3.30.83 \(\int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{(81 x^3+162 x^4+81 x^5) \log ^3(x)} \, dx\)

Optimal. Leaf size=20 \[ \frac {2 \left (8+\frac {1}{81 x^2 \log ^2(x)}\right )}{1+x} \]

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Rubi [F]  time = 0.42, 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 {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{\left (81 x^3+162 x^4+81 x^5\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4 - 4*x + (-4 - 6*x)*Log[x] - 1296*x^3*Log[x]^3)/((81*x^3 + 162*x^4 + 81*x^5)*Log[x]^3),x]

[Out]

16/(1 + x) - (4*Defer[Int][1/(x^3*(1 + x)*Log[x]^3), x])/81 - (2*Defer[Int][(2 + 3*x)/(x^3*(1 + x)^2*Log[x]^2)
, x])/81

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{x^3 \left (81+162 x+81 x^2\right ) \log ^3(x)} \, dx\\ &=\int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{81 x^3 (1+x)^2 \log ^3(x)} \, dx\\ &=\frac {1}{81} \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{x^3 (1+x)^2 \log ^3(x)} \, dx\\ &=\frac {1}{81} \int \left (-\frac {1296}{(1+x)^2}-\frac {4}{x^3 (1+x) \log ^3(x)}-\frac {2 (2+3 x)}{x^3 (1+x)^2 \log ^2(x)}\right ) \, dx\\ &=\frac {16}{1+x}-\frac {2}{81} \int \frac {2+3 x}{x^3 (1+x)^2 \log ^2(x)} \, dx-\frac {4}{81} \int \frac {1}{x^3 (1+x) \log ^3(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 19, normalized size = 0.95 \begin {gather*} \frac {2 \left (648+\frac {1}{x^2 \log ^2(x)}\right )}{81 (1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 4*x + (-4 - 6*x)*Log[x] - 1296*x^3*Log[x]^3)/((81*x^3 + 162*x^4 + 81*x^5)*Log[x]^3),x]

[Out]

(2*(648 + 1/(x^2*Log[x]^2)))/(81*(1 + x))

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fricas [A]  time = 0.56, size = 26, normalized size = 1.30 \begin {gather*} \frac {2 \, {\left (648 \, x^{2} \log \relax (x)^{2} + 1\right )}}{81 \, {\left (x^{3} + x^{2}\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1296*x^3*log(x)^3+(-6*x-4)*log(x)-4*x-4)/(81*x^5+162*x^4+81*x^3)/log(x)^3,x, algorithm="fricas")

[Out]

2/81*(648*x^2*log(x)^2 + 1)/((x^3 + x^2)*log(x)^2)

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giac [A]  time = 0.24, size = 29, normalized size = 1.45 \begin {gather*} \frac {2}{81 \, {\left (x^{3} \log \relax (x)^{2} + x^{2} \log \relax (x)^{2}\right )}} + \frac {16}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1296*x^3*log(x)^3+(-6*x-4)*log(x)-4*x-4)/(81*x^5+162*x^4+81*x^3)/log(x)^3,x, algorithm="giac")

[Out]

2/81/(x^3*log(x)^2 + x^2*log(x)^2) + 16/(x + 1)

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maple [A]  time = 0.04, size = 23, normalized size = 1.15

method	result	size
risch	\(\frac {16}{x +1}+\frac {2}{81 x^{2} \left (x +1\right ) \ln \relax (x )^{2}}\)	\(23\)
norman	\(\frac {\frac {2}{81}+16 x^{2} \ln \relax (x )^{2}}{x^{2} \left (x +1\right ) \ln \relax (x )^{2}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1296*x^3*ln(x)^3+(-6*x-4)*ln(x)-4*x-4)/(81*x^5+162*x^4+81*x^3)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

16/(x+1)+2/81/x^2/(x+1)/ln(x)^2

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maxima [A]  time = 0.44, size = 26, normalized size = 1.30 \begin {gather*} \frac {2 \, {\left (648 \, x^{2} \log \relax (x)^{2} + 1\right )}}{81 \, {\left (x^{3} + x^{2}\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1296*x^3*log(x)^3+(-6*x-4)*log(x)-4*x-4)/(81*x^5+162*x^4+81*x^3)/log(x)^3,x, algorithm="maxima")

[Out]

2/81*(648*x^2*log(x)^2 + 1)/((x^3 + x^2)*log(x)^2)

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mupad [B]  time = 1.83, size = 25, normalized size = 1.25 \begin {gather*} -\frac {16\,x^3\,{\ln \relax (x)}^2-\frac {2}{81}}{x^2\,{\ln \relax (x)}^2\,\left (x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + log(x)*(6*x + 4) + 1296*x^3*log(x)^3 + 4)/(log(x)^3*(81*x^3 + 162*x^4 + 81*x^5)),x)

[Out]

-(16*x^3*log(x)^2 - 2/81)/(x^2*log(x)^2*(x + 1))

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sympy [A]  time = 0.14, size = 20, normalized size = 1.00 \begin {gather*} \frac {2}{\left (81 x^{3} + 81 x^{2}\right ) \log {\relax (x )}^{2}} + \frac {16}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1296*x**3*ln(x)**3+(-6*x-4)*ln(x)-4*x-4)/(81*x**5+162*x**4+81*x**3)/ln(x)**3,x)

[Out]

2/((81*x**3 + 81*x**2)*log(x)**2) + 16/(x + 1)

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