\(\int \frac {-810+81 x^3+18 x^6 \log (x)+(180 x^4+36 x^6-18 x^7) \log ^2(x)+(-10 x^8+x^{11}) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx\) [4977]

Optimal result

Integrand size = 75, antiderivative size = 31 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=-\frac {49}{16}+x-\frac {-\frac {5}{x}+\frac {x}{1-\frac {9}{x^4 \log ^2(x)}}}{x} \]

[Out]

x-49/16-(x/(1-9/x^4/ln(x)^2)-5/x)/x

Rubi [F]

\[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=\int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx \]

[In]

Int[(-810 + 81*x^3 + 18*x^6*Log[x] + (180*x^4 + 36*x^6 - 18*x^7)*Log[x]^2 + (-10*x^8 + x^11)*Log[x]^4)/(81*x^3
- 18*x^7*Log[x]^2 + x^11*Log[x]^4),x]

[Out]

5/x^2 + x + 9*Defer[Int][1/(x*(-3 + x^2*Log[x])^2), x] + (3*Defer[Int][x/(-3 + x^2*Log[x])^2, x])/2 + 3*Defer[
Int][1/(x*(-3 + x^2*Log[x])), x] + 9*Defer[Int][1/(x*(3 + x^2*Log[x])^2), x] - (3*Defer[Int][x/(3 + x^2*Log[x]
)^2, x])/2 - 3*Defer[Int][1/(x*(3 + x^2*Log[x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{x^3 \left (9-x^4 \log ^2(x)\right )^2} \, dx \\ & = \int \left (\frac {-10+x^3}{x^3}+\frac {3 \left (6+x^2\right )}{2 x \left (-3+x^2 \log (x)\right )^2}+\frac {3}{x \left (-3+x^2 \log (x)\right )}-\frac {3 \left (-6+x^2\right )}{2 x \left (3+x^2 \log (x)\right )^2}-\frac {3}{x \left (3+x^2 \log (x)\right )}\right ) \, dx \\ & = \frac {3}{2} \int \frac {6+x^2}{x \left (-3+x^2 \log (x)\right )^2} \, dx-\frac {3}{2} \int \frac {-6+x^2}{x \left (3+x^2 \log (x)\right )^2} \, dx+3 \int \frac {1}{x \left (-3+x^2 \log (x)\right )} \, dx-3 \int \frac {1}{x \left (3+x^2 \log (x)\right )} \, dx+\int \frac {-10+x^3}{x^3} \, dx \\ & = \frac {3}{2} \int \left (\frac {6}{x \left (-3+x^2 \log (x)\right )^2}+\frac {x}{\left (-3+x^2 \log (x)\right )^2}\right ) \, dx-\frac {3}{2} \int \left (-\frac {6}{x \left (3+x^2 \log (x)\right )^2}+\frac {x}{\left (3+x^2 \log (x)\right )^2}\right ) \, dx+3 \int \frac {1}{x \left (-3+x^2 \log (x)\right )} \, dx-3 \int \frac {1}{x \left (3+x^2 \log (x)\right )} \, dx+\int \left (1-\frac {10}{x^3}\right ) \, dx \\ & = \frac {5}{x^2}+x+\frac {3}{2} \int \frac {x}{\left (-3+x^2 \log (x)\right )^2} \, dx-\frac {3}{2} \int \frac {x}{\left (3+x^2 \log (x)\right )^2} \, dx+3 \int \frac {1}{x \left (-3+x^2 \log (x)\right )} \, dx-3 \int \frac {1}{x \left (3+x^2 \log (x)\right )} \, dx+9 \int \frac {1}{x \left (-3+x^2 \log (x)\right )^2} \, dx+9 \int \frac {1}{x \left (3+x^2 \log (x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=\frac {5}{x^2}+x+\frac {9}{9-x^4 \log ^2(x)} \]

[In]

Integrate[(-810 + 81*x^3 + 18*x^6*Log[x] + (180*x^4 + 36*x^6 - 18*x^7)*Log[x]^2 + (-10*x^8 + x^11)*Log[x]^4)/(
81*x^3 - 18*x^7*Log[x]^2 + x^11*Log[x]^4),x]

[Out]

5/x^2 + x + 9/(9 - x^4*Log[x]^2)

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71

method	result	size
default	\(x +\frac {5}{x^{2}}-\frac {9}{x^{4} \ln \left (x \right )^{2}-9}\)	\(22\)
risch	\(\frac {x^{3}+5}{x^{2}}-\frac {9}{x^{4} \ln \left (x \right )^{2}-9}\)	\(25\)
parallelrisch	\(-\frac {-9 x^{7} \ln \left (x \right )^{2}+405+9 x^{6} \ln \left (x \right )^{2}-45 x^{4} \ln \left (x \right )^{2}+81 x^{3}}{9 x^{2} \left (x^{4} \ln \left (x \right )^{2}-9\right )}\)	\(52\)

[In]

int(((x^11-10*x^8)*ln(x)^4+(-18*x^7+36*x^6+180*x^4)*ln(x)^2+18*x^6*ln(x)+81*x^3-810)/(x^11*ln(x)^4-18*x^7*ln(x
)^2+81*x^3),x,method=_RETURNVERBOSE)

[Out]

x+5/x^2-9/(x^4*ln(x)^2-9)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=-\frac {9 \, x^{3} - {\left (x^{7} + 5 \, x^{4}\right )} \log \left (x\right )^{2} + 9 \, x^{2} + 45}{x^{6} \log \left (x\right )^{2} - 9 \, x^{2}} \]

[In]

integrate(((x^11-10*x^8)*log(x)^4+(-18*x^7+36*x^6+180*x^4)*log(x)^2+18*x^6*log(x)+81*x^3-810)/(x^11*log(x)^4-1
8*x^7*log(x)^2+81*x^3),x, algorithm="fricas")

[Out]

-(9*x^3 - (x^7 + 5*x^4)*log(x)^2 + 9*x^2 + 45)/(x^6*log(x)^2 - 9*x^2)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=x - \frac {9}{x^{4} \log {\left (x \right )}^{2} - 9} + \frac {5}{x^{2}} \]

[In]

integrate(((x**11-10*x**8)*ln(x)**4+(-18*x**7+36*x**6+180*x**4)*ln(x)**2+18*x**6*ln(x)+81*x**3-810)/(x**11*ln(
x)**4-18*x**7*ln(x)**2+81*x**3),x)

[Out]

x - 9/(x**4*log(x)**2 - 9) + 5/x**2

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=-\frac {9 \, x^{3} - {\left (x^{7} + 5 \, x^{4}\right )} \log \left (x\right )^{2} + 9 \, x^{2} + 45}{x^{6} \log \left (x\right )^{2} - 9 \, x^{2}} \]

[In]

integrate(((x^11-10*x^8)*log(x)^4+(-18*x^7+36*x^6+180*x^4)*log(x)^2+18*x^6*log(x)+81*x^3-810)/(x^11*log(x)^4-1
8*x^7*log(x)^2+81*x^3),x, algorithm="maxima")

[Out]

-(9*x^3 - (x^7 + 5*x^4)*log(x)^2 + 9*x^2 + 45)/(x^6*log(x)^2 - 9*x^2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=x - \frac {9}{x^{4} \log \left (x\right )^{2} - 9} + \frac {5}{x^{2}} \]

[In]

integrate(((x^11-10*x^8)*log(x)^4+(-18*x^7+36*x^6+180*x^4)*log(x)^2+18*x^6*log(x)+81*x^3-810)/(x^11*log(x)^4-1
8*x^7*log(x)^2+81*x^3),x, algorithm="giac")

[Out]

x - 9/(x^4*log(x)^2 - 9) + 5/x^2

Mupad [B] (verification not implemented)

Time = 11.53 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=x-\frac {9}{x^4\,{\ln \left (x\right )}^2-9}+\frac {5}{x^2} \]

[In]

int((18*x^6*log(x) - log(x)^4*(10*x^8 - x^11) + log(x)^2*(180*x^4 + 36*x^6 - 18*x^7) + 81*x^3 - 810)/(x^11*log
(x)^4 - 18*x^7*log(x)^2 + 81*x^3),x)

[Out]

x - 9/(x^4*log(x)^2 - 9) + 5/x^2