3.26.34 \(\int \frac {-4+7 x+x^2+x^3+(4-8 x+4 x^2) \log (x)+(-4-x^2) \log (\frac {x^2}{3})}{x^4+(2 x^3-2 x^4) \log (x)+(x^2-2 x^3+x^4) \log ^2(x)+(-2 x^3+(-2 x^2+2 x^3) \log (x)) \log (\frac {x^2}{3})+x^2 \log ^2(\frac {x^2}{3})} \, dx\)

Optimal. Leaf size=31 \[ -\frac {4+x}{x \left (\log (x)+\frac {-x+\log \left (\frac {x^2}{3}\right )}{-1+x}\right )} \]

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Rubi [F]  time = 2.16, 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+7 x+x^2+x^3+\left (4-8 x+4 x^2\right ) \log (x)+\left (-4-x^2\right ) \log \left (\frac {x^2}{3}\right )}{x^4+\left (2 x^3-2 x^4\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)+\left (-2 x^3+\left (-2 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {x^2}{3}\right )+x^2 \log ^2\left (\frac {x^2}{3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4 + 7*x + x^2 + x^3 + (4 - 8*x + 4*x^2)*Log[x] + (-4 - x^2)*Log[x^2/3])/(x^4 + (2*x^3 - 2*x^4)*Log[x] +
(x^2 - 2*x^3 + x^4)*Log[x]^2 + (-2*x^3 + (-2*x^2 + 2*x^3)*Log[x])*Log[x^2/3] + x^2*Log[x^2/3]^2),x]

[Out]

Defer[Int][(-x - Log[x] + x*Log[x] + Log[x^2/3])^(-2), x] + Log[3]*Defer[Int][(-x - Log[x] + x*Log[x] + Log[x^
2/3])^(-2), x] - (4 - Log[81])*Defer[Int][1/(x^2*(-x - Log[x] + x*Log[x] + Log[x^2/3])^2), x] + 7*Defer[Int][1
/(x*(-x - Log[x] + x*Log[x] + Log[x^2/3])^2), x] + Defer[Int][x/(-x - Log[x] + x*Log[x] + Log[x^2/3])^2, x] +
4*Defer[Int][Log[x]/(-x - Log[x] + x*Log[x] + Log[x^2/3])^2, x] + 4*Defer[Int][Log[x]/(x^2*(-x - Log[x] + x*Lo
g[x] + Log[x^2/3])^2), x] - 8*Defer[Int][Log[x]/(x*(-x - Log[x] + x*Log[x] + Log[x^2/3])^2), x] - Defer[Int][L
og[x^2]/(-x - Log[x] + x*Log[x] + Log[x^2/3])^2, x] - 4*Defer[Int][Log[x^2]/(x^2*(-x - Log[x] + x*Log[x] + Log
[x^2/3])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {7 x+x^2+x^3-4 (1-\log (3))+4 (-1+x)^2 \log (x)-x^2 \log \left (\frac {x^2}{3}\right )-4 \log \left (x^2\right )}{x^2 \left (x-(-1+x) \log (x)-\log \left (\frac {x^2}{3}\right )\right )^2} \, dx\\ &=\int \left (\frac {1}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {7}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {x}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {-4+\log (81)}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {4 (-1+x)^2 \log (x)}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {\log (3)-\log \left (x^2\right )}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}-\frac {4 \log \left (x^2\right )}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}\right ) \, dx\\ &=4 \int \frac {(-1+x)^2 \log (x)}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx-4 \int \frac {\log \left (x^2\right )}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+7 \int \frac {1}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+(-4+\log (81)) \int \frac {1}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {1}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {x}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {\log (3)-\log \left (x^2\right )}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx\\ &=4 \int \left (\frac {\log (x)}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {\log (x)}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}-\frac {2 \log (x)}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}\right ) \, dx-4 \int \frac {\log \left (x^2\right )}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+7 \int \frac {1}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+(-4+\log (81)) \int \frac {1}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {1}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {x}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \left (\frac {\log (3)}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}-\frac {\log \left (x^2\right )}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}\right ) \, dx\\ &=4 \int \frac {\log (x)}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+4 \int \frac {\log (x)}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx-4 \int \frac {\log \left (x^2\right )}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+7 \int \frac {1}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx-8 \int \frac {\log (x)}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\log (3) \int \frac {1}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+(-4+\log (81)) \int \frac {1}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {1}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {x}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx-\int \frac {\log \left (x^2\right )}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 2.93, size = 104, normalized size = 3.35 \begin {gather*} \frac {4+x-4 x^3-x^4-x \log (81)-2 x \left (-4+3 x+x^2\right ) \log (x)+x \left (-8+3 x+x^2\right ) \log \left (\frac {x^2}{3}\right )+4 x \log \left (x^2\right )}{x \left (-x+(-1+x) \log (x)+\log \left (\frac {x^2}{3}\right )\right ) \left (1+x+x^2+x \log (3)+2 x \log (x)-x \log \left (x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 7*x + x^2 + x^3 + (4 - 8*x + 4*x^2)*Log[x] + (-4 - x^2)*Log[x^2/3])/(x^4 + (2*x^3 - 2*x^4)*Log
[x] + (x^2 - 2*x^3 + x^4)*Log[x]^2 + (-2*x^3 + (-2*x^2 + 2*x^3)*Log[x])*Log[x^2/3] + x^2*Log[x^2/3]^2),x]

[Out]

(4 + x - 4*x^3 - x^4 - x*Log[81] - 2*x*(-4 + 3*x + x^2)*Log[x] + x*(-8 + 3*x + x^2)*Log[x^2/3] + 4*x*Log[x^2])
/(x*(-x + (-1 + x)*Log[x] + Log[x^2/3])*(1 + x + x^2 + x*Log[3] + 2*x*Log[x] - x*Log[x^2]))

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fricas [A]  time = 0.86, size = 28, normalized size = 0.90 \begin {gather*} \frac {x^{2} + 3 \, x - 4}{x^{2} + x \log \relax (3) - {\left (x^{2} + x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-4)*log(1/3*x^2)+(4*x^2-8*x+4)*log(x)+x^3+x^2+7*x-4)/(x^2*log(1/3*x^2)^2+((2*x^3-2*x^2)*log(x)
-2*x^3)*log(1/3*x^2)+(x^4-2*x^3+x^2)*log(x)^2+(-2*x^4+2*x^3)*log(x)+x^4),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.83, size = 33, normalized size = 1.06 \begin {gather*} -\frac {x^{2} + 3 \, x - 4}{x^{2} \log \relax (x) - x^{2} - x \log \relax (3) + x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-4)*log(1/3*x^2)+(4*x^2-8*x+4)*log(x)+x^3+x^2+7*x-4)/(x^2*log(1/3*x^2)^2+((2*x^3-2*x^2)*log(x)
-2*x^3)*log(1/3*x^2)+(x^4-2*x^3+x^2)*log(x)^2+(-2*x^4+2*x^3)*log(x)+x^4),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.15, size = 82, normalized size = 2.65

method	result	size
risch	\(-\frac {2 \left (x^{2}+3 x -4\right )}{x \left (-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 x \ln \relax (x )-2 \ln \relax (3)-2 x +2 \ln \relax (x )\right )}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2-4)*ln(1/3*x^2)+(4*x^2-8*x+4)*ln(x)+x^3+x^2+7*x-4)/(x^2*ln(1/3*x^2)^2+((2*x^3-2*x^2)*ln(x)-2*x^3)*ln
(1/3*x^2)+(x^4-2*x^3+x^2)*ln(x)^2+(-2*x^4+2*x^3)*ln(x)+x^4),x,method=_RETURNVERBOSE)

[Out]

-2*(x^2+3*x-4)/x/(-I*Pi*csgn(I*x)^2*csgn(I*x^2)+2*I*Pi*csgn(I*x)*csgn(I*x^2)^2-I*Pi*csgn(I*x^2)^3+2*x*ln(x)-2*
ln(3)-2*x+2*ln(x))

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maxima [A]  time = 0.55, size = 28, normalized size = 0.90 \begin {gather*} \frac {x^{2} + 3 \, x - 4}{x^{2} + x \log \relax (3) - {\left (x^{2} + x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-4)*log(1/3*x^2)+(4*x^2-8*x+4)*log(x)+x^3+x^2+7*x-4)/(x^2*log(1/3*x^2)^2+((2*x^3-2*x^2)*log(x)
-2*x^3)*log(1/3*x^2)+(x^4-2*x^3+x^2)*log(x)^2+(-2*x^4+2*x^3)*log(x)+x^4),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.87, size = 101, normalized size = 3.26 \begin {gather*} -\frac {x+\ln \left (\frac {x^2}{3}\right )\,\left (x^3+3\,x^2-4\,x\right )-4\,x^3-x^4-\ln \relax (x)\,\left (2\,x^3+6\,x^2-8\,x\right )+4}{\left (x-x^2\,\left (\ln \left (\frac {x^2}{3}\right )-2\,\ln \relax (x)\right )+x^2+x^3\right )\,\left (x-\ln \left (\frac {x^2}{3}\right )+2\,\ln \relax (x)-\ln \relax (x)\,\left (x+1\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x - log(x^2/3)*(x^2 + 4) + log(x)*(4*x^2 - 8*x + 4) + x^2 + x^3 - 4)/(log(x)*(2*x^3 - 2*x^4) + x^2*log(
x^2/3)^2 - log(x^2/3)*(log(x)*(2*x^2 - 2*x^3) + 2*x^3) + x^4 + log(x)^2*(x^2 - 2*x^3 + x^4)),x)

[Out]

-(x + log(x^2/3)*(3*x^2 - 4*x + x^3) - 4*x^3 - x^4 - log(x)*(6*x^2 - 8*x + 2*x^3) + 4)/((x - x^2*(log(x^2/3) -
2*log(x)) + x^2 + x^3)*(x - log(x^2/3) + 2*log(x) - log(x)*(x + 1)))

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sympy [A]  time = 0.38, size = 24, normalized size = 0.77 \begin {gather*} \frac {- x^{2} - 3 x + 4}{- x^{2} - x \log {\relax (3 )} + \left (x^{2} + x\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2-4)*ln(1/3*x**2)+(4*x**2-8*x+4)*ln(x)+x**3+x**2+7*x-4)/(x**2*ln(1/3*x**2)**2+((2*x**3-2*x**2)
*ln(x)-2*x**3)*ln(1/3*x**2)+(x**4-2*x**3+x**2)*ln(x)**2+(-2*x**4+2*x**3)*ln(x)+x**4),x)

[Out]

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

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