3.31.23 \(\int \frac {-5 \log (3)+(x-12 x^6-9 x^9) \log ^2(x)}{-5 x \log (3) \log (x)+(-x^2+2 x^7+x^{10}) \log ^2(x)} \, dx\)

Optimal. Leaf size=35 \[ \log \left (\frac {4}{\frac {x-x \left (-x^2+\left (x+x^4\right )^2\right )}{\log (3)}+\frac {5}{\log (x)}}\right ) \]

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Rubi [F]  time = 2.69, 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 {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5*Log[3] + (x - 12*x^6 - 9*x^9)*Log[x]^2)/(-5*x*Log[3]*Log[x] + (-x^2 + 2*x^7 + x^10)*Log[x]^2),x]

[Out]

-Log[-(x*(1 - 2*x^5 - x^8))] + Log[Log[x]] - Defer[Int][(5*Log[3] + x*Log[x] - 2*x^6*Log[x] - x^9*Log[x])^(-1)
, x] - 5*Log[3]*Defer[Int][1/(x*(-5*Log[3] - x*Log[x] + 2*x^6*Log[x] + x^9*Log[x])), x] - 2*Defer[Int][x^5/(-5
*Log[3] - x*Log[x] + 2*x^6*Log[x] + x^9*Log[x]), x] - Defer[Int][x^8/(-5*Log[3] - x*Log[x] + 2*x^6*Log[x] + x^
9*Log[x]), x] - 50*Log[3]*Defer[Int][x^4/((-1 + 2*x^5 + x^8)*(-5*Log[3] - x*Log[x] + 2*x^6*Log[x] + x^9*Log[x]
)), x] - 40*Log[3]*Defer[Int][x^7/((-1 + 2*x^5 + x^8)*(-5*Log[3] - x*Log[x] + 2*x^6*Log[x] + x^9*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \log (3)-\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{x \log (x) \left (5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)\right )} \, dx\\ &=\int \left (\frac {1-12 x^5-9 x^8}{x \left (-1+2 x^5+x^8\right )}+\frac {1}{x \log (x)}+\frac {1-2 x^5-x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)}-\frac {5 \left (-1+12 x^5+9 x^8\right ) \log (3)}{x \left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )}\right ) \, dx\\ &=-\left ((5 \log (3)) \int \frac {-1+12 x^5+9 x^8}{x \left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx\right )+\int \frac {1-12 x^5-9 x^8}{x \left (-1+2 x^5+x^8\right )} \, dx+\int \frac {1}{x \log (x)} \, dx+\int \frac {1-2 x^5-x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx\\ &=-\log \left (-x \left (1-2 x^5-x^8\right )\right )-(5 \log (3)) \int \left (\frac {1}{x \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )}+\frac {2 x^4 \left (5+4 x^3\right )}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )}\right ) \, dx+\int \left (-\frac {1}{5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)}-\frac {2 x^5}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)}-\frac {x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-\log \left (-x \left (1-2 x^5-x^8\right )\right )+\log (\log (x))-2 \int \frac {x^5}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx-(5 \log (3)) \int \frac {1}{x \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-(10 \log (3)) \int \frac {x^4 \left (5+4 x^3\right )}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-\int \frac {1}{5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)} \, dx-\int \frac {x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx\\ &=-\log \left (-x \left (1-2 x^5-x^8\right )\right )+\log (\log (x))-2 \int \frac {x^5}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx-(5 \log (3)) \int \frac {1}{x \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-(10 \log (3)) \int \left (\frac {5 x^4}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )}+\frac {4 x^7}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )}\right ) \, dx-\int \frac {1}{5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)} \, dx-\int \frac {x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx\\ &=-\log \left (-x \left (1-2 x^5-x^8\right )\right )+\log (\log (x))-2 \int \frac {x^5}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx-(5 \log (3)) \int \frac {1}{x \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-(40 \log (3)) \int \frac {x^7}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-(50 \log (3)) \int \frac {x^4}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-\int \frac {1}{5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)} \, dx-\int \frac {x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 64, normalized size = 1.83 \begin {gather*} -\log (x)-\log \left (1-2 x^5-x^8\right )+\log \left (x \left (1-2 x^5-x^8\right )\right )+\log (\log (x))-\log \left (5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*Log[3] + (x - 12*x^6 - 9*x^9)*Log[x]^2)/(-5*x*Log[3]*Log[x] + (-x^2 + 2*x^7 + x^10)*Log[x]^2),x]

[Out]

-Log[x] - Log[1 - 2*x^5 - x^8] + Log[x*(1 - 2*x^5 - x^8)] + Log[Log[x]] - Log[5*Log[3] + x*Log[x] - 2*x^6*Log[
x] - x^9*Log[x]]

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fricas [A]  time = 0.65, size = 57, normalized size = 1.63 \begin {gather*} -\log \left (x^{9} + 2 \, x^{6} - x\right ) - \log \left (\frac {{\left (x^{9} + 2 \, x^{6} - x\right )} \log \relax (x) - 5 \, \log \relax (3)}{x^{9} + 2 \, x^{6} - x}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^9-12*x^6+x)*log(x)^2-5*log(3))/((x^10+2*x^7-x^2)*log(x)^2-5*x*log(3)*log(x)),x, algorithm="fr
icas")

[Out]

-log(x^9 + 2*x^6 - x) - log(((x^9 + 2*x^6 - x)*log(x) - 5*log(3))/(x^9 + 2*x^6 - x)) + log(log(x))

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giac [A]  time = 0.23, size = 30, normalized size = 0.86 \begin {gather*} -\log \left (-x^{9} \log \relax (x) - 2 \, x^{6} \log \relax (x) + x \log \relax (x) + 5 \, \log \relax (3)\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^9-12*x^6+x)*log(x)^2-5*log(3))/((x^10+2*x^7-x^2)*log(x)^2-5*x*log(3)*log(x)),x, algorithm="gi
ac")

[Out]

-log(-x^9*log(x) - 2*x^6*log(x) + x*log(x) + 5*log(3)) + log(log(x))

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maple [A]  time = 0.04, size = 45, normalized size = 1.29

method	result	size
risch	\(-\ln \left (x^{9}+2 x^{6}-x \right )+\ln \left (\ln \relax (x )\right )-\ln \left (\ln \relax (x )-\frac {5 \ln \relax (3)}{x \left (x^{8}+2 x^{5}-1\right )}\right )\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-9*x^9-12*x^6+x)*ln(x)^2-5*ln(3))/((x^10+2*x^7-x^2)*ln(x)^2-5*x*ln(3)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-ln(x^9+2*x^6-x)+ln(ln(x))-ln(ln(x)-5*ln(3)/x/(x^8+2*x^5-1))

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maxima [A]  time = 0.97, size = 59, normalized size = 1.69 \begin {gather*} -\log \left (x^{8} + 2 \, x^{5} - 1\right ) - \log \relax (x) - \log \left (\frac {{\left (x^{9} + 2 \, x^{6} - x\right )} \log \relax (x) - 5 \, \log \relax (3)}{x^{9} + 2 \, x^{6} - x}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^9-12*x^6+x)*log(x)^2-5*log(3))/((x^10+2*x^7-x^2)*log(x)^2-5*x*log(3)*log(x)),x, algorithm="ma
xima")

[Out]

-log(x^8 + 2*x^5 - 1) - log(x) - log(((x^9 + 2*x^6 - x)*log(x) - 5*log(3))/(x^9 + 2*x^6 - x)) + log(log(x))

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mupad [B]  time = 26.46, size = 186, normalized size = 5.31 \begin {gather*} \ln \left (8\,x^6\,\ln \relax (x)+4\,x^9\,\ln \relax (x)-8\,x^{11}\,\ln \relax (x)-8\,x^{14}\,\ln \relax (x)-2\,x^{17}\,\ln \relax (x)+10\,\ln \relax (3)\,\ln \relax (x)-2\,x\,\ln \relax (x)-120\,x^5\,\ln \relax (3)\,\ln \relax (x)-90\,x^8\,\ln \relax (3)\,\ln \relax (x)\right )-\ln \left (10\,\ln \relax (3)-8\,x^6\,\ln \relax (x)-4\,x^9\,\ln \relax (x)+8\,x^{11}\,\ln \relax (x)+8\,x^{14}\,\ln \relax (x)+2\,x^{17}\,\ln \relax (x)-20\,x^5\,\ln \relax (3)-10\,x^8\,\ln \relax (3)+2\,x\,\ln \relax (x)\right )-\ln \left (x^{17}+4\,x^{14}+4\,x^{11}-2\,x^9+45\,\ln \relax (3)\,x^8-4\,x^6+60\,\ln \relax (3)\,x^5+x-5\,\ln \relax (3)\right )+\ln \left (x^8+2\,x^5-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*log(3) + log(x)^2*(12*x^6 - x + 9*x^9))/(log(x)^2*(2*x^7 - x^2 + x^10) - 5*x*log(3)*log(x)),x)

[Out]

log(8*x^6*log(x) + 4*x^9*log(x) - 8*x^11*log(x) - 8*x^14*log(x) - 2*x^17*log(x) + 10*log(3)*log(x) - 2*x*log(x
) - 120*x^5*log(3)*log(x) - 90*x^8*log(3)*log(x)) - log(10*log(3) - 8*x^6*log(x) - 4*x^9*log(x) + 8*x^11*log(x
) + 8*x^14*log(x) + 2*x^17*log(x) - 20*x^5*log(3) - 10*x^8*log(3) + 2*x*log(x)) - log(x - 5*log(3) + 60*x^5*lo
g(3) + 45*x^8*log(3) - 4*x^6 - 2*x^9 + 4*x^11 + 4*x^14 + x^17) + log(2*x^5 + x^8 - 1)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x**9-12*x**6+x)*ln(x)**2-5*ln(3))/((x**10+2*x**7-x**2)*ln(x)**2-5*x*ln(3)*ln(x)),x)

[Out]

Exception raised: PolynomialError

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