3.73.65 \(\int \frac {9 x+12 x^2+18 x^3-6 x^4-6 x^2 \log (x)+(24 x^2-12 x^4-12 x^2 \log (x)+(-36 x+18 x^3+18 x \log (x)) \log (-2+x^2+\log (x))) \log (-2 x+3 \log (-2+x^2+\log (x)))}{4 x-2 x^3-2 x \log (x)+(-6+3 x^2+3 \log (x)) \log (-2+x^2+\log (x))} \, dx\)

Optimal. Leaf size=20 \[ 3 x^2 \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right ) \]

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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+12 x^2+18 x^3-6 x^4-6 x^2 \log (x)+\left (24 x^2-12 x^4-12 x^2 \log (x)+\left (-36 x+18 x^3+18 x \log (x)\right ) \log \left (-2+x^2+\log (x)\right )\right ) \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right )}{4 x-2 x^3-2 x \log (x)+\left (-6+3 x^2+3 \log (x)\right ) \log \left (-2+x^2+\log (x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9*x + 12*x^2 + 18*x^3 - 6*x^4 - 6*x^2*Log[x] + (24*x^2 - 12*x^4 - 12*x^2*Log[x] + (-36*x + 18*x^3 + 18*x*
Log[x])*Log[-2 + x^2 + Log[x]])*Log[-2*x + 3*Log[-2 + x^2 + Log[x]]])/(4*x - 2*x^3 - 2*x*Log[x] + (-6 + 3*x^2
+ 3*Log[x])*Log[-2 + x^2 + Log[x]]),x]

[Out]

-9*Defer[Int][x/((-2 + x^2 + Log[x])*(2*x - 3*Log[-2 + x^2 + Log[x]])), x] - 12*Defer[Int][x^2/((-2 + x^2 + Lo
g[x])*(2*x - 3*Log[-2 + x^2 + Log[x]])), x] - 18*Defer[Int][x^3/((-2 + x^2 + Log[x])*(2*x - 3*Log[-2 + x^2 + L
og[x]])), x] + 6*Defer[Int][x^4/((-2 + x^2 + Log[x])*(2*x - 3*Log[-2 + x^2 + Log[x]])), x] + 6*Defer[Int][(x^2
*Log[x])/((-2 + x^2 + Log[x])*(2*x - 3*Log[-2 + x^2 + Log[x]])), x] + 6*Defer[Int][x*Log[-2*x + 3*Log[-2 + x^2
+ Log[x]]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 x+12 x^2+18 x^3-6 x^4-6 x^2 \log (x)+\left (24 x^2-12 x^4-12 x^2 \log (x)+\left (-36 x+18 x^3+18 x \log (x)\right ) \log \left (-2+x^2+\log (x)\right )\right ) \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right )}{\left (2-x^2-\log (x)\right ) \left (2 x-3 \log \left (-2+x^2+\log (x)\right )\right )} \, dx\\ &=\int \left (-\frac {9 x}{\left (-2+x^2+\log (x)\right ) \left (2 x-3 \log \left (-2+x^2+\log (x)\right )\right )}-\frac {12 x^2}{\left (-2+x^2+\log (x)\right ) \left (2 x-3 \log \left (-2+x^2+\log (x)\right )\right )}-\frac {18 x^3}{\left (-2+x^2+\log (x)\right ) \left (2 x-3 \log \left (-2+x^2+\log (x)\right )\right )}+\frac {6 x^4}{\left (-2+x^2+\log (x)\right ) \left (2 x-3 \log \left (-2+x^2+\log (x)\right )\right )}+\frac {6 x^2 \log (x)}{\left (-2+x^2+\log (x)\right ) \left (2 x-3 \log \left (-2+x^2+\log (x)\right )\right )}+6 x \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right )\right ) \, dx\\ &=6 \int \frac {x^4}{\left (-2+x^2+\log (x)\right ) \left (2 x-3 \log \left (-2+x^2+\log (x)\right )\right )} \, dx+6 \int \frac {x^2 \log (x)}{\left (-2+x^2+\log (x)\right ) \left (2 x-3 \log \left (-2+x^2+\log (x)\right )\right )} \, dx+6 \int x \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right ) \, dx-9 \int \frac {x}{\left (-2+x^2+\log (x)\right ) \left (2 x-3 \log \left (-2+x^2+\log (x)\right )\right )} \, dx-12 \int \frac {x^2}{\left (-2+x^2+\log (x)\right ) \left (2 x-3 \log \left (-2+x^2+\log (x)\right )\right )} \, dx-18 \int \frac {x^3}{\left (-2+x^2+\log (x)\right ) \left (2 x-3 \log \left (-2+x^2+\log (x)\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 20, normalized size = 1.00 \begin {gather*} 3 x^2 \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9*x + 12*x^2 + 18*x^3 - 6*x^4 - 6*x^2*Log[x] + (24*x^2 - 12*x^4 - 12*x^2*Log[x] + (-36*x + 18*x^3 +
18*x*Log[x])*Log[-2 + x^2 + Log[x]])*Log[-2*x + 3*Log[-2 + x^2 + Log[x]]])/(4*x - 2*x^3 - 2*x*Log[x] + (-6 +
3*x^2 + 3*Log[x])*Log[-2 + x^2 + Log[x]]),x]

[Out]

3*x^2*Log[-2*x + 3*Log[-2 + x^2 + Log[x]]]

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fricas [A]  time = 0.77, size = 20, normalized size = 1.00 \begin {gather*} 3 \, x^{2} \log \left (-2 \, x + 3 \, \log \left (x^{2} + \log \relax (x) - 2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x*log(x)+18*x^3-36*x)*log(log(x)+x^2-2)-12*x^2*log(x)-12*x^4+24*x^2)*log(3*log(log(x)+x^2-2)-2
*x)-6*x^2*log(x)-6*x^4+18*x^3+12*x^2+9*x)/((3*log(x)+3*x^2-6)*log(log(x)+x^2-2)-2*x*log(x)-2*x^3+4*x),x, algor
ithm="fricas")

[Out]

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

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giac [A]  time = 0.28, size = 20, normalized size = 1.00 \begin {gather*} 3 \, x^{2} \log \left (-2 \, x + 3 \, \log \left (x^{2} + \log \relax (x) - 2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x*log(x)+18*x^3-36*x)*log(log(x)+x^2-2)-12*x^2*log(x)-12*x^4+24*x^2)*log(3*log(log(x)+x^2-2)-2
*x)-6*x^2*log(x)-6*x^4+18*x^3+12*x^2+9*x)/((3*log(x)+3*x^2-6)*log(log(x)+x^2-2)-2*x*log(x)-2*x^3+4*x),x, algor
ithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 21, normalized size = 1.05

method	result	size
risch	\(3 x^{2} \ln \left (3 \ln \left (\ln \relax (x )+x^{2}-2\right )-2 x \right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((18*x*ln(x)+18*x^3-36*x)*ln(ln(x)+x^2-2)-12*x^2*ln(x)-12*x^4+24*x^2)*ln(3*ln(ln(x)+x^2-2)-2*x)-6*x^2*ln(
x)-6*x^4+18*x^3+12*x^2+9*x)/((3*ln(x)+3*x^2-6)*ln(ln(x)+x^2-2)-2*x*ln(x)-2*x^3+4*x),x,method=_RETURNVERBOSE)

[Out]

3*x^2*ln(3*ln(ln(x)+x^2-2)-2*x)

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maxima [A]  time = 0.39, size = 20, normalized size = 1.00 \begin {gather*} 3 \, x^{2} \log \left (-2 \, x + 3 \, \log \left (x^{2} + \log \relax (x) - 2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x*log(x)+18*x^3-36*x)*log(log(x)+x^2-2)-12*x^2*log(x)-12*x^4+24*x^2)*log(3*log(log(x)+x^2-2)-2
*x)-6*x^2*log(x)-6*x^4+18*x^3+12*x^2+9*x)/((3*log(x)+3*x^2-6)*log(log(x)+x^2-2)-2*x*log(x)-2*x^3+4*x),x, algor
ithm="maxima")

[Out]

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

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mupad [B]  time = 4.77, size = 20, normalized size = 1.00 \begin {gather*} 3\,x^2\,\ln \left (3\,\ln \left (\ln \relax (x)+x^2-2\right )-2\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x - 6*x^2*log(x) + 12*x^2 + 18*x^3 - 6*x^4 + log(3*log(log(x) + x^2 - 2) - 2*x)*(log(log(x) + x^2 - 2)*
(18*x*log(x) - 36*x + 18*x^3) - 12*x^2*log(x) + 24*x^2 - 12*x^4))/(4*x - 2*x*log(x) + log(log(x) + x^2 - 2)*(3
*log(x) + 3*x^2 - 6) - 2*x^3),x)

[Out]

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

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sympy [A]  time = 17.11, size = 20, normalized size = 1.00 \begin {gather*} 3 x^{2} \log {\left (- 2 x + 3 \log {\left (x^{2} + \log {\relax (x )} - 2 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x*ln(x)+18*x**3-36*x)*ln(ln(x)+x**2-2)-12*x**2*ln(x)-12*x**4+24*x**2)*ln(3*ln(ln(x)+x**2-2)-2*
x)-6*x**2*ln(x)-6*x**4+18*x**3+12*x**2+9*x)/((3*ln(x)+3*x**2-6)*ln(ln(x)+x**2-2)-2*x*ln(x)-2*x**3+4*x),x)

[Out]

3*x**2*log(-2*x + 3*log(x**2 + log(x) - 2))

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