3.38.71 \(\int \frac {-20 x-10 \log (x)-10 \log (2 x)}{12 x^3+12 x^2 \log (x) \log (2 x)+3 x \log ^2(x) \log ^2(2 x)} \, dx\)

Optimal. Leaf size=21 \[ \frac {10}{3 x \left (2+\frac {\log (x) \log (2 x)}{x}\right )} \]

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Rubi [F]  time = 0.75, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-20 x-10 \log (x)-10 \log (2 x)}{12 x^3+12 x^2 \log (x) \log (2 x)+3 x \log ^2(x) \log ^2(2 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-20*x - 10*Log[x] - 10*Log[2*x])/(12*x^3 + 12*x^2*Log[x]*Log[2*x] + 3*x*Log[x]^2*Log[2*x]^2),x]

[Out]

(-20*Defer[Int][(2*x + Log[x]*Log[2*x])^(-2), x])/3 + (20*Defer[Int][1/(Log[x]*(2*x + Log[x]*Log[2*x])^2), x])
/3 - (10*Defer[Int][Log[x]/(x*(2*x + Log[x]*Log[2*x])^2), x])/3 - (10*Defer[Int][1/(x*Log[x]*(2*x + Log[x]*Log
[2*x])), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 (-2 x-\log (x)-\log (2 x))}{3 x (2 x+\log (x) \log (2 x))^2} \, dx\\ &=\frac {10}{3} \int \frac {-2 x-\log (x)-\log (2 x)}{x (2 x+\log (x) \log (2 x))^2} \, dx\\ &=\frac {10}{3} \int \left (\frac {2 x-2 x \log (x)-\log ^2(x)}{x \log (x) (2 x+\log (x) \log (2 x))^2}-\frac {1}{x \log (x) (2 x+\log (x) \log (2 x))}\right ) \, dx\\ &=\frac {10}{3} \int \frac {2 x-2 x \log (x)-\log ^2(x)}{x \log (x) (2 x+\log (x) \log (2 x))^2} \, dx-\frac {10}{3} \int \frac {1}{x \log (x) (2 x+\log (x) \log (2 x))} \, dx\\ &=-\left (\frac {10}{3} \int \frac {1}{x \log (x) (2 x+\log (x) \log (2 x))} \, dx\right )+\frac {10}{3} \int \left (-\frac {2}{(2 x+\log (x) \log (2 x))^2}+\frac {2}{\log (x) (2 x+\log (x) \log (2 x))^2}-\frac {\log (x)}{x (2 x+\log (x) \log (2 x))^2}\right ) \, dx\\ &=-\left (\frac {10}{3} \int \frac {\log (x)}{x (2 x+\log (x) \log (2 x))^2} \, dx\right )-\frac {10}{3} \int \frac {1}{x \log (x) (2 x+\log (x) \log (2 x))} \, dx-\frac {20}{3} \int \frac {1}{(2 x+\log (x) \log (2 x))^2} \, dx+\frac {20}{3} \int \frac {1}{\log (x) (2 x+\log (x) \log (2 x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 17, normalized size = 0.81 \begin {gather*} \frac {10}{3 (2 x+\log (x) \log (2 x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20*x - 10*Log[x] - 10*Log[2*x])/(12*x^3 + 12*x^2*Log[x]*Log[2*x] + 3*x*Log[x]^2*Log[2*x]^2),x]

[Out]

10/(3*(2*x + Log[x]*Log[2*x]))

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fricas [A]  time = 0.98, size = 17, normalized size = 0.81 \begin {gather*} \frac {10}{3 \, {\left (\log \relax (2) \log \relax (x) + \log \relax (x)^{2} + 2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*log(2*x)-10*log(x)-20*x)/(3*x*log(x)^2*log(2*x)^2+12*x^2*log(x)*log(2*x)+12*x^3),x, algorithm="
fricas")

[Out]

10/3/(log(2)*log(x) + log(x)^2 + 2*x)

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giac [A]  time = 0.12, size = 17, normalized size = 0.81 \begin {gather*} \frac {10}{3 \, {\left (\log \relax (2) \log \relax (x) + \log \relax (x)^{2} + 2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*log(2*x)-10*log(x)-20*x)/(3*x*log(x)^2*log(2*x)^2+12*x^2*log(x)*log(2*x)+12*x^3),x, algorithm="
giac")

[Out]

10/3/(log(2)*log(x) + log(x)^2 + 2*x)

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maple [C]  time = 0.18, size = 25, normalized size = 1.19

method	result	size
risch	\(\frac {20 i}{3 \left (2 i \ln \relax (2) \ln \relax (x )+2 i \ln \relax (x )^{2}+4 i x \right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-10*ln(2*x)-10*ln(x)-20*x)/(3*x*ln(x)^2*ln(2*x)^2+12*x^2*ln(x)*ln(2*x)+12*x^3),x,method=_RETURNVERBOSE)

[Out]

20/3*I/(2*I*ln(2)*ln(x)+2*I*ln(x)^2+4*I*x)

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maxima [A]  time = 0.47, size = 17, normalized size = 0.81 \begin {gather*} \frac {10}{3 \, {\left (\log \relax (2) \log \relax (x) + \log \relax (x)^{2} + 2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*log(2*x)-10*log(x)-20*x)/(3*x*log(x)^2*log(2*x)^2+12*x^2*log(x)*log(2*x)+12*x^3),x, algorithm="
maxima")

[Out]

10/3/(log(2)*log(x) + log(x)^2 + 2*x)

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mupad [B]  time = 2.64, size = 20, normalized size = 0.95 \begin {gather*} \frac {10}{3\,\left ({\ln \relax (x)}^2+\ln \relax (2)\,\ln \relax (x)+2\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(20*x + 10*log(2*x) + 10*log(x))/(12*x^3 + 12*x^2*log(2*x)*log(x) + 3*x*log(2*x)^2*log(x)^2),x)

[Out]

10/(3*(2*x + log(x)^2 + log(2)*log(x)))

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sympy [A]  time = 0.27, size = 19, normalized size = 0.90 \begin {gather*} \frac {10}{6 x + 3 \log {\relax (x )}^{2} + 3 \log {\relax (2 )} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*ln(2*x)-10*ln(x)-20*x)/(3*x*ln(x)**2*ln(2*x)**2+12*x**2*ln(x)*ln(2*x)+12*x**3),x)

[Out]

10/(6*x + 3*log(x)**2 + 3*log(2)*log(x))

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