3.93.98 \(\int \frac {2+(2+4 x^3) \log (30 x)-2 \log (30 x) \log (-x \log (30 x))}{x^4 \log (30 x)+x \log (30 x) \log (-x \log (30 x))} \, dx\)

Optimal. Leaf size=19 \[ \log \left (\left (x^2+\frac {\log (-x \log (30 x))}{x}\right )^2\right ) \]

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Rubi [A]  time = 0.49, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6741, 6742, 6684} \begin {gather*} 2 \log \left (x^3+\log (-x \log (30 x))\right )-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + (2 + 4*x^3)*Log[30*x] - 2*Log[30*x]*Log[-(x*Log[30*x])])/(x^4*Log[30*x] + x*Log[30*x]*Log[-(x*Log[30*
x])]),x]

[Out]

-2*Log[x] + 2*Log[x^3 + Log[-(x*Log[30*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 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 {2+\left (2+4 x^3\right ) \log (30 x)-2 \log (30 x) \log (-x \log (30 x))}{x \log (30 x) \left (x^3+\log (-x \log (30 x))\right )} \, dx\\ &=\int \left (-\frac {2}{x}+\frac {2 \left (1+\log (30 x)+3 x^3 \log (30 x)\right )}{x \log (30 x) \left (x^3+\log (-x \log (30 x))\right )}\right ) \, dx\\ &=-2 \log (x)+2 \int \frac {1+\log (30 x)+3 x^3 \log (30 x)}{x \log (30 x) \left (x^3+\log (-x \log (30 x))\right )} \, dx\\ &=-2 \log (x)+2 \log \left (x^3+\log (-x \log (30 x))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 20, normalized size = 1.05 \begin {gather*} -2 \log (x)+2 \log \left (x^3+\log (-x \log (30 x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + (2 + 4*x^3)*Log[30*x] - 2*Log[30*x]*Log[-(x*Log[30*x])])/(x^4*Log[30*x] + x*Log[30*x]*Log[-(x*L
og[30*x])]),x]

[Out]

-2*Log[x] + 2*Log[x^3 + Log[-(x*Log[30*x])]]

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fricas [A]  time = 0.81, size = 22, normalized size = 1.16 \begin {gather*} 2 \, \log \left (x^{3} + \log \left (-x \log \left (30 \, x\right )\right )\right ) - 2 \, \log \left (30 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(30*x)*log(-x*log(30*x))+(4*x^3+2)*log(30*x)+2)/(x*log(30*x)*log(-x*log(30*x))+x^4*log(30*x))
,x, algorithm="fricas")

[Out]

2*log(x^3 + log(-x*log(30*x))) - 2*log(30*x)

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giac [A]  time = 0.21, size = 24, normalized size = 1.26 \begin {gather*} 2 \, \log \left (x^{3} + \log \relax (x) + \log \left (-\log \left (30\right ) - \log \relax (x)\right )\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(30*x)*log(-x*log(30*x))+(4*x^3+2)*log(30*x)+2)/(x*log(30*x)*log(-x*log(30*x))+x^4*log(30*x))
,x, algorithm="giac")

[Out]

2*log(x^3 + log(x) + log(-log(30) - log(x))) - 2*log(x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {-2 \ln \left (30 x \right ) \ln \left (-x \ln \left (30 x \right )\right )+\left (4 x^{3}+2\right ) \ln \left (30 x \right )+2}{x \ln \left (30 x \right ) \ln \left (-x \ln \left (30 x \right )\right )+x^{4} \ln \left (30 x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(30*x)*ln(-x*ln(30*x))+(4*x^3+2)*ln(30*x)+2)/(x*ln(30*x)*ln(-x*ln(30*x))+x^4*ln(30*x)),x)

[Out]

int((-2*ln(30*x)*ln(-x*ln(30*x))+(4*x^3+2)*ln(30*x)+2)/(x*ln(30*x)*ln(-x*ln(30*x))+x^4*ln(30*x)),x)

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maxima [A]  time = 0.48, size = 32, normalized size = 1.68 \begin {gather*} 2 \, \log \left (x^{3} + \log \relax (x) + \log \left (-\log \relax (5) - \log \relax (3) - \log \relax (2) - \log \relax (x)\right )\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(30*x)*log(-x*log(30*x))+(4*x^3+2)*log(30*x)+2)/(x*log(30*x)*log(-x*log(30*x))+x^4*log(30*x))
,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 7.51, size = 20, normalized size = 1.05 \begin {gather*} 2\,\ln \left (\ln \left (-x\,\ln \left (30\,x\right )\right )+x^3\right )-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(30*x)*(4*x^3 + 2) - 2*log(30*x)*log(-x*log(30*x)) + 2)/(x^4*log(30*x) + x*log(30*x)*log(-x*log(30*x))
),x)

[Out]

2*log(log(-x*log(30*x)) + x^3) - 2*log(x)

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sympy [A]  time = 0.37, size = 20, normalized size = 1.05 \begin {gather*} - 2 \log {\relax (x )} + 2 \log {\left (x^{3} + \log {\left (- x \log {\left (30 x \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(30*x)*ln(-x*ln(30*x))+(4*x**3+2)*ln(30*x)+2)/(x*ln(30*x)*ln(-x*ln(30*x))+x**4*ln(30*x)),x)

[Out]

-2*log(x) + 2*log(x**3 + log(-x*log(30*x)))

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