^{2} \left (x^{2}-\ln \left (x \right )\right )^{2}}\)	\(39\)
default	\(36 x^{4}-4 x +\frac {36}{x^{2} \ln \left (x \right )^{2}}+\frac {-72 x^{3} \ln \left (x \right )^{2}+72 x \ln \left (x \right )^{3}-36 x^{2}+72 \ln \left (x \right )}{\left (\ln \left (x \right )-x^{2}\right )^{2} \ln \left (x \right )^{2}}\)	\(61\)
parallelrisch	\(\frac {72 x^{10}-144 x^{8} \ln \left (x \right )+72 x^{6} \ln \left (x \right )^{2}-8 x^{7}+72+16 x^{5} \ln \left (x \right )-144 x^{5}-8 x^{3} \ln \left (x \right )^{2}+144 x^{3} \ln \left (x \right )}{2 x^{2} \left (x^{4}-2 x^{2} \ln \left (x \right )+\ln \left (x \right )^{2}\right )}\)	\(79\)

[In]

int(((144*x^6-4*x^3)*ln(x)^3+(-432*x^8+12*x^5+72*x^3)*ln(x)^2+(432*x^10-12*x^7-72*x^3-72)*ln(x)-144*x^12+4*x^9
-72*x^7+72*x^5+216*x^2-72)/(x^3*ln(x)^3-3*x^5*ln(x)^2+3*x^7*ln(x)-x^9),x,method=_RETURNVERBOSE)

[Out]

36*x^4-4*x-36*(2*x^5-2*x^3*ln(x)-1)/x^2/(x^2-ln(x))^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (28) = 56\).

Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81 \[ \int \frac {-72+216 x^2+72 x^5-72 x^7+4 x^9-144 x^{12}+\left (-72-72 x^3-12 x^7+432 x^{10}\right ) \log (x)+\left (72 x^3+12 x^5-432 x^8\right ) \log ^2(x)+\left (-4 x^3+144 x^6\right ) \log ^3(x)}{-x^9+3 x^7 \log (x)-3 x^5 \log ^2(x)+x^3 \log ^3(x)} \, dx=\frac {4 \, {\left (9 \, x^{10} - x^{7} - 18 \, x^{5} + {\left (9 \, x^{6} - x^{3}\right )} \log \left (x\right )^{2} - 2 \, {\left (9 \, x^{8} - x^{5} - 9 \, x^{3}\right )} \log \left (x\right ) + 9\right )}}{x^{6} - 2 \, x^{4} \log \left (x\right ) + x^{2} \log \left (x\right )^{2}} \]

[In]

integrate(((144*x^6-4*x^3)*log(x)^3+(-432*x^8+12*x^5+72*x^3)*log(x)^2+(432*x^10-12*x^7-72*x^3-72)*log(x)-144*x
^12+4*x^9-72*x^7+72*x^5+216*x^2-72)/(x^3*log(x)^3-3*x^5*log(x)^2+3*x^7*log(x)-x^9),x, algorithm="fricas")

[Out]

4*(9*x^10 - x^7 - 18*x^5 + (9*x^6 - x^3)*log(x)^2 - 2*(9*x^8 - x^5 - 9*x^3)*log(x) + 9)/(x^6 - 2*x^4*log(x) +
x^2*log(x)^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).

Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {-72+216 x^2+72 x^5-72 x^7+4 x^9-144 x^{12}+\left (-72-72 x^3-12 x^7+432 x^{10}\right ) \log (x)+\left (72 x^3+12 x^5-432 x^8\right ) \log ^2(x)+\left (-4 x^3+144 x^6\right ) \log ^3(x)}{-x^9+3 x^7 \log (x)-3 x^5 \log ^2(x)+x^3 \log ^3(x)} \, dx=36 x^{4} - 4 x + \frac {- 72 x^{5} + 72 x^{3} \log {\left (x \right )} + 36}{x^{6} - 2 x^{4} \log {\left (x \right )} + x^{2} \log {\left (x \right )}^{2}} \]

[In]

integrate(((144*x**6-4*x**3)*ln(x)**3+(-432*x**8+12*x**5+72*x**3)*ln(x)**2+(432*x**10-12*x**7-72*x**3-72)*ln(x
)-144*x**12+4*x**9-72*x**7+72*x**5+216*x**2-72)/(x**3*ln(x)**3-3*x**5*ln(x)**2+3*x**7*ln(x)-x**9),x)

[Out]

36*x**4 - 4*x + (-72*x**5 + 72*x**3*log(x) + 36)/(x**6 - 2*x**4*log(x) + x**2*log(x)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (28) = 56\).

Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81 \[ \int \frac {-72+216 x^2+72 x^5-72 x^7+4 x^9-144 x^{12}+\left (-72-72 x^3-12 x^7+432 x^{10}\right ) \log (x)+\left (72 x^3+12 x^5-432 x^8\right ) \log ^2(x)+\left (-4 x^3+144 x^6\right ) \log ^3(x)}{-x^9+3 x^7 \log (x)-3 x^5 \log ^2(x)+x^3 \log ^3(x)} \, dx=\frac {4 \, {\left (9 \, x^{10} - x^{7} - 18 \, x^{5} + {\left (9 \, x^{6} - x^{3}\right )} \log \left (x\right )^{2} - 2 \, {\left (9 \, x^{8} - x^{5} - 9 \, x^{3}\right )} \log \left (x\right ) + 9\right )}}{x^{6} - 2 \, x^{4} \log \left (x\right ) + x^{2} \log \left (x\right )^{2}} \]

[In]

integrate(((144*x^6-4*x^3)*log(x)^3+(-432*x^8+12*x^5+72*x^3)*log(x)^2+(432*x^10-12*x^7-72*x^3-72)*log(x)-144*x
^12+4*x^9-72*x^7+72*x^5+216*x^2-72)/(x^3*log(x)^3-3*x^5*log(x)^2+3*x^7*log(x)-x^9),x, algorithm="maxima")

[Out]

4*(9*x^10 - x^7 - 18*x^5 + (9*x^6 - x^3)*log(x)^2 - 2*(9*x^8 - x^5 - 9*x^3)*log(x) + 9)/(x^6 - 2*x^4*log(x) +
x^2*log(x)^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {-72+216 x^2+72 x^5-72 x^7+4 x^9-144 x^{12}+\left (-72-72 x^3-12 x^7+432 x^{10}\right ) \log (x)+\left (72 x^3+12 x^5-432 x^8\right ) \log ^2(x)+\left (-4 x^3+144 x^6\right ) \log ^3(x)}{-x^9+3 x^7 \log (x)-3 x^5 \log ^2(x)+x^3 \log ^3(x)} \, dx=36 \, x^{4} - 4 \, x - \frac {36 \, {\left (2 \, x^{5} - 2 \, x^{3} \log \left (x\right ) - 1\right )}}{x^{6} - 2 \, x^{4} \log \left (x\right ) + x^{2} \log \left (x\right )^{2}} \]

[In]

integrate(((144*x^6-4*x^3)*log(x)^3+(-432*x^8+12*x^5+72*x^3)*log(x)^2+(432*x^10-12*x^7-72*x^3-72)*log(x)-144*x
^12+4*x^9-72*x^7+72*x^5+216*x^2-72)/(x^3*log(x)^3-3*x^5*log(x)^2+3*x^7*log(x)-x^9),x, algorithm="giac")

[Out]

36*x^4 - 4*x - 36*(2*x^5 - 2*x^3*log(x) - 1)/(x^6 - 2*x^4*log(x) + x^2*log(x)^2)

Mupad [B] (verification not implemented)

Time = 10.70 (sec) , antiderivative size = 284, normalized size of antiderivative = 10.52 \[ \int \frac {-72+216 x^2+72 x^5-72 x^7+4 x^9-144 x^{12}+\left (-72-72 x^3-12 x^7+432 x^{10}\right ) \log (x)+\left (72 x^3+12 x^5-432 x^8\right ) \log ^2(x)+\left (-4 x^3+144 x^6\right ) \log ^3(x)}{-x^9+3 x^7 \log (x)-3 x^5 \log ^2(x)+x^3 \log ^3(x)} \, dx=\frac {\frac {36\,\left (6\,x^9-7\,x^7+5\,x^5+12\,x^4-x^3-6\,x^2+1\right )}{x^2\,{\left (2\,x^2-1\right )}^3}-\frac {36\,\ln \left (x\right )\,\left (6\,x^5-x^3+8\,x^2-2\right )}{x^2\,{\left (2\,x^2-1\right )}^3}+\frac {36\,x\,{\ln \left (x\right )}^2\,\left (2\,x^2+1\right )}{{\left (2\,x^2-1\right )}^3}}{\ln \left (x\right )-x^2}-4\,x+\frac {27\,x^7-\frac {81\,x^5}{2}+9\,x^3-36\,x^2+9}{-x^8+\frac {3\,x^6}{2}-\frac {3\,x^4}{4}+\frac {x^2}{8}}+36\,x^4+\frac {\frac {36\,x\,{\ln \left (x\right )}^2}{2\,x^2-1}+\frac {36\,\left (-x^7+x^5+3\,x^2-1\right )}{x^2\,\left (2\,x^2-1\right )}-\frac {36\,\ln \left (x\right )\,\left (x^3+1\right )}{x^2\,\left (2\,x^2-1\right )}}{x^4-2\,x^2\,\ln \left (x\right )+{\ln \left (x\right )}^2}-\frac {\ln \left (x\right )\,\left (9\,x^3+\frac {9\,x}{2}\right )}{x^6-\frac {3\,x^4}{2}+\frac {3\,x^2}{4}-\frac {1}{8}} \]

[In]

int(-(log(x)*(72*x^3 + 12*x^7 - 432*x^10 + 72) + log(x)^3*(4*x^3 - 144*x^6) - log(x)^2*(72*x^3 + 12*x^5 - 432*
x^8) - 216*x^2 - 72*x^5 + 72*x^7 - 4*x^9 + 144*x^12 + 72)/(3*x^7*log(x) + x^3*log(x)^3 - 3*x^5*log(x)^2 - x^9)
,x)

[Out]

((36*(12*x^4 - x^3 - 6*x^2 + 5*x^5 - 7*x^7 + 6*x^9 + 1))/(x^2*(2*x^2 - 1)^3) - (36*log(x)*(8*x^2 - x^3 + 6*x^5
- 2))/(x^2*(2*x^2 - 1)^3) + (36*x*log(x)^2*(2*x^2 + 1))/(2*x^2 - 1)^3)/(log(x) - x^2) - 4*x + (9*x^3 - 36*x^2
- (81*x^5)/2 + 27*x^7 + 9)/(x^2/8 - (3*x^4)/4 + (3*x^6)/2 - x^8) + 36*x^4 + ((36*x*log(x)^2)/(2*x^2 - 1) + (3
6*(3*x^2 + x^5 - x^7 - 1))/(x^2*(2*x^2 - 1)) - (36*log(x)*(x^3 + 1))/(x^2*(2*x^2 - 1)))/(log(x)^2 - 2*x^2*log(
x) + x^4) - (log(x)*((9*x)/2 + 9*x^3))/((3*x^2)/4 - (3*x^4)/2 + x^6 - 1/8)