3.29.30 \(\int \frac {216-36 x+(-351+e^3 (3-x)+117 x) \log (-\frac {x^2}{-3+x})+(108-18 x+(-162+54 x) \log (-\frac {x^2}{-3+x})) \log (\frac {x}{\log (-\frac {x^2}{-3+x})})+(-27+9 x) \log (-\frac {x^2}{-3+x}) \log ^2(\frac {x}{\log (-\frac {x^2}{-3+x})})}{(-9+3 x) \log (-\frac {x^2}{-3+x})} \, dx\)

Optimal. Leaf size=36 \[ \frac {1}{3} x \left (-e^3+9 \left (5+\left (2+\log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )\right )^2\right )\right ) \]

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Rubi [F]  time = 0.74, 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 {216-36 x+\left (-351+e^3 (3-x)+117 x\right ) \log \left (-\frac {x^2}{-3+x}\right )+\left (108-18 x+(-162+54 x) \log \left (-\frac {x^2}{-3+x}\right )\right ) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+(-27+9 x) \log \left (-\frac {x^2}{-3+x}\right ) \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-9+3 x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(216 - 36*x + (-351 + E^3*(3 - x) + 117*x)*Log[-(x^2/(-3 + x))] + (108 - 18*x + (-162 + 54*x)*Log[-(x^2/(-
3 + x))])*Log[x/Log[-(x^2/(-3 + x))]] + (-27 + 9*x)*Log[-(x^2/(-3 + x))]*Log[x/Log[-(x^2/(-3 + x))]]^2)/((-9 +
3*x)*Log[-(x^2/(-3 + x))]),x]

[Out]

-18*x + ((117 - E^3)*x)/3 + 18*x*Log[x/Log[x^2/(3 - x)]] - 18*Defer[Int][(6 - x)/((-3 + x)*Log[-(x^2/(-3 + x))
]), x] - 12*Defer[Int][(-6 + x)/((-3 + x)*Log[-(x^2/(-3 + x))]), x] - 6*Defer[Int][Log[x/Log[-(x^2/(-3 + x))]]
/Log[-(x^2/(-3 + x))], x] + 18*Defer[Int][Log[x/Log[-(x^2/(-3 + x))]]/((-3 + x)*Log[-(x^2/(-3 + x))]), x] + 3*
Defer[Int][Log[x/Log[-(x^2/(-3 + x))]]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (39-\frac {e^3}{3}+18 \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+3 \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )-\frac {6 (-6+x) \left (2+\log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx\\ &=\frac {1}{3} \left (117-e^3\right ) x+3 \int \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-6 \int \frac {(-6+x) \left (2+\log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx+18 \int \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx\\ &=\frac {1}{3} \left (117-e^3\right ) x+18 x \log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )+3 \int \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-6 \int \left (\frac {2 (-6+x)}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}+\frac {(-6+x) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-18 \int \frac {6-x+(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx\\ &=\frac {1}{3} \left (117-e^3\right ) x+18 x \log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )+3 \int \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-6 \int \frac {(-6+x) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx-12 \int \frac {-6+x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx-18 \int \left (1+\frac {6-x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx\\ &=-18 x+\frac {1}{3} \left (117-e^3\right ) x+18 x \log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )+3 \int \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-6 \int \left (\frac {\log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{\log \left (-\frac {x^2}{-3+x}\right )}-\frac {3 \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-12 \int \frac {-6+x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx-18 \int \frac {6-x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx\\ &=-18 x+\frac {1}{3} \left (117-e^3\right ) x+18 x \log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )+3 \int \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \, dx-6 \int \frac {\log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{\log \left (-\frac {x^2}{-3+x}\right )} \, dx-12 \int \frac {-6+x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx-18 \int \frac {6-x}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx+18 \int \frac {\log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-3+x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 51, normalized size = 1.42 \begin {gather*} -\frac {1}{3} \left (-81+e^3\right ) x+12 x \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+3 x \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(216 - 36*x + (-351 + E^3*(3 - x) + 117*x)*Log[-(x^2/(-3 + x))] + (108 - 18*x + (-162 + 54*x)*Log[-(
x^2/(-3 + x))])*Log[x/Log[-(x^2/(-3 + x))]] + (-27 + 9*x)*Log[-(x^2/(-3 + x))]*Log[x/Log[-(x^2/(-3 + x))]]^2)/
((-9 + 3*x)*Log[-(x^2/(-3 + x))]),x]

[Out]

-1/3*((-81 + E^3)*x) + 12*x*Log[x/Log[-(x^2/(-3 + x))]] + 3*x*Log[x/Log[-(x^2/(-3 + x))]]^2

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fricas [A]  time = 0.48, size = 49, normalized size = 1.36 \begin {gather*} 3 \, x \log \left (\frac {x}{\log \left (-\frac {x^{2}}{x - 3}\right )}\right )^{2} - \frac {1}{3} \, x e^{3} + 12 \, x \log \left (\frac {x}{\log \left (-\frac {x^{2}}{x - 3}\right )}\right ) + 27 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-27)*log(-x^2/(x-3))*log(x/log(-x^2/(x-3)))^2+((54*x-162)*log(-x^2/(x-3))-18*x+108)*log(x/log(-
x^2/(x-3)))+((3-x)*exp(3)+117*x-351)*log(-x^2/(x-3))-36*x+216)/(3*x-9)/log(-x^2/(x-3)),x, algorithm="fricas")

[Out]

3*x*log(x/log(-x^2/(x - 3)))^2 - 1/3*x*e^3 + 12*x*log(x/log(-x^2/(x - 3))) + 27*x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {9 \, {\left (x - 3\right )} \log \left (-\frac {x^{2}}{x - 3}\right ) \log \left (\frac {x}{\log \left (-\frac {x^{2}}{x - 3}\right )}\right )^{2} - {\left ({\left (x - 3\right )} e^{3} - 117 \, x + 351\right )} \log \left (-\frac {x^{2}}{x - 3}\right ) + 18 \, {\left (3 \, {\left (x - 3\right )} \log \left (-\frac {x^{2}}{x - 3}\right ) - x + 6\right )} \log \left (\frac {x}{\log \left (-\frac {x^{2}}{x - 3}\right )}\right ) - 36 \, x + 216}{3 \, {\left (x - 3\right )} \log \left (-\frac {x^{2}}{x - 3}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-27)*log(-x^2/(x-3))*log(x/log(-x^2/(x-3)))^2+((54*x-162)*log(-x^2/(x-3))-18*x+108)*log(x/log(-
x^2/(x-3)))+((3-x)*exp(3)+117*x-351)*log(-x^2/(x-3))-36*x+216)/(3*x-9)/log(-x^2/(x-3)),x, algorithm="giac")

[Out]

integrate(1/3*(9*(x - 3)*log(-x^2/(x - 3))*log(x/log(-x^2/(x - 3)))^2 - ((x - 3)*e^3 - 117*x + 351)*log(-x^2/(
x - 3)) + 18*(3*(x - 3)*log(-x^2/(x - 3)) - x + 6)*log(x/log(-x^2/(x - 3))) - 36*x + 216)/((x - 3)*log(-x^2/(x
- 3))), x)

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maple [A]  time = 0.42, size = 50, normalized size = 1.39

method	result	size
norman	\(\left (-\frac {{\mathrm e}^{3}}{3}+27\right ) x +3 x \ln \left (\frac {x}{\ln \left (-\frac {x^{2}}{x -3}\right )}\right )^{2}+12 \ln \left (\frac {x}{\ln \left (-\frac {x^{2}}{x -3}\right )}\right ) x\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9*x-27)*ln(-x^2/(x-3))*ln(x/ln(-x^2/(x-3)))^2+((54*x-162)*ln(-x^2/(x-3))-18*x+108)*ln(x/ln(-x^2/(x-3)))+
((3-x)*exp(3)+117*x-351)*ln(-x^2/(x-3))-36*x+216)/(3*x-9)/ln(-x^2/(x-3)),x,method=_RETURNVERBOSE)

[Out]

(-1/3*exp(3)+27)*x+3*x*ln(x/ln(-x^2/(x-3)))^2+12*ln(x/ln(-x^2/(x-3)))*x

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maxima [B]  time = 0.47, size = 63, normalized size = 1.75 \begin {gather*} 3 \, x \log \relax (x)^{2} + 3 \, x \log \left (2 \, \log \relax (x) - \log \left (-x + 3\right )\right )^{2} - \frac {1}{3} \, x {\left (e^{3} - 81\right )} + 12 \, x \log \relax (x) - 6 \, {\left (x \log \relax (x) + 2 \, x\right )} \log \left (2 \, \log \relax (x) - \log \left (-x + 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-27)*log(-x^2/(x-3))*log(x/log(-x^2/(x-3)))^2+((54*x-162)*log(-x^2/(x-3))-18*x+108)*log(x/log(-
x^2/(x-3)))+((3-x)*exp(3)+117*x-351)*log(-x^2/(x-3))-36*x+216)/(3*x-9)/log(-x^2/(x-3)),x, algorithm="maxima")

[Out]

3*x*log(x)^2 + 3*x*log(2*log(x) - log(-x + 3))^2 - 1/3*x*(e^3 - 81) + 12*x*log(x) - 6*(x*log(x) + 2*x)*log(2*l
og(x) - log(-x + 3))

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mupad [B]  time = 2.69, size = 50, normalized size = 1.39 \begin {gather*} 3\,x\,{\ln \left (\frac {x}{\ln \left (-\frac {x^2}{x-3}\right )}\right )}^2+12\,x\,\ln \left (\frac {x}{\ln \left (-\frac {x^2}{x-3}\right )}\right )-x\,\left (\frac {{\mathrm {e}}^3}{3}-27\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x/log(-x^2/(x - 3)))*(log(-x^2/(x - 3))*(54*x - 162) - 18*x + 108) - log(-x^2/(x - 3))*(exp(3)*(x - 3
) - 117*x + 351) - 36*x + log(x/log(-x^2/(x - 3)))^2*log(-x^2/(x - 3))*(9*x - 27) + 216)/(log(-x^2/(x - 3))*(3
*x - 9)),x)

[Out]

12*x*log(x/log(-x^2/(x - 3))) + 3*x*log(x/log(-x^2/(x - 3)))^2 - x*(exp(3)/3 - 27)

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sympy [A]  time = 0.75, size = 42, normalized size = 1.17 \begin {gather*} 3 x \log {\left (\frac {x}{\log {\left (- \frac {x^{2}}{x - 3} \right )}} \right )}^{2} + 12 x \log {\left (\frac {x}{\log {\left (- \frac {x^{2}}{x - 3} \right )}} \right )} + x \left (27 - \frac {e^{3}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-27)*ln(-x**2/(x-3))*ln(x/ln(-x**2/(x-3)))**2+((54*x-162)*ln(-x**2/(x-3))-18*x+108)*ln(x/ln(-x*
*2/(x-3)))+((3-x)*exp(3)+117*x-351)*ln(-x**2/(x-3))-36*x+216)/(3*x-9)/ln(-x**2/(x-3)),x)

[Out]

3*x*log(x/log(-x**2/(x - 3)))**2 + 12*x*log(x/log(-x**2/(x - 3))) + x*(27 - exp(3)/3)

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