3.3.85 \(\int \frac {(2 x^2+8 x^5) \log (x)+(6-26 x+8 x^2-2 x^4+8 x^5) \log ^3(x)+(6 x-2 x^2-2 x^5+(-6 x+2 x^2+2 x^5) \log (x)) \log (9-6 x+x^2-6 x^4+2 x^5+x^8)}{(-3+x+x^4) \log ^3(x)} \, dx\)

Optimal. Leaf size=25 \[ x (-2+4 x)+\frac {x^2 \log \left (\left (-3+x+x^4\right )^2\right )}{\log ^2(x)} \]

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Rubi [F]  time = 0.96, 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 {\left (2 x^2+8 x^5\right ) \log (x)+\left (6-26 x+8 x^2-2 x^4+8 x^5\right ) \log ^3(x)+\left (6 x-2 x^2-2 x^5+\left (-6 x+2 x^2+2 x^5\right ) \log (x)\right ) \log \left (9-6 x+x^2-6 x^4+2 x^5+x^8\right )}{\left (-3+x+x^4\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((2*x^2 + 8*x^5)*Log[x] + (6 - 26*x + 8*x^2 - 2*x^4 + 8*x^5)*Log[x]^3 + (6*x - 2*x^2 - 2*x^5 + (-6*x + 2*x
^2 + 2*x^5)*Log[x])*Log[9 - 6*x + x^2 - 6*x^4 + 2*x^5 + x^8])/((-3 + x + x^4)*Log[x]^3),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 \left (x^2+4 x^5\right ) \log (x)-\left (6-26 x+8 x^2-2 x^4+8 x^5\right ) \log ^3(x)-2 x \left (-3+x+x^4\right ) (-1+\log (x)) \log \left (\left (-3+x+x^4\right )^2\right )}{\left (3-x-x^4\right ) \log ^3(x)} \, dx\\ &=\int 2 \left (-1+4 x+\frac {x^2+4 x^5}{\left (-3+x+x^4\right ) \log ^2(x)}+\frac {x (-1+\log (x)) \log \left (\left (-3+x+x^4\right )^2\right )}{\log ^3(x)}\right ) \, dx\\ &=2 \int \left (-1+4 x+\frac {x^2+4 x^5}{\left (-3+x+x^4\right ) \log ^2(x)}+\frac {x (-1+\log (x)) \log \left (\left (-3+x+x^4\right )^2\right )}{\log ^3(x)}\right ) \, dx\\ &=-2 x+4 x^2+2 \int \frac {x^2+4 x^5}{\left (-3+x+x^4\right ) \log ^2(x)} \, dx+2 \int \frac {x (-1+\log (x)) \log \left (\left (-3+x+x^4\right )^2\right )}{\log ^3(x)} \, dx\\ &=-2 x+4 x^2+2 \int \frac {x^2 \left (1+4 x^3\right )}{\left (-3+x+x^4\right ) \log ^2(x)} \, dx+2 \int \left (-\frac {x \log \left (\left (-3+x+x^4\right )^2\right )}{\log ^3(x)}+\frac {x \log \left (\left (-3+x+x^4\right )^2\right )}{\log ^2(x)}\right ) \, dx\\ &=-2 x+4 x^2+2 \int \frac {x^2 \left (1+4 x^3\right )}{\left (-3+x+x^4\right ) \log ^2(x)} \, dx-2 \int \frac {x \log \left (\left (-3+x+x^4\right )^2\right )}{\log ^3(x)} \, dx+2 \int \frac {x \log \left (\left (-3+x+x^4\right )^2\right )}{\log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 26, normalized size = 1.04 \begin {gather*} -2 x+4 x^2+\frac {x^2 \log \left (\left (-3+x+x^4\right )^2\right )}{\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2*x^2 + 8*x^5)*Log[x] + (6 - 26*x + 8*x^2 - 2*x^4 + 8*x^5)*Log[x]^3 + (6*x - 2*x^2 - 2*x^5 + (-6*x
+ 2*x^2 + 2*x^5)*Log[x])*Log[9 - 6*x + x^2 - 6*x^4 + 2*x^5 + x^8])/((-3 + x + x^4)*Log[x]^3),x]

[Out]

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

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fricas [A]  time = 0.59, size = 47, normalized size = 1.88 \begin {gather*} \frac {x^{2} \log \left (x^{8} + 2 \, x^{5} - 6 \, x^{4} + x^{2} - 6 \, x + 9\right ) + 2 \, {\left (2 \, x^{2} - x\right )} \log \relax (x)^{2}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5+2*x^2-6*x)*log(x)-2*x^5-2*x^2+6*x)*log(x^8+2*x^5-6*x^4+x^2-6*x+9)+(8*x^5-2*x^4+8*x^2-26*x+6
)*log(x)^3+(8*x^5+2*x^2)*log(x))/(x^4+x-3)/log(x)^3,x, algorithm="fricas")

[Out]

(x^2*log(x^8 + 2*x^5 - 6*x^4 + x^2 - 6*x + 9) + 2*(2*x^2 - x)*log(x)^2)/log(x)^2

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giac [A]  time = 0.55, size = 39, normalized size = 1.56 \begin {gather*} 4 \, x^{2} - 2 \, x + \frac {x^{2} \log \left (x^{8} + 2 \, x^{5} - 6 \, x^{4} + x^{2} - 6 \, x + 9\right )}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5+2*x^2-6*x)*log(x)-2*x^5-2*x^2+6*x)*log(x^8+2*x^5-6*x^4+x^2-6*x+9)+(8*x^5-2*x^4+8*x^2-26*x+6
)*log(x)^3+(8*x^5+2*x^2)*log(x))/(x^4+x-3)/log(x)^3,x, algorithm="giac")

[Out]

4*x^2 - 2*x + x^2*log(x^8 + 2*x^5 - 6*x^4 + x^2 - 6*x + 9)/log(x)^2

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maple [C]  time = 0.09, size = 116, normalized size = 4.64

method	result	size
risch	\(\frac {2 x^{2} \ln \left (x^{4}+x -3\right )}{\ln \relax (x )^{2}}+\frac {x \left (-i \pi x \mathrm {csgn}\left (i \left (x^{4}+x -3\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{4}+x -3\right )^{2}\right )+2 i \pi x \,\mathrm {csgn}\left (i \left (x^{4}+x -3\right )\right ) \mathrm {csgn}\left (i \left (x^{4}+x -3\right )^{2}\right )^{2}-i \pi x \mathrm {csgn}\left (i \left (x^{4}+x -3\right )^{2}\right )^{3}+8 x \ln \relax (x )^{2}-4 \ln \relax (x )^{2}\right )}{2 \ln \relax (x )^{2}}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^5+2*x^2-6*x)*ln(x)-2*x^5-2*x^2+6*x)*ln(x^8+2*x^5-6*x^4+x^2-6*x+9)+(8*x^5-2*x^4+8*x^2-26*x+6)*ln(x)^
3+(8*x^5+2*x^2)*ln(x))/(x^4+x-3)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

2*x^2/ln(x)^2*ln(x^4+x-3)+1/2*x*(-I*Pi*x*csgn(I*(x^4+x-3))^2*csgn(I*(x^4+x-3)^2)+2*I*Pi*x*csgn(I*(x^4+x-3))*cs
gn(I*(x^4+x-3)^2)^2-I*Pi*x*csgn(I*(x^4+x-3)^2)^3+8*x*ln(x)^2-4*ln(x)^2)/ln(x)^2

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maxima [A]  time = 0.56, size = 32, normalized size = 1.28 \begin {gather*} \frac {2 \, {\left (x^{2} \log \left (x^{4} + x - 3\right ) + {\left (2 \, x^{2} - x\right )} \log \relax (x)^{2}\right )}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5+2*x^2-6*x)*log(x)-2*x^5-2*x^2+6*x)*log(x^8+2*x^5-6*x^4+x^2-6*x+9)+(8*x^5-2*x^4+8*x^2-26*x+6
)*log(x)^3+(8*x^5+2*x^2)*log(x))/(x^4+x-3)/log(x)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.48, size = 39, normalized size = 1.56 \begin {gather*} 4\,x^2-2\,x+\frac {x^2\,\ln \left (x^8+2\,x^5-6\,x^4+x^2-6\,x+9\right )}{{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(2*x^2 + 8*x^5) + log(x)^3*(8*x^2 - 26*x - 2*x^4 + 8*x^5 + 6) + log(x^2 - 6*x - 6*x^4 + 2*x^5 + x^
8 + 9)*(6*x - 2*x^2 - 2*x^5 + log(x)*(2*x^2 - 6*x + 2*x^5)))/(log(x)^3*(x + x^4 - 3)),x)

[Out]

4*x^2 - 2*x + (x^2*log(x^2 - 6*x - 6*x^4 + 2*x^5 + x^8 + 9))/log(x)^2

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sympy [A]  time = 0.56, size = 39, normalized size = 1.56 \begin {gather*} 4 x^{2} + \frac {x^{2} \log {\left (x^{8} + 2 x^{5} - 6 x^{4} + x^{2} - 6 x + 9 \right )}}{\log {\relax (x )}^{2}} - 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**5+2*x**2-6*x)*ln(x)-2*x**5-2*x**2+6*x)*ln(x**8+2*x**5-6*x**4+x**2-6*x+9)+(8*x**5-2*x**4+8*x*
*2-26*x+6)*ln(x)**3+(8*x**5+2*x**2)*ln(x))/(x**4+x-3)/ln(x)**3,x)

[Out]

4*x**2 + x**2*log(x**8 + 2*x**5 - 6*x**4 + x**2 - 6*x + 9)/log(x)**2 - 2*x

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