3.21.47 \(\int \frac {-8 x \log (9-x) \log (x)+(-36+4 x+(36-4 x) \log ^2(9-x)+(-36+4 x+(36-4 x) \log ^2(9-x)) \log (x)) \log (-1+\log ^2(9-x))}{9-x+(-9+x) \log ^2(9-x)} \, dx\)

Optimal. Leaf size=18 \[ -5-4 x \log (x) \log \left (-1+\log ^2(9-x)\right ) \]

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Rubi [F]  time = 3.18, 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 {-8 x \log (9-x) \log (x)+\left (-36+4 x+(36-4 x) \log ^2(9-x)+\left (-36+4 x+(36-4 x) \log ^2(9-x)\right ) \log (x)\right ) \log \left (-1+\log ^2(9-x)\right )}{9-x+(-9+x) \log ^2(9-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*x*Log[9 - x]*Log[x] + (-36 + 4*x + (36 - 4*x)*Log[9 - x]^2 + (-36 + 4*x + (36 - 4*x)*Log[9 - x]^2)*Log
[x])*Log[-1 + Log[9 - x]^2])/(9 - x + (-9 + x)*Log[9 - x]^2),x]

[Out]

-8*Defer[Int][(Log[9 - x]*Log[x])/(-1 + Log[9 - x]^2), x] - 72*Defer[Int][(Log[9 - x]*Log[x])/((-9 + x)*(-1 +
Log[9 - x]^2)), x] - 4*Defer[Int][Log[x]*Log[-1 + Log[9 - x]^2], x] + 4*Defer[Subst][Defer[Int][Log[-1 + Log[x
]^2], x], x, 9 - x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {8 x \log (9-x) \log (x)}{(-9+x) \left (-1+\log ^2(9-x)\right )}-4 (1+\log (x)) \log \left (-1+\log ^2(9-x)\right )\right ) \, dx\\ &=-\left (4 \int (1+\log (x)) \log \left (-1+\log ^2(9-x)\right ) \, dx\right )-8 \int \frac {x \log (9-x) \log (x)}{(-9+x) \left (-1+\log ^2(9-x)\right )} \, dx\\ &=-\left (4 \int \left (\log \left (-1+\log ^2(9-x)\right )+\log (x) \log \left (-1+\log ^2(9-x)\right )\right ) \, dx\right )-8 \int \left (\frac {\log (9-x) \log (x)}{-1+\log ^2(9-x)}+\frac {9 \log (9-x) \log (x)}{(-9+x) \left (-1+\log ^2(9-x)\right )}\right ) \, dx\\ &=-\left (4 \int \log \left (-1+\log ^2(9-x)\right ) \, dx\right )-4 \int \log (x) \log \left (-1+\log ^2(9-x)\right ) \, dx-8 \int \frac {\log (9-x) \log (x)}{-1+\log ^2(9-x)} \, dx-72 \int \frac {\log (9-x) \log (x)}{(-9+x) \left (-1+\log ^2(9-x)\right )} \, dx\\ &=-\left (4 \int \log (x) \log \left (-1+\log ^2(9-x)\right ) \, dx\right )+4 \operatorname {Subst}\left (\int \log \left (-1+\log ^2(x)\right ) \, dx,x,9-x\right )-8 \int \frac {\log (9-x) \log (x)}{-1+\log ^2(9-x)} \, dx-72 \int \frac {\log (9-x) \log (x)}{(-9+x) \left (-1+\log ^2(9-x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 16, normalized size = 0.89 \begin {gather*} -4 x \log (x) \log \left (-1+\log ^2(9-x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x*Log[9 - x]*Log[x] + (-36 + 4*x + (36 - 4*x)*Log[9 - x]^2 + (-36 + 4*x + (36 - 4*x)*Log[9 - x]^
2)*Log[x])*Log[-1 + Log[9 - x]^2])/(9 - x + (-9 + x)*Log[9 - x]^2),x]

[Out]

-4*x*Log[x]*Log[-1 + Log[9 - x]^2]

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fricas [A]  time = 0.59, size = 16, normalized size = 0.89 \begin {gather*} -4 \, x \log \left (\log \left (-x + 9\right )^{2} - 1\right ) \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x+36)*log(9-x)^2+4*x-36)*log(x)+(-4*x+36)*log(9-x)^2+4*x-36)*log(log(9-x)^2-1)-8*x*log(9-x)*l
og(x))/((x-9)*log(9-x)^2+9-x),x, algorithm="fricas")

[Out]

-4*x*log(log(-x + 9)^2 - 1)*log(x)

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giac [A]  time = 0.42, size = 16, normalized size = 0.89 \begin {gather*} -4 \, x \log \left (\log \left (-x + 9\right )^{2} - 1\right ) \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x+36)*log(9-x)^2+4*x-36)*log(x)+(-4*x+36)*log(9-x)^2+4*x-36)*log(log(9-x)^2-1)-8*x*log(9-x)*l
og(x))/((x-9)*log(9-x)^2+9-x),x, algorithm="giac")

[Out]

-4*x*log(log(-x + 9)^2 - 1)*log(x)

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maple [A]  time = 0.04, size = 17, normalized size = 0.94

method	result	size
risch	\(-4 x \ln \left (\ln \left (9-x \right )^{2}-1\right ) \ln \relax (x )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((-4*x+36)*ln(9-x)^2+4*x-36)*ln(x)+(-4*x+36)*ln(9-x)^2+4*x-36)*ln(ln(9-x)^2-1)-8*x*ln(9-x)*ln(x))/((x-9)
*ln(9-x)^2+9-x),x,method=_RETURNVERBOSE)

[Out]

-4*x*ln(ln(9-x)^2-1)*ln(x)

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maxima [A]  time = 0.87, size = 29, normalized size = 1.61 \begin {gather*} -4 \, x \log \relax (x) \log \left (\log \left (-x + 9\right ) + 1\right ) - 4 \, x \log \relax (x) \log \left (\log \left (-x + 9\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x+36)*log(9-x)^2+4*x-36)*log(x)+(-4*x+36)*log(9-x)^2+4*x-36)*log(log(9-x)^2-1)-8*x*log(9-x)*l
og(x))/((x-9)*log(9-x)^2+9-x),x, algorithm="maxima")

[Out]

-4*x*log(x)*log(log(-x + 9) + 1) - 4*x*log(x)*log(log(-x + 9) - 1)

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mupad [B]  time = 1.32, size = 16, normalized size = 0.89 \begin {gather*} -4\,x\,\ln \left ({\ln \left (9-x\right )}^2-1\right )\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(log(9 - x)^2 - 1)*(log(x)*(log(9 - x)^2*(4*x - 36) - 4*x + 36) - 4*x + log(9 - x)^2*(4*x - 36) + 36)
+ 8*x*log(9 - x)*log(x))/(log(9 - x)^2*(x - 9) - x + 9),x)

[Out]

-4*x*log(log(9 - x)^2 - 1)*log(x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x+36)*ln(9-x)**2+4*x-36)*ln(x)+(-4*x+36)*ln(9-x)**2+4*x-36)*ln(ln(9-x)**2-1)-8*x*ln(9-x)*ln(x
))/((x-9)*ln(9-x)**2+9-x),x)

[Out]

Timed out

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