3.22.24 \(\int \frac {-3 x+3 x^2-3 x^2 \log (-\frac {3}{2 x})+(3-6 x+18 x^2) \log ^2(-\frac {3}{2 x})}{x^2+(2 x-12 x^2) \log (-\frac {3}{2 x})+(1-12 x+36 x^2) \log ^2(-\frac {3}{2 x})} \, dx\)

Optimal. Leaf size=26 \[ \frac {3 (-1+x)}{6-\frac {1}{x}-\frac {1}{\log \left (-\frac {3}{2 x}\right )}} \]

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Rubi [F]  time = 1.74, 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+3 x^2-3 x^2 \log \left (-\frac {3}{2 x}\right )+\left (3-6 x+18 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )}{x^2+\left (2 x-12 x^2\right ) \log \left (-\frac {3}{2 x}\right )+\left (1-12 x+36 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*x + 3*x^2 - 3*x^2*Log[-3/(2*x)] + (3 - 6*x + 18*x^2)*Log[-3/(2*x)]^2)/(x^2 + (2*x - 12*x^2)*Log[-3/(2*
x)] + (1 - 12*x + 36*x^2)*Log[-3/(2*x)]^2),x]

[Out]

5/(12*(1 - 6*x)) + x/2 + Defer[Int][(x + (1 - 6*x)*Log[-3/(2*x)])^(-2), x]/18 + (5*Defer[Int][1/((1 - 6*x)^2*(
x + (1 - 6*x)*Log[-3/(2*x)])^2), x])/72 - (37*Defer[Int][x/(x + (1 - 6*x)*Log[-3/(2*x)])^2, x])/12 + 3*Defer[I
nt][x^2/(x + (1 - 6*x)*Log[-3/(2*x)])^2, x] + Defer[Int][1/((-1 + 6*x)*(x + (1 - 6*x)*Log[-3/(2*x)])^2), x]/8
+ (5*Defer[Int][1/((-1 + 6*x)^2*(-x - Log[-3/(2*x)] + 6*x*Log[-3/(2*x)])), x])/6 + (3*Defer[Int][1/((-1 + 6*x)
*(-x - Log[-3/(2*x)] + 6*x*Log[-3/(2*x)])), x])/4 - Defer[Int][(-x + (-1 + 6*x)*Log[-3/(2*x)])^(-1), x]/12 + D
efer[Int][x/(-x + (-1 + 6*x)*Log[-3/(2*x)]), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left ((-1+x) x-x^2 \log \left (-\frac {3}{2 x}\right )+\left (1-2 x+6 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )\right )}{\left (x+(1-6 x) \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx\\ &=3 \int \frac {(-1+x) x-x^2 \log \left (-\frac {3}{2 x}\right )+\left (1-2 x+6 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )}{\left (x+(1-6 x) \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx\\ &=3 \int \left (\frac {1-2 x+6 x^2}{(-1+6 x)^2}+\frac {x \left (-1+14 x-49 x^2+36 x^3\right )}{(-1+6 x)^2 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2}+\frac {x \left (2-3 x+6 x^2\right )}{(-1+6 x)^2 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )}\right ) \, dx\\ &=3 \int \frac {1-2 x+6 x^2}{(-1+6 x)^2} \, dx+3 \int \frac {x \left (-1+14 x-49 x^2+36 x^3\right )}{(-1+6 x)^2 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx+3 \int \frac {x \left (2-3 x+6 x^2\right )}{(-1+6 x)^2 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )} \, dx\\ &=3 \int \left (\frac {1}{6}+\frac {5}{6 (-1+6 x)^2}\right ) \, dx+3 \int \frac {x \left (-1+14 x-49 x^2+36 x^3\right )}{(1-6 x)^2 \left (x+(1-6 x) \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx+3 \int \left (-\frac {1}{36 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )}+\frac {x}{6 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )}+\frac {5}{18 (-1+6 x)^2 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )}+\frac {1}{4 (-1+6 x) \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )}\right ) \, dx\\ &=\frac {5}{12 (1-6 x)}+\frac {x}{2}-\frac {1}{12} \int \frac {1}{-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )} \, dx+\frac {1}{2} \int \frac {x}{-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )} \, dx+\frac {3}{4} \int \frac {1}{(-1+6 x) \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )} \, dx+\frac {5}{6} \int \frac {1}{(-1+6 x)^2 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )} \, dx+3 \int \left (\frac {1}{54 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2}-\frac {37 x}{36 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2}+\frac {x^2}{\left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2}+\frac {5}{216 (-1+6 x)^2 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2}+\frac {1}{24 (-1+6 x) \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2}\right ) \, dx\\ &=\frac {5}{12 (1-6 x)}+\frac {x}{2}+\frac {1}{18} \int \frac {1}{\left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx+\frac {5}{72} \int \frac {1}{(-1+6 x)^2 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx-\frac {1}{12} \int \frac {1}{-x+(-1+6 x) \log \left (-\frac {3}{2 x}\right )} \, dx+\frac {1}{8} \int \frac {1}{(-1+6 x) \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx+\frac {1}{2} \int \frac {x}{-x+(-1+6 x) \log \left (-\frac {3}{2 x}\right )} \, dx+\frac {3}{4} \int \frac {1}{(-1+6 x) \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )} \, dx+\frac {5}{6} \int \frac {1}{(-1+6 x)^2 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )} \, dx+3 \int \frac {x^2}{\left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx-\frac {37}{12} \int \frac {x}{\left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx\\ &=\frac {5}{12 (1-6 x)}+\frac {x}{2}+\frac {1}{18} \int \frac {1}{\left (x+(1-6 x) \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx+\frac {5}{72} \int \frac {1}{(1-6 x)^2 \left (x+(1-6 x) \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx-\frac {1}{12} \int \frac {1}{-x+(-1+6 x) \log \left (-\frac {3}{2 x}\right )} \, dx+\frac {1}{8} \int \frac {1}{(-1+6 x) \left (x+(1-6 x) \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx+\frac {1}{2} \int \frac {x}{-x+(-1+6 x) \log \left (-\frac {3}{2 x}\right )} \, dx+\frac {3}{4} \int \frac {1}{(-1+6 x) \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )} \, dx+\frac {5}{6} \int \frac {1}{(-1+6 x)^2 \left (-x-\log \left (-\frac {3}{2 x}\right )+6 x \log \left (-\frac {3}{2 x}\right )\right )} \, dx+3 \int \frac {x^2}{\left (x+(1-6 x) \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx-\frac {37}{12} \int \frac {x}{\left (x+(1-6 x) \log \left (-\frac {3}{2 x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.56, size = 47, normalized size = 1.81 \begin {gather*} \frac {-5 x+\left (-5-6 x+36 x^2\right ) \log \left (-\frac {3}{2 x}\right )}{12 \left (-x+(-1+6 x) \log \left (-\frac {3}{2 x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x + 3*x^2 - 3*x^2*Log[-3/(2*x)] + (3 - 6*x + 18*x^2)*Log[-3/(2*x)]^2)/(x^2 + (2*x - 12*x^2)*Log[
-3/(2*x)] + (1 - 12*x + 36*x^2)*Log[-3/(2*x)]^2),x]

[Out]

(-5*x + (-5 - 6*x + 36*x^2)*Log[-3/(2*x)])/(12*(-x + (-1 + 6*x)*Log[-3/(2*x)]))

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fricas [B]  time = 0.71, size = 41, normalized size = 1.58 \begin {gather*} \frac {{\left (36 \, x^{2} - 6 \, x - 5\right )} \log \left (-\frac {3}{2 \, x}\right ) - 5 \, x}{12 \, {\left ({\left (6 \, x - 1\right )} \log \left (-\frac {3}{2 \, x}\right ) - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^2-6*x+3)*log(-3/2/x)^2-3*x^2*log(-3/2/x)+3*x^2-3*x)/((36*x^2-12*x+1)*log(-3/2/x)^2+(-12*x^2+2
*x)*log(-3/2/x)+x^2),x, algorithm="fricas")

[Out]

1/12*((36*x^2 - 6*x - 5)*log(-3/2/x) - 5*x)/((6*x - 1)*log(-3/2/x) - x)

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giac [B]  time = 0.29, size = 63, normalized size = 2.42 \begin {gather*} \frac {1}{2} \, x - \frac {3 \, {\left (\frac {1}{x} - 1\right )}}{\frac {36 \, \log \left (-\frac {3}{2 \, x}\right )}{x} - \frac {6}{x} - \frac {12 \, \log \left (-\frac {3}{2 \, x}\right )}{x^{2}} + \frac {1}{x^{2}} + \frac {\log \left (-\frac {3}{2 \, x}\right )}{x^{3}}} + \frac {5}{2 \, {\left (\frac {1}{x} - 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^2-6*x+3)*log(-3/2/x)^2-3*x^2*log(-3/2/x)+3*x^2-3*x)/((36*x^2-12*x+1)*log(-3/2/x)^2+(-12*x^2+2
*x)*log(-3/2/x)+x^2),x, algorithm="giac")

[Out]

1/2*x - 3*(1/x - 1)/(36*log(-3/2/x)/x - 6/x - 12*log(-3/2/x)/x^2 + 1/x^2 + log(-3/2/x)/x^3) + 5/2/(1/x - 6)

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maple [A]  time = 0.14, size = 46, normalized size = 1.77

method	result	size
norman	\(\frac {-3 x \ln \left (-\frac {3}{2 x}\right )+3 x^{2} \ln \left (-\frac {3}{2 x}\right )}{6 x \ln \left (-\frac {3}{2 x}\right )-\ln \left (-\frac {3}{2 x}\right )-x}\)	\(46\)
risch	\(\frac {36 x^{2}-6 x -5}{72 x -12}+\frac {3 x^{2} \left (x -1\right )}{\left (6 x -1\right ) \left (6 x \ln \left (-\frac {3}{2 x}\right )-\ln \left (-\frac {3}{2 x}\right )-x \right )}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((18*x^2-6*x+3)*ln(-3/2/x)^2-3*x^2*ln(-3/2/x)+3*x^2-3*x)/((36*x^2-12*x+1)*ln(-3/2/x)^2+(-12*x^2+2*x)*ln(-3
/2/x)+x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.55, size = 84, normalized size = 3.23 \begin {gather*} \frac {36 \, x^{2} {\left (\log \relax (3) - \log \relax (2)\right )} - x {\left (6 \, \log \relax (3) - 6 \, \log \relax (2) + 5\right )} - {\left (36 \, x^{2} - 6 \, x - 5\right )} \log \left (-x\right ) - 5 \, \log \relax (3) + 5 \, \log \relax (2)}{12 \, {\left (x {\left (6 \, \log \relax (3) - 6 \, \log \relax (2) - 1\right )} - {\left (6 \, x - 1\right )} \log \left (-x\right ) - \log \relax (3) + \log \relax (2)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^2-6*x+3)*log(-3/2/x)^2-3*x^2*log(-3/2/x)+3*x^2-3*x)/((36*x^2-12*x+1)*log(-3/2/x)^2+(-12*x^2+2
*x)*log(-3/2/x)+x^2),x, algorithm="maxima")

[Out]

1/12*(36*x^2*(log(3) - log(2)) - x*(6*log(3) - 6*log(2) + 5) - (36*x^2 - 6*x - 5)*log(-x) - 5*log(3) + 5*log(2
))/(x*(6*log(3) - 6*log(2) - 1) - (6*x - 1)*log(-x) - log(3) + log(2))

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mupad [B]  time = 1.44, size = 31, normalized size = 1.19 \begin {gather*} -\frac {3\,x\,\ln \left (-\frac {3}{2\,x}\right )\,\left (x-1\right )}{x+\ln \left (-\frac {3}{2\,x}\right )-6\,x\,\ln \left (-\frac {3}{2\,x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x - log(-3/(2*x))^2*(18*x^2 - 6*x + 3) - 3*x^2 + 3*x^2*log(-3/(2*x)))/(log(-3/(2*x))*(2*x - 12*x^2) +
log(-3/(2*x))^2*(36*x^2 - 12*x + 1) + x^2),x)

[Out]

-(3*x*log(-3/(2*x))*(x - 1))/(x + log(-3/(2*x)) - 6*x*log(-3/(2*x)))

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sympy [B]  time = 0.33, size = 44, normalized size = 1.69 \begin {gather*} \frac {x}{2} + \frac {3 x^{3} - 3 x^{2}}{- 6 x^{2} + x + \left (36 x^{2} - 12 x + 1\right ) \log {\left (- \frac {3}{2 x} \right )}} - \frac {5}{72 x - 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x**2-6*x+3)*ln(-3/2/x)**2-3*x**2*ln(-3/2/x)+3*x**2-3*x)/((36*x**2-12*x+1)*ln(-3/2/x)**2+(-12*x*
*2+2*x)*ln(-3/2/x)+x**2),x)

[Out]

x/2 + (3*x**3 - 3*x**2)/(-6*x**2 + x + (36*x**2 - 12*x + 1)*log(-3/(2*x))) - 5/(72*x - 12)

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