3.81.37 \(\int \frac {-6-11 x+6 x^2-2 x^3+(6-2 x-12 x^2+2 x^3) \log (x)+(6 x-4 x^3) \log ^2(x)-2 x \log ^3(x)+(-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)) \log (2 x)}{9 x+12 x^2 \log (x)+4 x^3 \log ^2(x)} \, dx\)

Optimal. Leaf size=31 \[ x \left (\frac {x+\frac {(-1+\log (x))^2}{x}}{3+2 x \log (x)}-\log (2 x)\right ) \]

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Rubi [F]  time = 2.08, 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 {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{9 x+12 x^2 \log (x)+4 x^3 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 - 11*x + 6*x^2 - 2*x^3 + (6 - 2*x - 12*x^2 + 2*x^3)*Log[x] + (6*x - 4*x^3)*Log[x]^2 - 2*x*Log[x]^3 + (
-9*x - 12*x^2*Log[x] - 4*x^3*Log[x]^2)*Log[2*x])/(9*x + 12*x^2*Log[x] + 4*x^3*Log[x]^2),x]

[Out]

-3/(4*x^2) - x^(-1) + Log[x]/(2*x) - x*Log[2*x] - 2*Defer[Int][(3 + 2*x*Log[x])^(-2), x] + (27*Defer[Int][1/(x
^3*(3 + 2*x*Log[x])^2), x])/4 + (9*Defer[Int][1/(x^2*(3 + 2*x*Log[x])^2), x])/2 - 3*Defer[Int][1/(x*(3 + 2*x*L
og[x])^2), x] + 3*Defer[Int][x/(3 + 2*x*Log[x])^2, x] - 2*Defer[Int][x^2/(3 + 2*x*Log[x])^2, x] - (27*Defer[In
t][1/(x^3*(3 + 2*x*Log[x])), x])/4 - 6*Defer[Int][1/(x^2*(3 + 2*x*Log[x])), x] - Defer[Int][1/(x*(3 + 2*x*Log[
x])), x] + Defer[Int][x/(3 + 2*x*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{x (3+2 x \log (x))^2} \, dx\\ &=\int \left (-\frac {11}{(3+2 x \log (x))^2}-\frac {6}{x (3+2 x \log (x))^2}+\frac {6 x}{(3+2 x \log (x))^2}-\frac {2 x^2}{(3+2 x \log (x))^2}+\frac {2 \left (3-x-6 x^2+x^3\right ) \log (x)}{x (3+2 x \log (x))^2}-\frac {2 \left (-3+2 x^2\right ) \log ^2(x)}{(3+2 x \log (x))^2}-\frac {2 \log ^3(x)}{(3+2 x \log (x))^2}-\log (2 x)\right ) \, dx\\ &=-\left (2 \int \frac {x^2}{(3+2 x \log (x))^2} \, dx\right )+2 \int \frac {\left (3-x-6 x^2+x^3\right ) \log (x)}{x (3+2 x \log (x))^2} \, dx-2 \int \frac {\left (-3+2 x^2\right ) \log ^2(x)}{(3+2 x \log (x))^2} \, dx-2 \int \frac {\log ^3(x)}{(3+2 x \log (x))^2} \, dx-6 \int \frac {1}{x (3+2 x \log (x))^2} \, dx+6 \int \frac {x}{(3+2 x \log (x))^2} \, dx-11 \int \frac {1}{(3+2 x \log (x))^2} \, dx-\int \log (2 x) \, dx\\ &=x-x \log (2 x)-2 \int \frac {x^2}{(3+2 x \log (x))^2} \, dx-2 \int \left (-\frac {3}{4 x^3}+\frac {\log (x)}{4 x^2}-\frac {27}{8 x^3 (3+2 x \log (x))^2}+\frac {27}{8 x^3 (3+2 x \log (x))}\right ) \, dx-2 \int \left (\frac {-3+2 x^2}{4 x^2}+\frac {9 \left (-3+2 x^2\right )}{4 x^2 (3+2 x \log (x))^2}-\frac {3 \left (-3+2 x^2\right )}{2 x^2 (3+2 x \log (x))}\right ) \, dx+2 \int \left (-\frac {3 \left (3-x-6 x^2+x^3\right )}{2 x^2 (3+2 x \log (x))^2}+\frac {3-x-6 x^2+x^3}{2 x^2 (3+2 x \log (x))}\right ) \, dx-6 \int \frac {1}{x (3+2 x \log (x))^2} \, dx+6 \int \frac {x}{(3+2 x \log (x))^2} \, dx-11 \int \frac {1}{(3+2 x \log (x))^2} \, dx\\ &=-\frac {3}{4 x^2}+x-x \log (2 x)-\frac {1}{2} \int \frac {-3+2 x^2}{x^2} \, dx-\frac {1}{2} \int \frac {\log (x)}{x^2} \, dx-2 \int \frac {x^2}{(3+2 x \log (x))^2} \, dx-3 \int \frac {3-x-6 x^2+x^3}{x^2 (3+2 x \log (x))^2} \, dx+3 \int \frac {-3+2 x^2}{x^2 (3+2 x \log (x))} \, dx-\frac {9}{2} \int \frac {-3+2 x^2}{x^2 (3+2 x \log (x))^2} \, dx-6 \int \frac {1}{x (3+2 x \log (x))^2} \, dx+6 \int \frac {x}{(3+2 x \log (x))^2} \, dx+\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))^2} \, dx-\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))} \, dx-11 \int \frac {1}{(3+2 x \log (x))^2} \, dx+\int \frac {3-x-6 x^2+x^3}{x^2 (3+2 x \log (x))} \, dx\\ &=-\frac {3}{4 x^2}+\frac {1}{2 x}+x+\frac {\log (x)}{2 x}-x \log (2 x)-\frac {1}{2} \int \left (2-\frac {3}{x^2}\right ) \, dx-2 \int \frac {x^2}{(3+2 x \log (x))^2} \, dx-3 \int \left (-\frac {6}{(3+2 x \log (x))^2}+\frac {3}{x^2 (3+2 x \log (x))^2}-\frac {1}{x (3+2 x \log (x))^2}+\frac {x}{(3+2 x \log (x))^2}\right ) \, dx+3 \int \left (\frac {2}{3+2 x \log (x)}-\frac {3}{x^2 (3+2 x \log (x))}\right ) \, dx-\frac {9}{2} \int \left (\frac {2}{(3+2 x \log (x))^2}-\frac {3}{x^2 (3+2 x \log (x))^2}\right ) \, dx-6 \int \frac {1}{x (3+2 x \log (x))^2} \, dx+6 \int \frac {x}{(3+2 x \log (x))^2} \, dx+\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))^2} \, dx-\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))} \, dx-11 \int \frac {1}{(3+2 x \log (x))^2} \, dx+\int \left (-\frac {6}{3+2 x \log (x)}+\frac {3}{x^2 (3+2 x \log (x))}-\frac {1}{x (3+2 x \log (x))}+\frac {x}{3+2 x \log (x)}\right ) \, dx\\ &=-\frac {3}{4 x^2}-\frac {1}{x}+\frac {\log (x)}{2 x}-x \log (2 x)-2 \int \frac {x^2}{(3+2 x \log (x))^2} \, dx+3 \int \frac {1}{x (3+2 x \log (x))^2} \, dx-3 \int \frac {x}{(3+2 x \log (x))^2} \, dx+3 \int \frac {1}{x^2 (3+2 x \log (x))} \, dx-6 \int \frac {1}{x (3+2 x \log (x))^2} \, dx+6 \int \frac {x}{(3+2 x \log (x))^2} \, dx+\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))^2} \, dx-\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))} \, dx-9 \int \frac {1}{(3+2 x \log (x))^2} \, dx-9 \int \frac {1}{x^2 (3+2 x \log (x))^2} \, dx-9 \int \frac {1}{x^2 (3+2 x \log (x))} \, dx-11 \int \frac {1}{(3+2 x \log (x))^2} \, dx+\frac {27}{2} \int \frac {1}{x^2 (3+2 x \log (x))^2} \, dx+18 \int \frac {1}{(3+2 x \log (x))^2} \, dx-\int \frac {1}{x (3+2 x \log (x))} \, dx+\int \frac {x}{3+2 x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 40, normalized size = 1.29 \begin {gather*} \frac {1+x^2+\log ^2(x)-3 x \log (2 x)-2 \log (x) \left (1+x^2 \log (2 x)\right )}{3+2 x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 11*x + 6*x^2 - 2*x^3 + (6 - 2*x - 12*x^2 + 2*x^3)*Log[x] + (6*x - 4*x^3)*Log[x]^2 - 2*x*Log[x]
^3 + (-9*x - 12*x^2*Log[x] - 4*x^3*Log[x]^2)*Log[2*x])/(9*x + 12*x^2*Log[x] + 4*x^3*Log[x]^2),x]

[Out]

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

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fricas [A]  time = 0.88, size = 50, normalized size = 1.61 \begin {gather*} -\frac {{\left (2 \, x^{2} - 1\right )} \log \relax (x)^{2} - x^{2} + 3 \, x \log \relax (2) + {\left (2 \, x^{2} \log \relax (2) + 3 \, x + 2\right )} \log \relax (x) - 1}{2 \, x \log \relax (x) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*log(x)^2-12*x^2*log(x)-9*x)*log(2*x)-2*x*log(x)^3+(-4*x^3+6*x)*log(x)^2+(2*x^3-12*x^2-2*x+6
)*log(x)-2*x^3+6*x^2-11*x-6)/(4*x^3*log(x)^2+12*x^2*log(x)+9*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.24, size = 61, normalized size = 1.97 \begin {gather*} -x \log \relax (2) - \frac {1}{2} \, {\left (2 \, x - \frac {1}{x}\right )} \log \relax (x) + \frac {4 \, x^{4} + 4 \, x^{2} + 12 \, x + 9}{4 \, {\left (2 \, x^{3} \log \relax (x) + 3 \, x^{2}\right )}} - \frac {4 \, x + 3}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*log(x)^2-12*x^2*log(x)-9*x)*log(2*x)-2*x*log(x)^3+(-4*x^3+6*x)*log(x)^2+(2*x^3-12*x^2-2*x+6
)*log(x)-2*x^3+6*x^2-11*x-6)/(4*x^3*log(x)^2+12*x^2*log(x)+9*x),x, algorithm="giac")

[Out]

-x*log(2) - 1/2*(2*x - 1/x)*log(x) + 1/4*(4*x^4 + 4*x^2 + 12*x + 9)/(2*x^3*log(x) + 3*x^2) - 1/4*(4*x + 3)/x^2

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maple [A]  time = 0.10, size = 43, normalized size = 1.39

method	result	size
default	\(\frac {x^{2}+\ln \relax (x )^{2}-2 \ln \relax (x )-\frac {11 x \ln \relax (x )}{3}-2 x^{2} \ln \relax (x )^{2}}{3+2 x \ln \relax (x )}-x \ln \relax (2)\)	\(43\)
risch	\(-\frac {\left (2 x^{2}-1\right ) \ln \relax (x )}{2 x}-\frac {3+4 x^{3} \ln \relax (2)+4 x}{4 x^{2}}+\frac {4 x^{4}+4 x^{2}+12 x +9}{4 x^{2} \left (3+2 x \ln \relax (x )\right )}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3*ln(x)^2-12*x^2*ln(x)-9*x)*ln(2*x)-2*x*ln(x)^3+(-4*x^3+6*x)*ln(x)^2+(2*x^3-12*x^2-2*x+6)*ln(x)-2*x
^3+6*x^2-11*x-6)/(4*x^3*ln(x)^2+12*x^2*ln(x)+9*x),x,method=_RETURNVERBOSE)

[Out]

(x^2+ln(x)^2-2*ln(x)-11/3*x*ln(x)-2*x^2*ln(x)^2)/(3+2*x*ln(x))-x*ln(2)

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maxima [A]  time = 0.48, size = 50, normalized size = 1.61 \begin {gather*} -\frac {{\left (2 \, x^{2} - 1\right )} \log \relax (x)^{2} - x^{2} + 3 \, x \log \relax (2) + {\left (2 \, x^{2} \log \relax (2) + 3 \, x + 2\right )} \log \relax (x) - 1}{2 \, x \log \relax (x) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*log(x)^2-12*x^2*log(x)-9*x)*log(2*x)-2*x*log(x)^3+(-4*x^3+6*x)*log(x)^2+(2*x^3-12*x^2-2*x+6
)*log(x)-2*x^3+6*x^2-11*x-6)/(4*x^3*log(x)^2+12*x^2*log(x)+9*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.41, size = 85, normalized size = 2.74 \begin {gather*} -\frac {2\,x^2\,\ln \relax (x)+\frac {11\,x^3\,\ln \relax (x)}{3}+3\,x^3\,\left (\ln \left (2\,x\right )-\ln \relax (x)\right )-x^2\,{\ln \relax (x)}^2+2\,x^4\,{\ln \relax (x)}^2-x^4+2\,x^4\,\ln \relax (x)\,\left (\ln \left (2\,x\right )-\ln \relax (x)\right )}{2\,x^3\,\ln \relax (x)+3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(11*x - log(x)^2*(6*x - 4*x^3) + 2*x*log(x)^3 + log(2*x)*(9*x + 12*x^2*log(x) + 4*x^3*log(x)^2) - 6*x^2 +
2*x^3 + log(x)*(2*x + 12*x^2 - 2*x^3 - 6) + 6)/(9*x + 12*x^2*log(x) + 4*x^3*log(x)^2),x)

[Out]

-(2*x^2*log(x) + (11*x^3*log(x))/3 + 3*x^3*(log(2*x) - log(x)) - x^2*log(x)^2 + 2*x^4*log(x)^2 - x^4 + 2*x^4*l
og(x)*(log(2*x) - log(x)))/(2*x^3*log(x) + 3*x^2)

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sympy [B]  time = 0.36, size = 56, normalized size = 1.81 \begin {gather*} - x \log {\relax (2 )} + \frac {4 x^{4} + 4 x^{2} + 12 x + 9}{8 x^{3} \log {\relax (x )} + 12 x^{2}} + \frac {\left (1 - 2 x^{2}\right ) \log {\relax (x )}}{2 x} - \frac {4 x + 3}{4 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3*ln(x)**2-12*x**2*ln(x)-9*x)*ln(2*x)-2*x*ln(x)**3+(-4*x**3+6*x)*ln(x)**2+(2*x**3-12*x**2-2*
x+6)*ln(x)-2*x**3+6*x**2-11*x-6)/(4*x**3*ln(x)**2+12*x**2*ln(x)+9*x),x)

[Out]

-x*log(2) + (4*x**4 + 4*x**2 + 12*x + 9)/(8*x**3*log(x) + 12*x**2) + (1 - 2*x**2)*log(x)/(2*x) - (4*x + 3)/(4*
x**2)

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