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

Optimal. Leaf size=20 $$3x^2\log (-2x+3\log (-2+x^2+\log (x)))$$

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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 = {} $$\begin{array}{c}\int \frac{9x+12x^2+18x^3-6x^4-6x^2\log (x)+(24x^2-12x^4-12x^2\log (x)+(-36x+18x^3+18x\log (x))\log (-2+x^2+\log (x)))\log (-2x+3\log (-2+x^2+\log (x)))}{4x-2x^3-2x\log (x)+(-6+3x^2+3\log (x))\log (-2+x^2+\log (x))}dx\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{9x+12x^2+18x^3-6x^4-6x^2\log (x)+(24x^2-12x^4-12x^2\log (x)+(-36x+18x^3+18x\log (x))\log (-2+x^2+\log (x)))\log (-2x+3\log (-2+x^2+\log (x)))}{(2-x^2-\log (x))(2x-3\log (-2+x^2+\log (x)))}dx\\ & =\int (-\frac{9x}{(-2+x^2+\log (x))(2x-3\log (-2+x^2+\log (x)))}-\frac{12x^2}{(-2+x^2+\log (x))(2x-3\log (-2+x^2+\log (x)))}-\frac{18x^3}{(-2+x^2+\log (x))(2x-3\log (-2+x^2+\log (x)))}+\frac{6x^4}{(-2+x^2+\log (x))(2x-3\log (-2+x^2+\log (x)))}+\frac{6x^2\log (x)}{(-2+x^2+\log (x))(2x-3\log (-2+x^2+\log (x)))}+6x\log (-2x+3\log (-2+x^2+\log (x))))dx\\ & =6\int \frac{x^4}{(-2+x^2+\log (x))(2x-3\log (-2+x^2+\log (x)))}dx+6\int \frac{x^2\log (x)}{(-2+x^2+\log (x))(2x-3\log (-2+x^2+\log (x)))}dx+6\int x\log (-2x+3\log (-2+x^2+\log (x)))dx-9\int \frac{x}{(-2+x^2+\log (x))(2x-3\log (-2+x^2+\log (x)))}dx-12\int \frac{x^2}{(-2+x^2+\log (x))(2x-3\log (-2+x^2+\log (x)))}dx-18\int \frac{x^3}{(-2+x^2+\log (x))(2x-3\log (-2+x^2+\log (x)))}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.09, size = 20, normalized size = 1.00 $$\begin{array}{c}3x^2\log (-2x+3\log (-2+x^2+\log (x)))\end{array}$$

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 $$\begin{array}{c}3x^2\log (-2x+3\log (x^2+\log (x)-2))\end{array}$$

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 $$\begin{array}{c}3x^2\log (-2x+3\log (x^2+\log (x)-2))\end{array}$$

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	$3x^2\ln (3\ln (\ln (x)+x^2-2)-2x)$	$21$

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 $$\begin{array}{c}3x^2\log (-2x+3\log (x^2+\log (x)-2))\end{array}$$

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 $$\begin{array}{c}3x^2\ln (3\ln (\ln (x)+x^2-2)-2x)\end{array}$$

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 $$\begin{array}{c}3x^2\log (-2x+3\log (x^2+\log (x)-2))\end{array}$$

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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