3.39.17 \(\int \frac {6 x+18 x^2-4 x^4+(12 x+4 x^4) \log (x)}{9-12 x^3+4 x^6} \, dx\)

Optimal. Leaf size=21 \[ 2 \left (6-\frac {x+\log (x)}{-\frac {3}{x^2}+2 x}\right ) \]

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Rubi [F]  time = 0.83, 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 {6 x+18 x^2-4 x^4+\left (12 x+4 x^4\right ) \log (x)}{9-12 x^3+4 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6*x + 18*x^2 - 4*x^4 + (12*x + 4*x^4)*Log[x])/(9 - 12*x^3 + 4*x^6),x]

[Out]

(2*x^3)/(3 - 2*x^3) - (2^(1/3)*ArcTan[(3^(1/3) + 2*2^(1/3)*x)/3^(5/6)])/3^(5/6) - ((-2/3)^(1/3)*Log[x]*Log[1 +
(-2/3)^(1/3)*x])/3 + ((-1)^(2/3)*(2/3)^(1/3)*Log[x]*Log[1 - (-1)^(2/3)*(2/3)^(1/3)*x])/3 - ((2/3)^(1/3)*Log[3
^(1/3) - 2^(1/3)*x])/3 + ((2/3)^(1/3)*Log[3/2]*Log[3^(1/3) - 2^(1/3)*x])/9 + Log[3^(2/3) + 6^(1/3)*x + 2^(2/3)
*x^2]/(3*2^(2/3)*3^(1/3)) - ((-2/3)^(1/3)*PolyLog[2, -((-2/3)^(1/3)*x)])/3 + ((-1)^(2/3)*(2/3)^(1/3)*PolyLog[2
, (-1)^(2/3)*(2/3)^(1/3)*x])/3 - ((2/3)^(1/3)*PolyLog[2, 1 - (2/3)^(1/3)*x])/3 + 18*Defer[Int][(x*Log[x])/(-3
+ 2*x^3)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 \int \frac {6 x+18 x^2-4 x^4+\left (12 x+4 x^4\right ) \log (x)}{\left (-6+4 x^3\right )^2} \, dx\\ &=4 \int \frac {2 x \left (3+9 x-2 x^3+6 \log (x)+2 x^3 \log (x)\right )}{\left (6-4 x^3\right )^2} \, dx\\ &=8 \int \frac {x \left (3+9 x-2 x^3+6 \log (x)+2 x^3 \log (x)\right )}{\left (6-4 x^3\right )^2} \, dx\\ &=8 \int \left (-\frac {x \left (-3-9 x+2 x^3\right )}{4 \left (-3+2 x^3\right )^2}+\frac {x \left (3+x^3\right ) \log (x)}{2 \left (-3+2 x^3\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x \left (-3-9 x+2 x^3\right )}{\left (-3+2 x^3\right )^2} \, dx\right )+4 \int \frac {x \left (3+x^3\right ) \log (x)}{\left (-3+2 x^3\right )^2} \, dx\\ &=\frac {2 x^3}{3-2 x^3}-\frac {1}{18} \int \frac {36 x}{-3+2 x^3} \, dx+4 \int \left (\frac {9 x \log (x)}{2 \left (-3+2 x^3\right )^2}+\frac {x \log (x)}{2 \left (-3+2 x^3\right )}\right ) \, dx\\ &=\frac {2 x^3}{3-2 x^3}-2 \int \frac {x}{-3+2 x^3} \, dx+2 \int \frac {x \log (x)}{-3+2 x^3} \, dx+18 \int \frac {x \log (x)}{\left (-3+2 x^3\right )^2} \, dx\\ &=\frac {2 x^3}{3-2 x^3}+2 \int \left (-\frac {(-1)^{2/3} \log (x)}{3 \sqrt [3]{6} \left (\sqrt [3]{3}+\sqrt [3]{-2} x\right )}-\frac {\log (x)}{3 \sqrt [3]{6} \left (\sqrt [3]{3}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{-\frac {1}{6}} \log (x)}{3 \left (\sqrt [3]{3}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx+18 \int \frac {x \log (x)}{\left (-3+2 x^3\right )^2} \, dx-\frac {2^{2/3} \int \frac {1}{-\sqrt [3]{3}+\sqrt [3]{2} x} \, dx}{3 \sqrt [3]{3}}+\frac {2^{2/3} \int \frac {-\sqrt [3]{3}+\sqrt [3]{2} x}{3^{2/3}+\sqrt [3]{6} x+2^{2/3} x^2} \, dx}{3 \sqrt [3]{3}}\\ &=\frac {2 x^3}{3-2 x^3}-\frac {1}{3} \sqrt [3]{\frac {2}{3}} \log \left (\sqrt [3]{3}-\sqrt [3]{2} x\right )+18 \int \frac {x \log (x)}{\left (-3+2 x^3\right )^2} \, dx-\frac {\int \frac {1}{3^{2/3}+\sqrt [3]{6} x+2^{2/3} x^2} \, dx}{\sqrt [3]{2}}+\frac {1}{3} \left (\sqrt [3]{-\frac {1}{3}} 2^{2/3}\right ) \int \frac {\log (x)}{\sqrt [3]{3}-(-1)^{2/3} \sqrt [3]{2} x} \, dx-\frac {(-2)^{2/3} \int \frac {\log (x)}{\sqrt [3]{3}+\sqrt [3]{-2} x} \, dx}{3 \sqrt [3]{3}}+\frac {\int \frac {\sqrt [3]{6}+2\ 2^{2/3} x}{3^{2/3}+\sqrt [3]{6} x+2^{2/3} x^2} \, dx}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {2^{2/3} \int \frac {\log (x)}{\sqrt [3]{3}-\sqrt [3]{2} x} \, dx}{3 \sqrt [3]{3}}\\ &=\frac {2 x^3}{3-2 x^3}-\frac {1}{3} \sqrt [3]{-\frac {2}{3}} \log (x) \log \left (1+\sqrt [3]{-\frac {2}{3}} x\right )+\frac {1}{3} (-1)^{2/3} \sqrt [3]{\frac {2}{3}} \log (x) \log \left (1-(-1)^{2/3} \sqrt [3]{\frac {2}{3}} x\right )-\frac {1}{3} \sqrt [3]{\frac {2}{3}} \log \left (\sqrt [3]{3}-\sqrt [3]{2} x\right )+\frac {1}{9} \sqrt [3]{\frac {2}{3}} \log \left (\frac {3}{2}\right ) \log \left (\sqrt [3]{3}-\sqrt [3]{2} x\right )+\frac {\log \left (3^{2/3}+\sqrt [3]{6} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt [3]{3}}+18 \int \frac {x \log (x)}{\left (-3+2 x^3\right )^2} \, dx+\frac {1}{3} \sqrt [3]{-\frac {2}{3}} \int \frac {\log \left (1+\sqrt [3]{-\frac {2}{3}} x\right )}{x} \, dx+\sqrt [3]{\frac {2}{3}} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\frac {2}{3}} x\right )-\frac {1}{3} \left ((-1)^{2/3} \sqrt [3]{\frac {2}{3}}\right ) \int \frac {\log \left (1-(-1)^{2/3} \sqrt [3]{\frac {2}{3}} x\right )}{x} \, dx-\frac {2^{2/3} \int \frac {\log \left (\sqrt [3]{\frac {2}{3}} x\right )}{\sqrt [3]{3}-\sqrt [3]{2} x} \, dx}{3 \sqrt [3]{3}}\\ &=\frac {2 x^3}{3-2 x^3}-\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt [3]{3}+2 \sqrt [3]{2} x}{3^{5/6}}\right )}{3^{5/6}}-\frac {1}{3} \sqrt [3]{-\frac {2}{3}} \log (x) \log \left (1+\sqrt [3]{-\frac {2}{3}} x\right )+\frac {1}{3} (-1)^{2/3} \sqrt [3]{\frac {2}{3}} \log (x) \log \left (1-(-1)^{2/3} \sqrt [3]{\frac {2}{3}} x\right )-\frac {1}{3} \sqrt [3]{\frac {2}{3}} \log \left (\sqrt [3]{3}-\sqrt [3]{2} x\right )+\frac {1}{9} \sqrt [3]{\frac {2}{3}} \log \left (\frac {3}{2}\right ) \log \left (\sqrt [3]{3}-\sqrt [3]{2} x\right )+\frac {\log \left (3^{2/3}+\sqrt [3]{6} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {1}{3} \sqrt [3]{-\frac {2}{3}} \text {Li}_2\left (-\sqrt [3]{-\frac {2}{3}} x\right )+\frac {1}{3} (-1)^{2/3} \sqrt [3]{\frac {2}{3}} \text {Li}_2\left ((-1)^{2/3} \sqrt [3]{\frac {2}{3}} x\right )-\frac {1}{3} \sqrt [3]{\frac {2}{3}} \text {Li}_2\left (1-\sqrt [3]{\frac {2}{3}} x\right )+18 \int \frac {x \log (x)}{\left (-3+2 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 20, normalized size = 0.95 \begin {gather*} \frac {2 \left (3+2 x^2 \log (x)\right )}{6-4 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*x + 18*x^2 - 4*x^4 + (12*x + 4*x^4)*Log[x])/(9 - 12*x^3 + 4*x^6),x]

[Out]

(2*(3 + 2*x^2*Log[x]))/(6 - 4*x^3)

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fricas [A]  time = 0.68, size = 20, normalized size = 0.95 \begin {gather*} -\frac {2 \, x^{2} \log \relax (x) + 3}{2 \, x^{3} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+12*x)*log(x)-4*x^4+18*x^2+6*x)/(4*x^6-12*x^3+9),x, algorithm="fricas")

[Out]

-(2*x^2*log(x) + 3)/(2*x^3 - 3)

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giac [A]  time = 0.20, size = 28, normalized size = 1.33 \begin {gather*} -\frac {2 \, x^{2} \log \relax (x)}{2 \, x^{3} - 3} - \frac {3}{2 \, x^{3} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+12*x)*log(x)-4*x^4+18*x^2+6*x)/(4*x^6-12*x^3+9),x, algorithm="giac")

[Out]

-2*x^2*log(x)/(2*x^3 - 3) - 3/(2*x^3 - 3)

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maple [A]  time = 0.05, size = 20, normalized size = 0.95

method	result	size
norman	\(\frac {-2 x^{2} \ln \relax (x )-3}{2 x^{3}-3}\)	\(20\)
default	\(-\frac {3}{2 \left (x^{3}-\frac {3}{2}\right )}-\frac {2 \ln \relax (x ) x^{2}}{2 x^{3}-3}\)	\(27\)
risch	\(-\frac {2 \ln \relax (x ) x^{2}}{2 x^{3}-3}-\frac {3}{2 x^{3}-3}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^4+12*x)*ln(x)-4*x^4+18*x^2+6*x)/(4*x^6-12*x^3+9),x,method=_RETURNVERBOSE)

[Out]

(-2*x^2*ln(x)-3)/(2*x^3-3)

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maxima [A]  time = 1.18, size = 28, normalized size = 1.33 \begin {gather*} -\frac {2 \, x^{2} \log \relax (x)}{2 \, x^{3} - 3} - \frac {3}{2 \, x^{3} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+12*x)*log(x)-4*x^4+18*x^2+6*x)/(4*x^6-12*x^3+9),x, algorithm="maxima")

[Out]

-2*x^2*log(x)/(2*x^3 - 3) - 3/(2*x^3 - 3)

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mupad [B]  time = 2.32, size = 20, normalized size = 0.95 \begin {gather*} -\frac {2\,x^2\,\ln \relax (x)+3}{2\,x^3-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + log(x)*(12*x + 4*x^4) + 18*x^2 - 4*x^4)/(4*x^6 - 12*x^3 + 9),x)

[Out]

-(2*x^2*log(x) + 3)/(2*x^3 - 3)

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sympy [A]  time = 0.16, size = 24, normalized size = 1.14 \begin {gather*} - \frac {2 x^{2} \log {\relax (x )}}{2 x^{3} - 3} - \frac {18}{12 x^{3} - 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**4+12*x)*ln(x)-4*x**4+18*x**2+6*x)/(4*x**6-12*x**3+9),x)

[Out]

-2*x**2*log(x)/(2*x**3 - 3) - 18/(12*x**3 - 18)

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