\(\int \frac {240 x+2 x^3-x^4+4 x^6+(-240 x-120 x^2+1440 x^4) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)}{-x^4+4 x^6+(-120 x^2+1440 x^4) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)} \, dx\) [2320]

Optimal result

Integrand size = 94, antiderivative size = 18 \[ \int \frac {240 x+2 x^3-x^4+4 x^6+\left (-240 x-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)}{-x^4+4 x^6+\left (-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)} \, dx=x+\log \left (4-\frac {1}{\left (x+\frac {120 \log (x)}{x}\right )^2}\right ) \]

[Out]

x+ln(4-1/(120*ln(x)/x+x)^2)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6874, 6816} \[ \int \frac {240 x+2 x^3-x^4+4 x^6+\left (-240 x-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)}{-x^4+4 x^6+\left (-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)} \, dx=\log \left (-2 x^2+x-240 \log (x)\right )-2 \log \left (x^2+120 \log (x)\right )+\log \left (2 x^2+x+240 \log (x)\right )+x \]

[In]

Int[(240*x + 2*x^3 - x^4 + 4*x^6 + (-240*x - 120*x^2 + 1440*x^4)*Log[x] + 172800*x^2*Log[x]^2 + 6912000*Log[x]
^3)/(-x^4 + 4*x^6 + (-120*x^2 + 1440*x^4)*Log[x] + 172800*x^2*Log[x]^2 + 6912000*Log[x]^3),x]

[Out]

x + Log[x - 2*x^2 - 240*Log[x]] - 2*Log[x^2 + 120*Log[x]] + Log[x + 2*x^2 + 240*Log[x]]

Rule 6816

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (1-\frac {4 \left (60+x^2\right )}{x \left (x^2+120 \log (x)\right )}+\frac {240-x+4 x^2}{x \left (-x+2 x^2+240 \log (x)\right )}+\frac {240+x+4 x^2}{x \left (x+2 x^2+240 \log (x)\right )}\right ) \, dx \\ & = x-4 \int \frac {60+x^2}{x \left (x^2+120 \log (x)\right )} \, dx+\int \frac {240-x+4 x^2}{x \left (-x+2 x^2+240 \log (x)\right )} \, dx+\int \frac {240+x+4 x^2}{x \left (x+2 x^2+240 \log (x)\right )} \, dx \\ & = x+\log \left (x-2 x^2-240 \log (x)\right )-2 \log \left (x^2+120 \log (x)\right )+\log \left (x+2 x^2+240 \log (x)\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(39\) vs. \(2(18)=36\).

Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.17 \[ \int \frac {240 x+2 x^3-x^4+4 x^6+\left (-240 x-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)}{-x^4+4 x^6+\left (-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)} \, dx=x-2 \log \left (x^2+120 \log (x)\right )+\log \left (-x+2 x^2+240 \log (x)\right )+\log \left (x+2 x^2+240 \log (x)\right ) \]

[In]

Integrate[(240*x + 2*x^3 - x^4 + 4*x^6 + (-240*x - 120*x^2 + 1440*x^4)*Log[x] + 172800*x^2*Log[x]^2 + 6912000*
Log[x]^3)/(-x^4 + 4*x^6 + (-120*x^2 + 1440*x^4)*Log[x] + 172800*x^2*Log[x]^2 + 6912000*Log[x]^3),x]

[Out]

x - 2*Log[x^2 + 120*Log[x]] + Log[-x + 2*x^2 + 240*Log[x]] + Log[x + 2*x^2 + 240*Log[x]]

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.94

method	result	size
default	\(x -2 \ln \left (x^{2}+120 \ln \left (x \right )\right )+\ln \left (x^{4}+\left (240 \ln \left (x \right )-\frac {1}{4}\right ) x^{2}+14400 \ln \left (x \right )^{2}\right )\)	\(35\)
risch	\(x -2 \ln \left (\frac {x^{2}}{120}+\ln \left (x \right )\right )+\ln \left (\frac {x^{4}}{14400}+\frac {x^{2} \ln \left (x \right )}{60}-\frac {x^{2}}{57600}+\ln \left (x \right )^{2}\right )\)	\(37\)
parallelrisch	\(\ln \left (x^{2}-\frac {x}{2}+120 \ln \left (x \right )\right )+\ln \left (x^{2}+\frac {x}{2}+120 \ln \left (x \right )\right )-2 \ln \left (x^{2}+120 \ln \left (x \right )\right )+x\)	\(38\)

[In]

int((6912000*ln(x)^3+172800*x^2*ln(x)^2+(1440*x^4-120*x^2-240*x)*ln(x)+4*x^6-x^4+2*x^3+240*x)/(6912000*ln(x)^3
+172800*x^2*ln(x)^2+(1440*x^4-120*x^2)*ln(x)+4*x^6-x^4),x,method=_RETURNVERBOSE)

[Out]

x-2*ln(x^2+120*ln(x))+ln(x^4+(240*ln(x)-1/4)*x^2+14400*ln(x)^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (18) = 36\).

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.11 \[ \int \frac {240 x+2 x^3-x^4+4 x^6+\left (-240 x-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)}{-x^4+4 x^6+\left (-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)} \, dx=x + \log \left (4 \, x^{4} + 960 \, x^{2} \log \left (x\right ) - x^{2} + 57600 \, \log \left (x\right )^{2}\right ) - 2 \, \log \left (x^{2} + 120 \, \log \left (x\right )\right ) \]

[In]

integrate((6912000*log(x)^3+172800*x^2*log(x)^2+(1440*x^4-120*x^2-240*x)*log(x)+4*x^6-x^4+2*x^3+240*x)/(691200
0*log(x)^3+172800*x^2*log(x)^2+(1440*x^4-120*x^2)*log(x)+4*x^6-x^4),x, algorithm="fricas")

[Out]

x + log(4*x^4 + 960*x^2*log(x) - x^2 + 57600*log(x)^2) - 2*log(x^2 + 120*log(x))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).

Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {240 x+2 x^3-x^4+4 x^6+\left (-240 x-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)}{-x^4+4 x^6+\left (-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)} \, dx=x - 2 \log {\left (\frac {x^{2}}{120} + \log {\left (x \right )} \right )} + \log {\left (\frac {x^{4}}{14400} + \frac {x^{2} \log {\left (x \right )}}{60} - \frac {x^{2}}{57600} + \log {\left (x \right )}^{2} \right )} \]

[In]

integrate((6912000*ln(x)**3+172800*x**2*ln(x)**2+(1440*x**4-120*x**2-240*x)*ln(x)+4*x**6-x**4+2*x**3+240*x)/(6
912000*ln(x)**3+172800*x**2*ln(x)**2+(1440*x**4-120*x**2)*ln(x)+4*x**6-x**4),x)

[Out]

x - 2*log(x**2/120 + log(x)) + log(x**4/14400 + x**2*log(x)/60 - x**2/57600 + log(x)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (18) = 36\).

Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {240 x+2 x^3-x^4+4 x^6+\left (-240 x-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)}{-x^4+4 x^6+\left (-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)} \, dx=x + \log \left (\frac {1}{120} \, x^{2} + \frac {1}{240} \, x + \log \left (x\right )\right ) + \log \left (\frac {1}{120} \, x^{2} - \frac {1}{240} \, x + \log \left (x\right )\right ) - 2 \, \log \left (\frac {1}{120} \, x^{2} + \log \left (x\right )\right ) \]

[In]

integrate((6912000*log(x)^3+172800*x^2*log(x)^2+(1440*x^4-120*x^2-240*x)*log(x)+4*x^6-x^4+2*x^3+240*x)/(691200
0*log(x)^3+172800*x^2*log(x)^2+(1440*x^4-120*x^2)*log(x)+4*x^6-x^4),x, algorithm="maxima")

[Out]

x + log(1/120*x^2 + 1/240*x + log(x)) + log(1/120*x^2 - 1/240*x + log(x)) - 2*log(1/120*x^2 + log(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (18) = 36\).

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.11 \[ \int \frac {240 x+2 x^3-x^4+4 x^6+\left (-240 x-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)}{-x^4+4 x^6+\left (-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)} \, dx=x + \log \left (4 \, x^{4} + 960 \, x^{2} \log \left (x\right ) - x^{2} + 57600 \, \log \left (x\right )^{2}\right ) - 2 \, \log \left (x^{2} + 120 \, \log \left (x\right )\right ) \]

[In]

integrate((6912000*log(x)^3+172800*x^2*log(x)^2+(1440*x^4-120*x^2-240*x)*log(x)+4*x^6-x^4+2*x^3+240*x)/(691200
0*log(x)^3+172800*x^2*log(x)^2+(1440*x^4-120*x^2)*log(x)+4*x^6-x^4),x, algorithm="giac")

[Out]

x + log(4*x^4 + 960*x^2*log(x) - x^2 + 57600*log(x)^2) - 2*log(x^2 + 120*log(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {240 x+2 x^3-x^4+4 x^6+\left (-240 x-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)}{-x^4+4 x^6+\left (-120 x^2+1440 x^4\right ) \log (x)+172800 x^2 \log ^2(x)+6912000 \log ^3(x)} \, dx=\int \frac {240\,x+6912000\,{\ln \left (x\right )}^3+172800\,x^2\,{\ln \left (x\right )}^2+2\,x^3-x^4+4\,x^6-\ln \left (x\right )\,\left (-1440\,x^4+120\,x^2+240\,x\right )}{6912000\,{\ln \left (x\right )}^3-\ln \left (x\right )\,\left (120\,x^2-1440\,x^4\right )+172800\,x^2\,{\ln \left (x\right )}^2-x^4+4\,x^6} \,d x \]

[In]

int((240*x + 6912000*log(x)^3 + 172800*x^2*log(x)^2 + 2*x^3 - x^4 + 4*x^6 - log(x)*(240*x + 120*x^2 - 1440*x^4
))/(6912000*log(x)^3 - log(x)*(120*x^2 - 1440*x^4) + 172800*x^2*log(x)^2 - x^4 + 4*x^6),x)

[Out]

int((240*x + 6912000*log(x)^3 + 172800*x^2*log(x)^2 + 2*x^3 - x^4 + 4*x^6 - log(x)*(240*x + 120*x^2 - 1440*x^4
))/(6912000*log(x)^3 - log(x)*(120*x^2 - 1440*x^4) + 172800*x^2*log(x)^2 - x^4 + 4*x^6), x)