∫ −1+(−x+x2+x3) log(x)+x log(x) log( 120__ log (x)) ___________________________________________________________(−x2 +x3) log(x)+x log(x) log( 120_

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

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Rubi [A]
time = 0.44, antiderivative size = 19, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6873, 6820, 6874, 6816} \begin {gather*} \log \left (-x^2+x-\log \left (\frac {120}{\log (x)}\right )\right )+x \end {gather*}

Int[(-1 + (-x + x^2 + x^3)*Log[x] + x*Log[x]*Log[120/Log[x]])/((-x^2 + x^3)*Log[x] + x*Log[x]*Log[120/Log[x]])

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

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

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

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1-\left (-x+x^2+x^3\right ) \log (x)-x \log (x) \log \left (\frac {120}{\log (x)}\right )}{x \log (x) \left (x-x^2-\log \left (\frac {120}{\log (x)}\right )\right )} \, dx\\ &=\int \frac {-1+x \log (x) \left (-1+x+x^2+\log \left (\frac {120}{\log (x)}\right )\right )}{x \log (x) \left ((-1+x) x+\log \left (\frac {120}{\log (x)}\right )\right )} \, dx\\ &=\int \left (1+\frac {-1-x \log (x)+2 x^2 \log (x)}{x \log (x) \left (-x+x^2+\log \left (\frac {120}{\log (x)}\right )\right )}\right ) \, dx\\ &=x+\int \frac {-1-x \log (x)+2 x^2 \log (x)}{x \log (x) \left (-x+x^2+\log \left (\frac {120}{\log (x)}\right )\right )} \, dx\\ &=x+\log \left (x-x^2-\log \left (\frac {120}{\log (x)}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.05, size = 17, normalized size = 0.94 \begin {gather*} x+\log \left (-x+x^2+\log \left (\frac {120}{\log (x)}\right )\right ) \end {gather*}

Integrate[(-1 + (-x + x^2 + x^3)*Log[x] + x*Log[x]*Log[120/Log[x]])/((-x^2 + x^3)*Log[x] + x*Log[x]*Log[120/Lo

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Maple [C] Result contains complex when optimal does not.
time = 2.26, size = 37, normalized size = 2.06


method	result	size

risch	\(x +\ln \left (\ln \left (\ln \left (x \right )\right )+\frac {i \left (2 i x^{2}+2 i \ln \left (5\right )+2 i \ln \left (3\right )+6 i \ln \left (2\right )-2 i x \right )}{2}\right )\)	\(37\)

int((x*ln(x)*ln(120/ln(x))+(x^3+x^2-x)*ln(x)-1)/(x*ln(x)*ln(120/ln(x))+(x^3-x^2)*ln(x)),x,method=_RETURNVERBOS

x+ln(ln(ln(x))+1/2*I*(2*I*x^2+2*I*ln(5)+2*I*ln(3)+6*I*ln(2)-2*I*x))

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Maxima [A]
time = 0.53, size = 25, normalized size = 1.39 \begin {gather*} x + \log \left (-x^{2} + x - \log \left (5\right ) - \log \left (3\right ) - 3 \, \log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) \end {gather*}

integrate((x*log(x)*log(120/log(x))+(x^3+x^2-x)*log(x)-1)/(x*log(x)*log(120/log(x))+(x^3-x^2)*log(x)),x, algor

x + log(-x^2 + x - log(5) - log(3) - 3*log(2) + log(log(x)))

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Fricas [A]
time = 0.34, size = 17, normalized size = 0.94 \begin {gather*} x + \log \left (x^{2} - x + \log \left (\frac {120}{\log \left (x\right )}\right )\right ) \end {gather*}

integrate((x*log(x)*log(120/log(x))+(x^3+x^2-x)*log(x)-1)/(x*log(x)*log(120/log(x))+(x^3-x^2)*log(x)),x, algor

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Sympy [A]
time = 0.10, size = 14, normalized size = 0.78 \begin {gather*} x + \log {\left (x^{2} - x + \log {\left (\frac {120}{\log {\left (x \right )}} \right )} \right )} \end {gather*}

integrate((x*ln(x)*ln(120/ln(x))+(x**3+x**2-x)*ln(x)-1)/(x*ln(x)*ln(120/ln(x))+(x**3-x**2)*ln(x)),x)

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Giac [A]
time = 0.41, size = 17, normalized size = 0.94 \begin {gather*} x + \log \left (-x^{2} + x - \log \left (120\right ) + \log \left (\log \left (x\right )\right )\right ) \end {gather*}

integrate((x*log(x)*log(120/log(x))+(x^3+x^2-x)*log(x)-1)/(x*log(x)*log(120/log(x))+(x^3-x^2)*log(x)),x, algor

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} -\int \frac {\ln \left (x\right )\,\left (x^3+x^2-x\right )+x\,\ln \left (\frac {120}{\ln \left (x\right )}\right )\,\ln \left (x\right )-1}{\ln \left (x\right )\,\left (x^2-x^3\right )-x\,\ln \left (\frac {120}{\ln \left (x\right )}\right )\,\ln \left (x\right )} \,d x \end {gather*}

int(-(log(x)*(x^2 - x + x^3) + x*log(120/log(x))*log(x) - 1)/(log(x)*(x^2 - x^3) - x*log(120/log(x))*log(x)),x

-int((log(x)*(x^2 - x + x^3) + x*log(120/log(x))*log(x) - 1)/(log(x)*(x^2 - x^3) - x*log(120/log(x))*log(x)),

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3.6.84 \(\int \frac {-1+(-x+x^2+x^3) \log (x)+x \log (x) \log (\frac {120}{\log (x)})}{(-x^2+x^3) \log (x)+x \log (x) \log (\frac {120}{\log (x)})} \, dx\) [584]