3.75.61 \(\int \frac {1-81 x^2+162 x^3+(-1-x^2) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+(-2 x-160 x^2+164 x^3+160 x^4-162 x^5) \log (x)+(1-2 x^2+x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ \frac {x}{(1-x) \left (x+81 x^2-\log (x)-x \log (x)\right )} \]

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Rubi [F]  time = 1.30, 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 {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 - 81*x^2 + 162*x^3 + (-1 - x^2)*Log[x])/(x^2 + 160*x^3 + 6238*x^4 - 12960*x^5 + 6561*x^6 + (-2*x - 160*
x^2 + 164*x^3 + 160*x^4 - 162*x^5)*Log[x] + (1 - 2*x^2 + x^4)*Log[x]^2),x]

[Out]

161*Defer[Int][(x + 81*x^2 - Log[x] - x*Log[x])^(-2), x] + 120*Defer[Int][1/((-1 + x)*(x + 81*x^2 - Log[x] - x
*Log[x])^2), x] + 81*Defer[Int][x/(x + 81*x^2 - Log[x] - x*Log[x])^2, x] - 40*Defer[Int][1/((1 + x)*(x + 81*x^
2 - Log[x] - x*Log[x])^2), x] + Defer[Int][1/((-1 + x)^2*(x + 81*x^2 - Log[x] - x*Log[x])), x] + Defer[Int][1/
((-1 + x)*(x + 81*x^2 - Log[x] - x*Log[x])), x]/2 + Defer[Int][1/((1 + x)*(x + 81*x^2 - Log[x] - x*Log[x])), x
]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1-81 x^2+162 x^3-\left (1+x^2\right ) \log (x)}{(1-x)^2 (x (1+81 x)-(1+x) \log (x))^2} \, dx\\ &=\int \left (\frac {-1-x+161 x^2+81 x^3}{\left (-1+x^2\right ) \left (x+81 x^2-\log (x)-x \log (x)\right )^2}+\frac {1+x^2}{(-1+x)^2 (1+x) \left (x+81 x^2-\log (x)-x \log (x)\right )}\right ) \, dx\\ &=\int \frac {-1-x+161 x^2+81 x^3}{\left (-1+x^2\right ) \left (x+81 x^2-\log (x)-x \log (x)\right )^2} \, dx+\int \frac {1+x^2}{(-1+x)^2 (1+x) \left (x+81 x^2-\log (x)-x \log (x)\right )} \, dx\\ &=\int \left (\frac {161}{\left (x+81 x^2-\log (x)-x \log (x)\right )^2}+\frac {120}{(-1+x) \left (x+81 x^2-\log (x)-x \log (x)\right )^2}+\frac {81 x}{\left (x+81 x^2-\log (x)-x \log (x)\right )^2}-\frac {40}{(1+x) \left (x+81 x^2-\log (x)-x \log (x)\right )^2}\right ) \, dx+\int \left (\frac {1}{(-1+x)^2 \left (x+81 x^2-\log (x)-x \log (x)\right )}+\frac {1}{2 (-1+x) \left (x+81 x^2-\log (x)-x \log (x)\right )}+\frac {1}{2 (1+x) \left (x+81 x^2-\log (x)-x \log (x)\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {1}{(-1+x) \left (x+81 x^2-\log (x)-x \log (x)\right )} \, dx+\frac {1}{2} \int \frac {1}{(1+x) \left (x+81 x^2-\log (x)-x \log (x)\right )} \, dx-40 \int \frac {1}{(1+x) \left (x+81 x^2-\log (x)-x \log (x)\right )^2} \, dx+81 \int \frac {x}{\left (x+81 x^2-\log (x)-x \log (x)\right )^2} \, dx+120 \int \frac {1}{(-1+x) \left (x+81 x^2-\log (x)-x \log (x)\right )^2} \, dx+161 \int \frac {1}{\left (x+81 x^2-\log (x)-x \log (x)\right )^2} \, dx+\int \frac {1}{(-1+x)^2 \left (x+81 x^2-\log (x)-x \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.41, size = 24, normalized size = 0.89 \begin {gather*} \frac {x}{(-1+x) \left (-x-81 x^2+\log (x)+x \log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 81*x^2 + 162*x^3 + (-1 - x^2)*Log[x])/(x^2 + 160*x^3 + 6238*x^4 - 12960*x^5 + 6561*x^6 + (-2*x
- 160*x^2 + 164*x^3 + 160*x^4 - 162*x^5)*Log[x] + (1 - 2*x^2 + x^4)*Log[x]^2),x]

[Out]

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

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fricas [A]  time = 0.64, size = 28, normalized size = 1.04 \begin {gather*} -\frac {x}{81 \, x^{3} - 80 \, x^{2} - {\left (x^{2} - 1\right )} \log \relax (x) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-1)*log(x)+162*x^3-81*x^2+1)/((x^4-2*x^2+1)*log(x)^2+(-162*x^5+160*x^4+164*x^3-160*x^2-2*x)*lo
g(x)+6561*x^6-12960*x^5+6238*x^4+160*x^3+x^2),x, algorithm="fricas")

[Out]

-x/(81*x^3 - 80*x^2 - (x^2 - 1)*log(x) - x)

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giac [A]  time = 0.17, size = 28, normalized size = 1.04 \begin {gather*} -\frac {x}{81 \, x^{3} - x^{2} \log \relax (x) - 80 \, x^{2} - x + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-1)*log(x)+162*x^3-81*x^2+1)/((x^4-2*x^2+1)*log(x)^2+(-162*x^5+160*x^4+164*x^3-160*x^2-2*x)*lo
g(x)+6561*x^6-12960*x^5+6238*x^4+160*x^3+x^2),x, algorithm="giac")

[Out]

-x/(81*x^3 - x^2*log(x) - 80*x^2 - x + log(x))

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maple [A]  time = 0.08, size = 27, normalized size = 1.00

method	result	size
risch	\(-\frac {x}{\left (x -1\right ) \left (81 x^{2}-\ln \relax (x )-x \ln \relax (x )+x \right )}\)	\(27\)
norman	\(-\frac {x}{81 x^{3}-x^{2} \ln \relax (x )-80 x^{2}-x +\ln \relax (x )}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2-1)*ln(x)+162*x^3-81*x^2+1)/((x^4-2*x^2+1)*ln(x)^2+(-162*x^5+160*x^4+164*x^3-160*x^2-2*x)*ln(x)+6561
*x^6-12960*x^5+6238*x^4+160*x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

-x/(x-1)/(81*x^2-ln(x)-x*ln(x)+x)

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maxima [A]  time = 0.41, size = 28, normalized size = 1.04 \begin {gather*} -\frac {x}{81 \, x^{3} - 80 \, x^{2} - {\left (x^{2} - 1\right )} \log \relax (x) - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-1)*log(x)+162*x^3-81*x^2+1)/((x^4-2*x^2+1)*log(x)^2+(-162*x^5+160*x^4+164*x^3-160*x^2-2*x)*lo
g(x)+6561*x^6-12960*x^5+6238*x^4+160*x^3+x^2),x, algorithm="maxima")

[Out]

-x/(81*x^3 - 80*x^2 - (x^2 - 1)*log(x) - x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \relax (x)\,\left (x^2+1\right )+81\,x^2-162\,x^3-1}{{\ln \relax (x)}^2\,\left (x^4-2\,x^2+1\right )-\ln \relax (x)\,\left (162\,x^5-160\,x^4-164\,x^3+160\,x^2+2\,x\right )+x^2+160\,x^3+6238\,x^4-12960\,x^5+6561\,x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(x^2 + 1) + 81*x^2 - 162*x^3 - 1)/(log(x)^2*(x^4 - 2*x^2 + 1) - log(x)*(2*x + 160*x^2 - 164*x^3 -
160*x^4 + 162*x^5) + x^2 + 160*x^3 + 6238*x^4 - 12960*x^5 + 6561*x^6),x)

[Out]

int(-(log(x)*(x^2 + 1) + 81*x^2 - 162*x^3 - 1)/(log(x)^2*(x^4 - 2*x^2 + 1) - log(x)*(2*x + 160*x^2 - 164*x^3 -
160*x^4 + 162*x^5) + x^2 + 160*x^3 + 6238*x^4 - 12960*x^5 + 6561*x^6), x)

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sympy [A]  time = 0.22, size = 20, normalized size = 0.74 \begin {gather*} \frac {x}{- 81 x^{3} + 80 x^{2} + x + \left (x^{2} - 1\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2-1)*ln(x)+162*x**3-81*x**2+1)/((x**4-2*x**2+1)*ln(x)**2+(-162*x**5+160*x**4+164*x**3-160*x**2
-2*x)*ln(x)+6561*x**6-12960*x**5+6238*x**4+160*x**3+x**2),x)

[Out]

x/(-81*x**3 + 80*x**2 + x + (x**2 - 1)*log(x))

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