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

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 = 0.68, 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 = {6741, 6688, 6742, 6684} \begin {gather*} \log \left (-x^2+x-\log \left (\frac {120}{\log (x)}\right )\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

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]])
,x]

[Out]

x + Log[x - x^2 - Log[120/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 6688

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

Rule 6741

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

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 \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 = 0.17, size = 17, normalized size = 0.94 \begin {gather*} x+\log \left (-x+x^2+\log \left (\frac {120}{\log (x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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
g[x]]),x]

[Out]

x + Log[-x + x^2 + Log[120/Log[x]]]

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fricas [A]  time = 0.67, size = 17, normalized size = 0.94 \begin {gather*} x + \log \left (x^{2} - x + \log \left (\frac {120}{\log \relax (x)}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
ithm="fricas")

[Out]

x + log(x^2 - x + log(120/log(x)))

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giac [A]  time = 0.53, size = 17, normalized size = 0.94 \begin {gather*} x + \log \left (-x^{2} + x - \log \left (120\right ) + \log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
ithm="giac")

[Out]

x + log(-x^2 + x - log(120) + log(log(x)))

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maple [C]  time = 0.28, size = 37, normalized size = 2.06

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
E)

[Out]

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

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maxima [A]  time = 0.92, size = 25, normalized size = 1.39 \begin {gather*} x + \log \left (-x^{2} + x - \log \relax (5) - \log \relax (3) - 3 \, \log \relax (2) + \log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
ithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
)

[Out]

-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)

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sympy [A]  time = 0.32, size = 14, normalized size = 0.78 \begin {gather*} x + \log {\left (x^{2} - x + \log {\left (\frac {120}{\log {\relax (x )}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

x + log(x**2 - x + log(120/log(x)))

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