3.44.55 \(\int \frac {4+8 x+14 x^2+24 x^3+36 x^4+32 x^5+18 x^6+6 x^7+x^8}{-2 x-2 x^2-x^3+(8 x^3+24 x^4+36 x^5+32 x^6+18 x^7+6 x^8+x^9) \log (x)} \, dx\)

Optimal. Leaf size=19 \[ \log \left (-\frac {1}{x^2 \left (2+2 x+x^2\right )^2}+\log (x)\right ) \]

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Rubi [F]  time = 3.66, 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+8 x+14 x^2+24 x^3+36 x^4+32 x^5+18 x^6+6 x^7+x^8}{-2 x-2 x^2-x^3+\left (8 x^3+24 x^4+36 x^5+32 x^6+18 x^7+6 x^8+x^9\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 + 8*x + 14*x^2 + 24*x^3 + 36*x^4 + 32*x^5 + 18*x^6 + 6*x^7 + x^8)/(-2*x - 2*x^2 - x^3 + (8*x^3 + 24*x^4
+ 36*x^5 + 32*x^6 + 18*x^7 + 6*x^8 + x^9)*Log[x]),x]

[Out]

(4 - 4*I)*Defer[Int][1/(((-2 - 2*I) - 2*x)*(1 - x^2*(2 + 2*x + x^2)^2*Log[x])), x] + 4*Defer[Int][1/(((-2 + 2*
I) - 2*x)*(1 - x^2*(2 + 2*x + x^2)^2*Log[x])), x] - (4*I)*Defer[Int][1/(((2 + 2*I) + 2*x)*(1 - x^2*(2 + 2*x +
x^2)^2*Log[x])), x] + 2*Defer[Int][1/(x*(-1 + x^2*(2 + 2*x + x^2)^2*Log[x])), x] + 4*Defer[Int][x/(-1 + x^2*(2
+ 2*x + x^2)^2*Log[x]), x] + 8*Defer[Int][x^2/(-1 + x^2*(2 + 2*x + x^2)^2*Log[x]), x] + 8*Defer[Int][x^3/(-1
+ x^2*(2 + 2*x + x^2)^2*Log[x]), x] + 4*Defer[Int][x^4/(-1 + x^2*(2 + 2*x + x^2)^2*Log[x]), x] + Defer[Int][x^
5/(-1 + x^2*(2 + 2*x + x^2)^2*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4-8 x-14 x^2-24 x^3-36 x^4-32 x^5-18 x^6-6 x^7-x^8}{x \left (2+2 x+x^2\right ) \left (1-4 x^2 \log (x)-8 x^3 \log (x)-8 x^4 \log (x)-4 x^5 \log (x)-x^6 \log (x)\right )} \, dx\\ &=\int \frac {-4-8 x-14 x^2-24 x^3-36 x^4-32 x^5-18 x^6-6 x^7-x^8}{x \left (2+2 x+x^2\right ) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx\\ &=\int \left (\frac {2}{x \left (-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)\right )}+\frac {4 x}{-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)}+\frac {8 x^2}{-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)}+\frac {8 x^3}{-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)}+\frac {4 x^4}{-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)}+\frac {x^5}{-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)}+\frac {4 (1+x)}{\left (2+2 x+x^2\right ) \left (-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)\right )}\right ) \, dx\\ &=2 \int \frac {1}{x \left (-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)\right )} \, dx+4 \int \frac {x}{-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)} \, dx+4 \int \frac {x^4}{-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)} \, dx+4 \int \frac {1+x}{\left (2+2 x+x^2\right ) \left (-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)\right )} \, dx+8 \int \frac {x^2}{-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)} \, dx+8 \int \frac {x^3}{-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)} \, dx+\int \frac {x^5}{-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)} \, dx\\ &=2 \int \frac {1}{x \left (-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+4 \int \frac {-1-x}{\left (2+2 x+x^2\right ) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+4 \int \frac {x}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+4 \int \frac {x^4}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+8 \int \frac {x^2}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+8 \int \frac {x^3}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+\int \frac {x^5}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx\\ &=2 \int \frac {1}{x \left (-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+4 \int \frac {x}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+4 \int \frac {x^4}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+4 \int \left (\frac {1}{\left (2+2 x+x^2\right ) \left (-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)\right )}+\frac {x}{\left (2+2 x+x^2\right ) \left (-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)\right )}\right ) \, dx+8 \int \frac {x^2}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+8 \int \frac {x^3}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+\int \frac {x^5}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx\\ &=2 \int \frac {1}{x \left (-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+4 \int \frac {1}{\left (2+2 x+x^2\right ) \left (-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)\right )} \, dx+4 \int \frac {x}{\left (2+2 x+x^2\right ) \left (-1+4 x^2 \log (x)+8 x^3 \log (x)+8 x^4 \log (x)+4 x^5 \log (x)+x^6 \log (x)\right )} \, dx+4 \int \frac {x}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+4 \int \frac {x^4}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+8 \int \frac {x^2}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+8 \int \frac {x^3}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+\int \frac {x^5}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx\\ &=2 \int \frac {1}{x \left (-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+4 \int \frac {1}{\left (-2-2 x-x^2\right ) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+4 \int \frac {x}{\left (-2-2 x-x^2\right ) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+4 \int \frac {x}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+4 \int \frac {x^4}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+8 \int \frac {x^2}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+8 \int \frac {x^3}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+\int \frac {x^5}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx\\ &=2 \int \frac {1}{x \left (-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+4 \int \frac {x}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+4 \int \frac {x^4}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+4 \int \left (\frac {1-i}{((-2-2 i)-2 x) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )}+\frac {1+i}{((-2+2 i)-2 x) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )}\right ) \, dx+4 \int \left (-\frac {i}{((-2+2 i)-2 x) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )}-\frac {i}{((2+2 i)+2 x) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )}\right ) \, dx+8 \int \frac {x^2}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+8 \int \frac {x^3}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+\int \frac {x^5}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx\\ &=-\left (4 i \int \frac {1}{((-2+2 i)-2 x) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx\right )-4 i \int \frac {1}{((2+2 i)+2 x) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+2 \int \frac {1}{x \left (-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+4 \int \frac {x}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+4 \int \frac {x^4}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+(4-4 i) \int \frac {1}{((-2-2 i)-2 x) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+(4+4 i) \int \frac {1}{((-2+2 i)-2 x) \left (1-x^2 \left (2+2 x+x^2\right )^2 \log (x)\right )} \, dx+8 \int \frac {x^2}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+8 \int \frac {x^3}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx+\int \frac {x^5}{-1+x^2 \left (2+2 x+x^2\right )^2 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.74, size = 54, normalized size = 2.84 \begin {gather*} -2 \log (x)-2 \log \left (2+2 x+x^2\right )+\log \left (1-4 x^2 \log (x)-8 x^3 \log (x)-8 x^4 \log (x)-4 x^5 \log (x)-x^6 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 8*x + 14*x^2 + 24*x^3 + 36*x^4 + 32*x^5 + 18*x^6 + 6*x^7 + x^8)/(-2*x - 2*x^2 - x^3 + (8*x^3 +
24*x^4 + 36*x^5 + 32*x^6 + 18*x^7 + 6*x^8 + x^9)*Log[x]),x]

[Out]

-2*Log[x] - 2*Log[2 + 2*x + x^2] + Log[1 - 4*x^2*Log[x] - 8*x^3*Log[x] - 8*x^4*Log[x] - 4*x^5*Log[x] - x^6*Log
[x]]

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fricas [B]  time = 0.74, size = 57, normalized size = 3.00 \begin {gather*} \log \left (\frac {{\left (x^{6} + 4 \, x^{5} + 8 \, x^{4} + 8 \, x^{3} + 4 \, x^{2}\right )} \log \relax (x) - 1}{x^{6} + 4 \, x^{5} + 8 \, x^{4} + 8 \, x^{3} + 4 \, x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8+6*x^7+18*x^6+32*x^5+36*x^4+24*x^3+14*x^2+8*x+4)/((x^9+6*x^8+18*x^7+32*x^6+36*x^5+24*x^4+8*x^3)*
log(x)-x^3-2*x^2-2*x),x, algorithm="fricas")

[Out]

log(((x^6 + 4*x^5 + 8*x^4 + 8*x^3 + 4*x^2)*log(x) - 1)/(x^6 + 4*x^5 + 8*x^4 + 8*x^3 + 4*x^2))

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giac [B]  time = 0.19, size = 53, normalized size = 2.79 \begin {gather*} \log \left (x^{6} \log \relax (x) + 4 \, x^{5} \log \relax (x) + 8 \, x^{4} \log \relax (x) + 8 \, x^{3} \log \relax (x) + 4 \, x^{2} \log \relax (x) - 1\right ) - 2 \, \log \left (x^{2} + 2 \, x + 2\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8+6*x^7+18*x^6+32*x^5+36*x^4+24*x^3+14*x^2+8*x+4)/((x^9+6*x^8+18*x^7+32*x^6+36*x^5+24*x^4+8*x^3)*
log(x)-x^3-2*x^2-2*x),x, algorithm="giac")

[Out]

log(x^6*log(x) + 4*x^5*log(x) + 8*x^4*log(x) + 8*x^3*log(x) + 4*x^2*log(x) - 1) - 2*log(x^2 + 2*x + 2) - 2*log
(x)

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maple [A]  time = 0.07, size = 30, normalized size = 1.58

method	result	size
risch	\(\ln \left (\ln \relax (x )-\frac {1}{x^{2} \left (x^{4}+4 x^{3}+8 x^{2}+8 x +4\right )}\right )\)	\(30\)
norman	\(-2 \ln \relax (x )-2 \ln \left (x^{2}+2 x +2\right )+\ln \left (x^{6} \ln \relax (x )+4 x^{5} \ln \relax (x )+8 x^{4} \ln \relax (x )+8 x^{3} \ln \relax (x )+4 x^{2} \ln \relax (x )-1\right )\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8+6*x^7+18*x^6+32*x^5+36*x^4+24*x^3+14*x^2+8*x+4)/((x^9+6*x^8+18*x^7+32*x^6+36*x^5+24*x^4+8*x^3)*ln(x)-
x^3-2*x^2-2*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.38, size = 57, normalized size = 3.00 \begin {gather*} \log \left (\frac {{\left (x^{6} + 4 \, x^{5} + 8 \, x^{4} + 8 \, x^{3} + 4 \, x^{2}\right )} \log \relax (x) - 1}{x^{6} + 4 \, x^{5} + 8 \, x^{4} + 8 \, x^{3} + 4 \, x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8+6*x^7+18*x^6+32*x^5+36*x^4+24*x^3+14*x^2+8*x+4)/((x^9+6*x^8+18*x^7+32*x^6+36*x^5+24*x^4+8*x^3)*
log(x)-x^3-2*x^2-2*x),x, algorithm="maxima")

[Out]

log(((x^6 + 4*x^5 + 8*x^4 + 8*x^3 + 4*x^2)*log(x) - 1)/(x^6 + 4*x^5 + 8*x^4 + 8*x^3 + 4*x^2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} -\int \frac {x^8+6\,x^7+18\,x^6+32\,x^5+36\,x^4+24\,x^3+14\,x^2+8\,x+4}{2\,x-\ln \relax (x)\,\left (x^9+6\,x^8+18\,x^7+32\,x^6+36\,x^5+24\,x^4+8\,x^3\right )+2\,x^2+x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x + 14*x^2 + 24*x^3 + 36*x^4 + 32*x^5 + 18*x^6 + 6*x^7 + x^8 + 4)/(2*x - log(x)*(8*x^3 + 24*x^4 + 36*x
^5 + 32*x^6 + 18*x^7 + 6*x^8 + x^9) + 2*x^2 + x^3),x)

[Out]

-int((8*x + 14*x^2 + 24*x^3 + 36*x^4 + 32*x^5 + 18*x^6 + 6*x^7 + x^8 + 4)/(2*x - log(x)*(8*x^3 + 24*x^4 + 36*x
^5 + 32*x^6 + 18*x^7 + 6*x^8 + x^9) + 2*x^2 + x^3), x)

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sympy [A]  time = 0.64, size = 29, normalized size = 1.53 \begin {gather*} \log {\left (\log {\relax (x )} - \frac {1}{x^{6} + 4 x^{5} + 8 x^{4} + 8 x^{3} + 4 x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**8+6*x**7+18*x**6+32*x**5+36*x**4+24*x**3+14*x**2+8*x+4)/((x**9+6*x**8+18*x**7+32*x**6+36*x**5+24
*x**4+8*x**3)*ln(x)-x**3-2*x**2-2*x),x)

[Out]

log(log(x) - 1/(x**6 + 4*x**5 + 8*x**4 + 8*x**3 + 4*x**2))

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