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

Optimal. Leaf size=18 \[ x+\log \left (4-\frac {1}{\left (x+\frac {120 \log (x)}{x}\right )^2}\right ) \]

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Rubi [B]  time = 0.43, antiderivative size = 37, normalized size of antiderivative = 2.06, number of steps used = 5, number of rules used = 2, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6742, 6684} \begin {gather*} \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 \end {gather*}

Antiderivative was successfully verified.

[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 6684

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 39, normalized size = 2.17 \begin {gather*} 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 ) \end {gather*}

Antiderivative was successfully verified.

[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]]

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fricas [B]  time = 0.85, size = 38, normalized size = 2.11 \begin {gather*} x + \log \left (4 \, x^{4} + 960 \, x^{2} \log \relax (x) - x^{2} + 57600 \, \log \relax (x)^{2}\right ) - 2 \, \log \left (x^{2} + 120 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [B]  time = 0.46, size = 38, normalized size = 2.11 \begin {gather*} x + \log \left (4 \, x^{4} + 960 \, x^{2} \log \relax (x) - x^{2} + 57600 \, \log \relax (x)^{2}\right ) - 2 \, \log \left (x^{2} + 120 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.04, size = 37, normalized size = 2.06

method	result	size
risch	\(x -2 \ln \left (\frac {x^{2}}{120}+\ln \relax (x )\right )+\ln \left (\frac {x^{4}}{14400}+\frac {x^{2} \ln \relax (x )}{60}-\frac {x^{2}}{57600}+\ln \relax (x )^{2}\right )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[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(1/120*x^2+ln(x))+ln(1/14400*x^4+1/60*x^2*ln(x)-1/57600*x^2+ln(x)^2)

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maxima [B]  time = 0.43, size = 37, normalized size = 2.06 \begin {gather*} x + \log \left (\frac {1}{120} \, x^{2} + \frac {1}{240} \, x + \log \relax (x)\right ) + \log \left (\frac {1}{120} \, x^{2} - \frac {1}{240} \, x + \log \relax (x)\right ) - 2 \, \log \left (\frac {1}{120} \, x^{2} + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {240\,x+6912000\,{\ln \relax (x)}^3+172800\,x^2\,{\ln \relax (x)}^2+2\,x^3-x^4+4\,x^6-\ln \relax (x)\,\left (-1440\,x^4+120\,x^2+240\,x\right )}{6912000\,{\ln \relax (x)}^3-\ln \relax (x)\,\left (120\,x^2-1440\,x^4\right )+172800\,x^2\,{\ln \relax (x)}^2-x^4+4\,x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.35, size = 37, normalized size = 2.06 \begin {gather*} x - 2 \log {\left (\frac {x^{2}}{120} + \log {\relax (x )} \right )} + \log {\left (\frac {x^{4}}{14400} + \frac {x^{2} \log {\relax (x )}}{60} - \frac {x^{2}}{57600} + \log {\relax (x )}^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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