3.62.9 \(\int \frac {3 x^2+3 x^3 \log (\frac {3}{2})+(-12 x^2-9 x^3 \log (\frac {3}{2})) \log (x)+(-36+27 x-(18 x-18 x^2) \log (\frac {3}{2})) \log ^2(x)}{-x^7 \log (\frac {3}{2})-(12 x^5-6 x^6) \log (\frac {3}{2}) \log (x)-(36 x^3-36 x^4+9 x^5) \log (\frac {3}{2}) \log ^2(x)} \, dx\)

Optimal. Leaf size=32 \[ \frac {-3-\frac {3}{x \log \left (\frac {3}{2}\right )}}{x^2 \left (-3+\frac {6}{x}+\frac {x}{\log (x)}\right )} \]

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Rubi [F]  time = 1.73, 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 {3 x^2+3 x^3 \log \left (\frac {3}{2}\right )+\left (-12 x^2-9 x^3 \log \left (\frac {3}{2}\right )\right ) \log (x)+\left (-36+27 x-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )\right ) \log ^2(x)}{-x^7 \log \left (\frac {3}{2}\right )-\left (12 x^5-6 x^6\right ) \log \left (\frac {3}{2}\right ) \log (x)-\left (36 x^3-36 x^4+9 x^5\right ) \log \left (\frac {3}{2}\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*x^2 + 3*x^3*Log[3/2] + (-12*x^2 - 9*x^3*Log[3/2])*Log[x] + (-36 + 27*x - (18*x - 18*x^2)*Log[3/2])*Log[
x]^2)/(-(x^7*Log[3/2]) - (12*x^5 - 6*x^6)*Log[3/2]*Log[x] - (36*x^3 - 36*x^4 + 9*x^5)*Log[3/2]*Log[x]^2),x]

[Out]

-1/2*1/(x^2*Log[3/2]) - (1 + Log[9/4])/(4*(2 - x)*Log[3/2]) - (1 + Log[9/4])/(4*x*Log[3/2]) + ((1 - Log[27/8])
*Defer[Int][(x^2 + 6*Log[x] - 3*x*Log[x])^(-2), x])/Log[3/2] - ((4 + Log[6561/256])*Defer[Int][1/((-2 + x)^2*(
x^2 + 6*Log[x] - 3*x*Log[x])^2), x])/Log[3/2] - (Log[81/16]*Defer[Int][1/((-2 + x)*(x^2 + 6*Log[x] - 3*x*Log[x
])^2), x])/Log[3/2] - (3*Defer[Int][1/(x*(x^2 + 6*Log[x] - 3*x*Log[x])^2), x])/Log[3/2] + Defer[Int][x/(x^2 +
6*Log[x] - 3*x*Log[x])^2, x] + ((2 + Log[81/16])*Defer[Int][1/((-2 + x)^2*(x^2 + 6*Log[x] - 3*x*Log[x])), x])/
Log[3/2] + Defer[Int][1/((-2 + x)*(x^2 + 6*Log[x] - 3*x*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (-x^2 \left (1+x \log \left (\frac {3}{2}\right )\right )+x^2 \left (4+x \log \left (\frac {27}{8}\right )\right ) \log (x)+\left (12-6 x^2 \log \left (\frac {3}{2}\right )+x \left (-9+6 \log \left (\frac {3}{2}\right )\right )\right ) \log ^2(x)\right )}{x^3 \log \left (\frac {3}{2}\right ) \left (x^2-3 (-2+x) \log (x)\right )^2} \, dx\\ &=\frac {3 \int \frac {-x^2 \left (1+x \log \left (\frac {3}{2}\right )\right )+x^2 \left (4+x \log \left (\frac {27}{8}\right )\right ) \log (x)+\left (12-6 x^2 \log \left (\frac {3}{2}\right )+x \left (-9+6 \log \left (\frac {3}{2}\right )\right )\right ) \log ^2(x)}{x^3 \left (x^2-3 (-2+x) \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}\\ &=\frac {3 \int \left (\frac {4-x \left (3-\log \left (\frac {9}{4}\right )\right )-x^2 \log \left (\frac {9}{4}\right )}{3 (2-x)^2 x^3}+\frac {-12-x^2 \left (7-12 \log \left (\frac {3}{2}\right )\right )+12 x \left (1-\log \left (\frac {3}{2}\right )\right )+x^4 \log \left (\frac {3}{2}\right )+x^3 \left (1-\log \left (\frac {2187}{128}\right )\right )}{3 (2-x)^2 x \left (x^2+6 \log (x)-3 x \log (x)\right )^2}+\frac {2+x \log \left (\frac {3}{2}\right )+\log \left (\frac {9}{4}\right )}{3 (-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )}\right ) \, dx}{\log \left (\frac {3}{2}\right )}\\ &=\frac {\int \frac {4-x \left (3-\log \left (\frac {9}{4}\right )\right )-x^2 \log \left (\frac {9}{4}\right )}{(2-x)^2 x^3} \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\int \frac {-12-x^2 \left (7-12 \log \left (\frac {3}{2}\right )\right )+12 x \left (1-\log \left (\frac {3}{2}\right )\right )+x^4 \log \left (\frac {3}{2}\right )+x^3 \left (1-\log \left (\frac {2187}{128}\right )\right )}{(2-x)^2 x \left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\int \frac {2+x \log \left (\frac {3}{2}\right )+\log \left (\frac {9}{4}\right )}{(-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )} \, dx}{\log \left (\frac {3}{2}\right )}\\ &=\frac {\int \left (\frac {1}{x^3}+\frac {-1-\log \left (\frac {9}{4}\right )}{4 (-2+x)^2}+\frac {1+\log \left (\frac {9}{4}\right )}{4 x^2}\right ) \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\int \left (-\frac {3}{x \left (x^2+6 \log (x)-3 x \log (x)\right )^2}+\frac {x \log \left (\frac {3}{2}\right )}{\left (x^2+6 \log (x)-3 x \log (x)\right )^2}+\frac {1-\log \left (\frac {27}{8}\right )}{\left (x^2+6 \log (x)-3 x \log (x)\right )^2}-\frac {\log \left (\frac {81}{16}\right )}{(-2+x) \left (x^2+6 \log (x)-3 x \log (x)\right )^2}+\frac {-4-\log \left (\frac {6561}{256}\right )}{(-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )^2}\right ) \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\int \left (\frac {\log \left (\frac {3}{2}\right )}{(-2+x) \left (x^2+6 \log (x)-3 x \log (x)\right )}+\frac {2+\log \left (\frac {81}{16}\right )}{(-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )}\right ) \, dx}{\log \left (\frac {3}{2}\right )}\\ &=-\frac {1}{2 x^2 \log \left (\frac {3}{2}\right )}-\frac {1+\log \left (\frac {9}{4}\right )}{4 (2-x) \log \left (\frac {3}{2}\right )}-\frac {1+\log \left (\frac {9}{4}\right )}{4 x \log \left (\frac {3}{2}\right )}-\frac {3 \int \frac {1}{x \left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\left (1-\log \left (\frac {27}{8}\right )\right ) \int \frac {1}{\left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}-\frac {\log \left (\frac {81}{16}\right ) \int \frac {1}{(-2+x) \left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}+\frac {\left (2+\log \left (\frac {81}{16}\right )\right ) \int \frac {1}{(-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )} \, dx}{\log \left (\frac {3}{2}\right )}-\frac {\left (4+\log \left (\frac {6561}{256}\right )\right ) \int \frac {1}{(-2+x)^2 \left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx}{\log \left (\frac {3}{2}\right )}+\int \frac {x}{\left (x^2+6 \log (x)-3 x \log (x)\right )^2} \, dx+\int \frac {1}{(-2+x) \left (x^2+6 \log (x)-3 x \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.58, size = 34, normalized size = 1.06 \begin {gather*} -\frac {3 \left (1+x \log \left (\frac {3}{2}\right )\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \left (x^4-3 (-2+x) x^2 \log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x^2 + 3*x^3*Log[3/2] + (-12*x^2 - 9*x^3*Log[3/2])*Log[x] + (-36 + 27*x - (18*x - 18*x^2)*Log[3/2]
)*Log[x]^2)/(-(x^7*Log[3/2]) - (12*x^5 - 6*x^6)*Log[3/2]*Log[x] - (36*x^3 - 36*x^4 + 9*x^5)*Log[3/2]*Log[x]^2)
,x]

[Out]

(-3*(1 + x*Log[3/2])*Log[x])/(Log[3/2]*(x^4 - 3*(-2 + x)*x^2*Log[x]))

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fricas [A]  time = 0.57, size = 34, normalized size = 1.06 \begin {gather*} -\frac {3 \, {\left (x \log \left (\frac {2}{3}\right ) - 1\right )} \log \relax (x)}{x^{4} \log \left (\frac {2}{3}\right ) - 3 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (\frac {2}{3}\right ) \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-18*x^2+18*x)*log(2/3)+27*x-36)*log(x)^2+(9*x^3*log(2/3)-12*x^2)*log(x)-3*x^3*log(2/3)+3*x^2)/((9
*x^5-36*x^4+36*x^3)*log(2/3)*log(x)^2+(-6*x^6+12*x^5)*log(2/3)*log(x)+x^7*log(2/3)),x, algorithm="fricas")

[Out]

-3*(x*log(2/3) - 1)*log(x)/(x^4*log(2/3) - 3*(x^3 - 2*x^2)*log(2/3)*log(x))

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giac [B]  time = 2.65, size = 151, normalized size = 4.72 \begin {gather*} -\frac {2 \, x \log \relax (3) - 2 \, x \log \relax (2) + x + 2}{4 \, {\left (x^{2} \log \relax (3) - x^{2} \log \relax (2)\right )}} - \frac {x \log \relax (3) - x \log \relax (2) + 1}{x^{3} \log \relax (3) - x^{3} \log \relax (2) - 3 \, x^{2} \log \relax (3) \log \relax (x) + 3 \, x^{2} \log \relax (2) \log \relax (x) - 2 \, x^{2} \log \relax (3) + 2 \, x^{2} \log \relax (2) + 12 \, x \log \relax (3) \log \relax (x) - 12 \, x \log \relax (2) \log \relax (x) - 12 \, \log \relax (3) \log \relax (x) + 12 \, \log \relax (2) \log \relax (x)} + \frac {2 \, \log \relax (3) - 2 \, \log \relax (2) + 1}{4 \, {\left (x \log \relax (3) - x \log \relax (2) - 2 \, \log \relax (3) + 2 \, \log \relax (2)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-18*x^2+18*x)*log(2/3)+27*x-36)*log(x)^2+(9*x^3*log(2/3)-12*x^2)*log(x)-3*x^3*log(2/3)+3*x^2)/((9
*x^5-36*x^4+36*x^3)*log(2/3)*log(x)^2+(-6*x^6+12*x^5)*log(2/3)*log(x)+x^7*log(2/3)),x, algorithm="giac")

[Out]

-1/4*(2*x*log(3) - 2*x*log(2) + x + 2)/(x^2*log(3) - x^2*log(2)) - (x*log(3) - x*log(2) + 1)/(x^3*log(3) - x^3
*log(2) - 3*x^2*log(3)*log(x) + 3*x^2*log(2)*log(x) - 2*x^2*log(3) + 2*x^2*log(2) + 12*x*log(3)*log(x) - 12*x*
log(2)*log(x) - 12*log(3)*log(x) + 12*log(2)*log(x)) + 1/4*(2*log(3) - 2*log(2) + 1)/(x*log(3) - x*log(2) - 2*
log(3) + 2*log(2))

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maple [A]  time = 0.48, size = 39, normalized size = 1.22

method	result	size
norman	\(\frac {-3 x \ln \relax (x )+\frac {3 \ln \relax (x )}{\ln \relax (2)-\ln \relax (3)}}{x^{2} \left (x^{2}-3 x \ln \relax (x )+6 \ln \relax (x )\right )}\)	\(39\)
risch	\(\frac {x \ln \relax (2)-x \ln \relax (3)-1}{x^{2} \left (x \ln \relax (2)-x \ln \relax (3)-2 \ln \relax (2)+2 \ln \relax (3)\right )}-\frac {x \ln \relax (2)-x \ln \relax (3)-1}{\left (x -2\right ) \left (\ln \relax (2)-\ln \relax (3)\right ) \left (x^{2}-3 x \ln \relax (x )+6 \ln \relax (x )\right )}\)	\(79\)
default	\(-\frac {3 \left (\frac {\ln \relax (2) \ln \relax (x )}{x \left (x^{2}-3 x \ln \relax (x )+6 \ln \relax (x )\right )}-\frac {\ln \relax (x )}{x^{2} \left (x^{2}-3 x \ln \relax (x )+6 \ln \relax (x )\right )}-\frac {\ln \relax (3) \ln \relax (x )}{x \left (x^{2}-3 x \ln \relax (x )+6 \ln \relax (x )\right )}\right )}{\ln \relax (2)-\ln \relax (3)}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-18*x^2+18*x)*ln(2/3)+27*x-36)*ln(x)^2+(9*x^3*ln(2/3)-12*x^2)*ln(x)-3*x^3*ln(2/3)+3*x^2)/((9*x^5-36*x^4
+36*x^3)*ln(2/3)*ln(x)^2+(-6*x^6+12*x^5)*ln(2/3)*ln(x)+x^7*ln(2/3)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.58, size = 57, normalized size = 1.78 \begin {gather*} -\frac {3 \, {\left (x {\left (\log \relax (3) - \log \relax (2)\right )} + 1\right )} \log \relax (x)}{x^{4} {\left (\log \relax (3) - \log \relax (2)\right )} - 3 \, {\left (x^{3} {\left (\log \relax (3) - \log \relax (2)\right )} - 2 \, x^{2} {\left (\log \relax (3) - \log \relax (2)\right )}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-18*x^2+18*x)*log(2/3)+27*x-36)*log(x)^2+(9*x^3*log(2/3)-12*x^2)*log(x)-3*x^3*log(2/3)+3*x^2)/((9
*x^5-36*x^4+36*x^3)*log(2/3)*log(x)^2+(-6*x^6+12*x^5)*log(2/3)*log(x)+x^7*log(2/3)),x, algorithm="maxima")

[Out]

-3*(x*(log(3) - log(2)) + 1)*log(x)/(x^4*(log(3) - log(2)) - 3*(x^3*(log(3) - log(2)) - 2*x^2*(log(3) - log(2)
))*log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\ln \relax (x)}^2\,\left (27\,x+\ln \left (\frac {2}{3}\right )\,\left (18\,x-18\,x^2\right )-36\right )+\ln \relax (x)\,\left (9\,x^3\,\ln \left (\frac {2}{3}\right )-12\,x^2\right )-3\,x^3\,\ln \left (\frac {2}{3}\right )+3\,x^2}{x^7\,\ln \left (\frac {2}{3}\right )+\ln \left (\frac {2}{3}\right )\,{\ln \relax (x)}^2\,\left (9\,x^5-36\,x^4+36\,x^3\right )+\ln \left (\frac {2}{3}\right )\,\ln \relax (x)\,\left (12\,x^5-6\,x^6\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(27*x + log(2/3)*(18*x - 18*x^2) - 36) + log(x)*(9*x^3*log(2/3) - 12*x^2) - 3*x^3*log(2/3) + 3*x
^2)/(x^7*log(2/3) + log(2/3)*log(x)^2*(36*x^3 - 36*x^4 + 9*x^5) + log(2/3)*log(x)*(12*x^5 - 6*x^6)),x)

[Out]

int((log(x)^2*(27*x + log(2/3)*(18*x - 18*x^2) - 36) + log(x)*(9*x^3*log(2/3) - 12*x^2) - 3*x^3*log(2/3) + 3*x
^2)/(x^7*log(2/3) + log(2/3)*log(x)^2*(36*x^3 - 36*x^4 + 9*x^5) + log(2/3)*log(x)*(12*x^5 - 6*x^6)), x)

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sympy [B]  time = 1.65, size = 119, normalized size = 3.72 \begin {gather*} - \frac {x \left (- \log {\relax (2 )} + \log {\relax (3 )}\right ) + 1}{x^{3} \left (- \log {\relax (3 )} + \log {\relax (2 )}\right ) + x^{2} \left (- 2 \log {\relax (2 )} + 2 \log {\relax (3 )}\right )} + \frac {- x \log {\relax (3 )} + x \log {\relax (2 )} - 1}{- x^{3} \log {\relax (2 )} + x^{3} \log {\relax (3 )} - 2 x^{2} \log {\relax (3 )} + 2 x^{2} \log {\relax (2 )} + \left (- 3 x^{2} \log {\relax (3 )} + 3 x^{2} \log {\relax (2 )} - 12 x \log {\relax (2 )} + 12 x \log {\relax (3 )} - 12 \log {\relax (3 )} + 12 \log {\relax (2 )}\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-18*x**2+18*x)*ln(2/3)+27*x-36)*ln(x)**2+(9*x**3*ln(2/3)-12*x**2)*ln(x)-3*x**3*ln(2/3)+3*x**2)/((
9*x**5-36*x**4+36*x**3)*ln(2/3)*ln(x)**2+(-6*x**6+12*x**5)*ln(2/3)*ln(x)+x**7*ln(2/3)),x)

[Out]

-(x*(-log(2) + log(3)) + 1)/(x**3*(-log(3) + log(2)) + x**2*(-2*log(2) + 2*log(3))) + (-x*log(3) + x*log(2) -
1)/(-x**3*log(2) + x**3*log(3) - 2*x**2*log(3) + 2*x**2*log(2) + (-3*x**2*log(3) + 3*x**2*log(2) - 12*x*log(2)
+ 12*x*log(3) - 12*log(3) + 12*log(2))*log(x))

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