3.35.91 \(\int \frac {9+x+(-27-4 x) \log (x)+(-3+5 x) \log ^2(x)+(6-2 x) \log ^3(x)}{(9 x+x^2) \log (x)+(-3 x-2 x^2) \log ^2(x)+(-6 x+x^2) \log ^3(x)} \, dx\)

Optimal. Leaf size=27 \[ \log \left (\frac {4 \log (x)}{x \left (-2+\frac {x}{3}+\frac {5}{1-\log (x)}\right )}\right ) \]

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Rubi [F]  time = 1.76, 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 {9+x+(-27-4 x) \log (x)+(-3+5 x) \log ^2(x)+(6-2 x) \log ^3(x)}{\left (9 x+x^2\right ) \log (x)+\left (-3 x-2 x^2\right ) \log ^2(x)+\left (-6 x+x^2\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 + x + (-27 - 4*x)*Log[x] + (-3 + 5*x)*Log[x]^2 + (6 - 2*x)*Log[x]^3)/((9*x + x^2)*Log[x] + (-3*x - 2*x^
2)*Log[x]^2 + (-6*x + x^2)*Log[x]^3),x]

[Out]

-Log[6 - x] - Log[x] + Log[1 - Log[x]] + Log[Log[x]] - Defer[Int][(-9 - x - 6*Log[x] + x*Log[x])^(-1), x] - 15
*Defer[Int][1/((-6 + x)*(-9 - x - 6*Log[x] + x*Log[x])), x] + 6*Defer[Int][1/(x*(-9 - x - 6*Log[x] + x*Log[x])
), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9+x+(-27-4 x) \log (x)+(-3+5 x) \log ^2(x)+(6-2 x) \log ^3(x)}{x (1-\log (x)) \log (x) (9+x+6 \log (x)-x \log (x))} \, dx\\ &=\int \left (-\frac {2 (-3+x)}{(-6+x) x}+\frac {1}{x (-1+\log (x))}+\frac {1}{x \log (x)}+\frac {-36-3 x-x^2}{(-6+x) x (-9-x-6 \log (x)+x \log (x))}\right ) \, dx\\ &=-\left (2 \int \frac {-3+x}{(-6+x) x} \, dx\right )+\int \frac {1}{x (-1+\log (x))} \, dx+\int \frac {1}{x \log (x)} \, dx+\int \frac {-36-3 x-x^2}{(-6+x) x (-9-x-6 \log (x)+x \log (x))} \, dx\\ &=-\left (2 \int \left (\frac {1}{2 (-6+x)}+\frac {1}{2 x}\right ) \, dx\right )+\int \left (-\frac {1}{-9-x-6 \log (x)+x \log (x)}-\frac {15}{(-6+x) (-9-x-6 \log (x)+x \log (x))}+\frac {6}{x (-9-x-6 \log (x)+x \log (x))}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,-1+\log (x)\right )+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-\log (6-x)-\log (x)+\log (1-\log (x))+\log (\log (x))+6 \int \frac {1}{x (-9-x-6 \log (x)+x \log (x))} \, dx-15 \int \frac {1}{(-6+x) (-9-x-6 \log (x)+x \log (x))} \, dx-\int \frac {1}{-9-x-6 \log (x)+x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 30, normalized size = 1.11 \begin {gather*} -\log (x)+\log (1-\log (x))+\log (\log (x))-\log (9+x+6 \log (x)-x \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 + x + (-27 - 4*x)*Log[x] + (-3 + 5*x)*Log[x]^2 + (6 - 2*x)*Log[x]^3)/((9*x + x^2)*Log[x] + (-3*x
- 2*x^2)*Log[x]^2 + (-6*x + x^2)*Log[x]^3),x]

[Out]

-Log[x] + Log[1 - Log[x]] + Log[Log[x]] - Log[9 + x + 6*Log[x] - x*Log[x]]

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fricas [A]  time = 0.94, size = 39, normalized size = 1.44 \begin {gather*} -\log \left (x^{2} - 6 \, x\right ) - \log \left (\frac {{\left (x - 6\right )} \log \relax (x) - x - 9}{x - 6}\right ) + \log \left (\log \relax (x) - 1\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6-2*x)*log(x)^3+(5*x-3)*log(x)^2+(-4*x-27)*log(x)+x+9)/((x^2-6*x)*log(x)^3+(-2*x^2-3*x)*log(x)^2+(
x^2+9*x)*log(x)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.34, size = 29, normalized size = 1.07 \begin {gather*} -\log \left (x \log \relax (x) - x - 6 \, \log \relax (x) - 9\right ) - \log \relax (x) + \log \left (\log \relax (x) - 1\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6-2*x)*log(x)^3+(5*x-3)*log(x)^2+(-4*x-27)*log(x)+x+9)/((x^2-6*x)*log(x)^3+(-2*x^2-3*x)*log(x)^2+(
x^2+9*x)*log(x)),x, algorithm="giac")

[Out]

-log(x*log(x) - x - 6*log(x) - 9) - log(x) + log(log(x) - 1) + log(log(x))

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maple [A]  time = 0.05, size = 30, normalized size = 1.11

method	result	size
norman	\(-\ln \relax (x )-\ln \left (x \ln \relax (x )-x -6 \ln \relax (x )-9\right )+\ln \left (\ln \relax (x )\right )+\ln \left (\ln \relax (x )-1\right )\)	\(30\)
risch	\(-\ln \left (x^{2}-6 x \right )+\ln \left (\ln \relax (x )^{2}-\ln \relax (x )\right )-\ln \left (\ln \relax (x )-\frac {x +9}{x -6}\right )\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6-2*x)*ln(x)^3+(5*x-3)*ln(x)^2+(-4*x-27)*ln(x)+x+9)/((x^2-6*x)*ln(x)^3+(-2*x^2-3*x)*ln(x)^2+(x^2+9*x)*ln
(x)),x,method=_RETURNVERBOSE)

[Out]

-ln(x)-ln(x*ln(x)-x-6*ln(x)-9)+ln(ln(x))+ln(ln(x)-1)

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maxima [A]  time = 0.54, size = 39, normalized size = 1.44 \begin {gather*} -\log \left (x - 6\right ) - \log \relax (x) - \log \left (\frac {{\left (x - 6\right )} \log \relax (x) - x - 9}{x - 6}\right ) + \log \left (\log \relax (x) - 1\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6-2*x)*log(x)^3+(5*x-3)*log(x)^2+(-4*x-27)*log(x)+x+9)/((x^2-6*x)*log(x)^3+(-2*x^2-3*x)*log(x)^2+(
x^2+9*x)*log(x)),x, algorithm="maxima")

[Out]

-log(x - 6) - log(x) - log(((x - 6)*log(x) - x - 9)/(x - 6)) + log(log(x) - 1) + log(log(x))

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mupad [B]  time = 2.34, size = 28, normalized size = 1.04 \begin {gather*} \ln \left (\ln \relax (x)\right )-\ln \left (x+6\,\ln \relax (x)-x\,\ln \relax (x)+9\right )+\ln \left (\ln \relax (x)-1\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - log(x)*(4*x + 27) + log(x)^2*(5*x - 3) - log(x)^3*(2*x - 6) + 9)/(log(x)^2*(3*x + 2*x^2) + log(x)^3*
(6*x - x^2) - log(x)*(9*x + x^2)),x)

[Out]

log(log(x)) - log(x + 6*log(x) - x*log(x) + 9) + log(log(x) - 1) - log(x)

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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(((6-2*x)*ln(x)**3+(5*x-3)*ln(x)**2+(-4*x-27)*ln(x)+x+9)/((x**2-6*x)*ln(x)**3+(-2*x**2-3*x)*ln(x)**2+
(x**2+9*x)*ln(x)),x)

[Out]

Exception raised: PolynomialError

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