3.63.61 \(\int \frac {(-36+12 x) \log (x) \log (3 x^2)+(-9+3 x-9 \log (x)) \log ^2(3 x^2)+(-12+7 x-x^2+(-12+x^2) \log (x)) \log ^4(3 x^2)}{81-54 x+9 x^2+(216-198 x+60 x^2-6 x^3) \log ^2(3 x^2)+(144-168 x+73 x^2-14 x^3+x^4) \log ^4(3 x^2)} \, dx\) [6261]

Optimal. Leaf size=25 \[ -\frac {x \log (x)}{(-3+x) \left (-4+x-\frac {3}{\log ^2\left (3 x^2\right )}\right )} \]

[Out]

-1/(x-3/ln(3*x^2)^2-4)/(-3+x)*x*ln(x)

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Rubi [F]
time = 131.54, 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 {(-36+12 x) \log (x) \log \left (3 x^2\right )+(-9+3 x-9 \log (x)) \log ^2\left (3 x^2\right )+\left (-12+7 x-x^2+\left (-12+x^2\right ) \log (x)\right ) \log ^4\left (3 x^2\right )}{81-54 x+9 x^2+\left (216-198 x+60 x^2-6 x^3\right ) \log ^2\left (3 x^2\right )+\left (144-168 x+73 x^2-14 x^3+x^4\right ) \log ^4\left (3 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-36 + 12*x)*Log[x]*Log[3*x^2] + (-9 + 3*x - 9*Log[x])*Log[3*x^2]^2 + (-12 + 7*x - x^2 + (-12 + x^2)*Log[
x])*Log[3*x^2]^4)/(81 - 54*x + 9*x^2 + (216 - 198*x + 60*x^2 - 6*x^3)*Log[3*x^2]^2 + (144 - 168*x + 73*x^2 - 1
4*x^3 + x^4)*Log[3*x^2]^4),x]

[Out]

-((x*Log[x])/(3 - x)) + (x*Log[x])/(4 - x) - 27*Defer[Int][Log[x]/((-4 + x)*(3 - (-4 + x)*Log[3*x^2]^2)^2), x]
+ 27*Defer[Int][Log[x]/((-3 + x)*(3 - (-4 + x)*Log[3*x^2]^2)^2), x] + 12*Defer[Int][(Log[x]*Log[3*x^2])/((-3
+ x)*(3 - (-4 + x)*Log[3*x^2]^2)^2), x] + 36*Defer[Int][Log[x]/((-4 + x)^2*(-3 - 4*Log[3*x^2]^2 + x*Log[3*x^2]
^2)^2), x] - 3*Defer[Int][1/((-4 + x)*(-3 - 4*Log[3*x^2]^2 + x*Log[3*x^2]^2)), x] + 3*Defer[Int][1/((-3 + x)*(
-3 - 4*Log[3*x^2]^2 + x*Log[3*x^2]^2)), x] + 24*Defer[Int][Log[x]/((-4 + x)^2*(-3 - 4*Log[3*x^2]^2 + x*Log[3*x
^2]^2)), x] - 9*Defer[Int][Log[x]/((-4 + x)*(-3 - 4*Log[3*x^2]^2 + x*Log[3*x^2]^2)), x] - 9*Defer[Int][Log[x]/
((-3 + x)^2*(-3 - 4*Log[3*x^2]^2 + x*Log[3*x^2]^2)), x] + 9*Defer[Int][Log[x]/((-3 + x)*(-3 - 4*Log[3*x^2]^2 +
x*Log[3*x^2]^2)), x]

Rubi steps

Aborted

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Mathematica [A]
time = 0.09, size = 34, normalized size = 1.36 \begin {gather*} -\frac {x \log (x) \log ^2\left (3 x^2\right )}{(-3+x) \left (-3+(-4+x) \log ^2\left (3 x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-36 + 12*x)*Log[x]*Log[3*x^2] + (-9 + 3*x - 9*Log[x])*Log[3*x^2]^2 + (-12 + 7*x - x^2 + (-12 + x^2
)*Log[x])*Log[3*x^2]^4)/(81 - 54*x + 9*x^2 + (216 - 198*x + 60*x^2 - 6*x^3)*Log[3*x^2]^2 + (144 - 168*x + 73*x
^2 - 14*x^3 + x^4)*Log[3*x^2]^4),x]

[Out]

-((x*Log[x]*Log[3*x^2]^2)/((-3 + x)*(-3 + (-4 + x)*Log[3*x^2]^2)))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 305.00, size = 498, normalized size = 19.92

method	result	size
risch	\(-\frac {x \ln \left (x \right )}{x^{2}-7 x +12}-\frac {12 \ln \left (x \right ) x}{\left (x^{2}-7 x +12\right ) \left (-12+16 x \ln \left (3\right ) \ln \left (x \right )+4 x \ln \left (3\right )^{2}-64 \ln \left (3\right ) \ln \left (x \right )+16 x \ln \left (x \right )^{2}+32 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3}-16 \ln \left (3\right )^{2}-64 \ln \left (x \right )^{2}+32 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-64 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+16 i \ln \left (3\right ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-6 \pi ^{2} x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}+4 \pi ^{2} x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}-\pi ^{2} x \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}+4 \pi ^{2} x \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}+16 i \ln \left (3\right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+16 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \ln \left (x \right )-8 i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \ln \left (x \right )-4 i x \ln \left (3\right ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+8 i x \ln \left (3\right ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+4 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{6}+4 \pi ^{2} \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}-16 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}+24 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}-16 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}-\pi ^{2} x \mathrm {csgn}\left (i x^{2}\right )^{6}-8 i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3} \ln \left (x \right )-32 i \ln \left (3\right ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-4 i x \ln \left (3\right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}\)	\(498\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-12)*ln(x)-x^2+7*x-12)*ln(3*x^2)^4+(-9*ln(x)+3*x-9)*ln(3*x^2)^2+(12*x-36)*ln(x)*ln(3*x^2))/((x^4-14*
x^3+73*x^2-168*x+144)*ln(3*x^2)^4+(-6*x^3+60*x^2-198*x+216)*ln(3*x^2)^2+9*x^2-54*x+81),x,method=_RETURNVERBOSE
)

[Out]

-x/(x^2-7*x+12)*ln(x)-12*ln(x)*x/(x^2-7*x+12)/(-12+16*x*ln(3)*ln(x)-8*I*Pi*x*csgn(I*x^2)^3*ln(x)+4*x*ln(3)^2-6
4*ln(3)*ln(x)+16*x*ln(x)^2-6*Pi^2*x*csgn(I*x)^2*csgn(I*x^2)^4+4*Pi^2*x*csgn(I*x)*csgn(I*x^2)^5-Pi^2*x*csgn(I*x
)^4*csgn(I*x^2)^2+4*Pi^2*x*csgn(I*x)^3*csgn(I*x^2)^3+16*I*Pi*x*csgn(I*x)*csgn(I*x^2)^2*ln(x)-8*I*Pi*x*csgn(I*x
)^2*csgn(I*x^2)*ln(x)+4*Pi^2*csgn(I*x^2)^6-16*ln(3)^2-64*ln(x)^2+4*Pi^2*csgn(I*x)^4*csgn(I*x^2)^2-16*Pi^2*csgn
(I*x)^3*csgn(I*x^2)^3+24*Pi^2*csgn(I*x)^2*csgn(I*x^2)^4-16*Pi^2*csgn(I*x)*csgn(I*x^2)^5-Pi^2*x*csgn(I*x^2)^6-3
2*I*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+32*I*ln(x)*Pi*csgn(I*x)^2*csgn(I*x^2)-64*I*ln(x)*Pi*csgn(I*x)*csgn(I*x^2)
^2-4*I*x*ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)-4*I*x*ln(3)*Pi*csgn(I*x^2)^3+16*I*ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)+8
*I*x*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+16*I*ln(3)*Pi*csgn(I*x^2)^3+32*I*ln(x)*Pi*csgn(I*x^2)^3)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (25) = 50\).
time = 0.72, size = 140, normalized size = 5.60 \begin {gather*} -\frac {4 \, {\left (x^{2} - 4 \, x\right )} \log \left (x\right )^{3} + 4 \, {\left (x^{2} \log \left (3\right ) - 4 \, x \log \left (3\right )\right )} \log \left (x\right )^{2} + {\left (x^{2} \log \left (3\right )^{2} - 4 \, x \log \left (3\right )^{2}\right )} \log \left (x\right )}{x^{3} \log \left (3\right )^{2} - {\left (11 \, \log \left (3\right )^{2} + 3\right )} x^{2} + 4 \, {\left (x^{3} - 11 \, x^{2} + 40 \, x - 48\right )} \log \left (x\right )^{2} + {\left (40 \, \log \left (3\right )^{2} + 21\right )} x - 48 \, \log \left (3\right )^{2} + 4 \, {\left (x^{3} \log \left (3\right ) - 11 \, x^{2} \log \left (3\right ) + 40 \, x \log \left (3\right ) - 48 \, \log \left (3\right )\right )} \log \left (x\right ) - 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-12)*log(x)-x^2+7*x-12)*log(3*x^2)^4+(-9*log(x)+3*x-9)*log(3*x^2)^2+(12*x-36)*log(x)*log(3*x^2
))/((x^4-14*x^3+73*x^2-168*x+144)*log(3*x^2)^4+(-6*x^3+60*x^2-198*x+216)*log(3*x^2)^2+9*x^2-54*x+81),x, algori
thm="maxima")

[Out]

-(4*(x^2 - 4*x)*log(x)^3 + 4*(x^2*log(3) - 4*x*log(3))*log(x)^2 + (x^2*log(3)^2 - 4*x*log(3)^2)*log(x))/(x^3*l
og(3)^2 - (11*log(3)^2 + 3)*x^2 + 4*(x^3 - 11*x^2 + 40*x - 48)*log(x)^2 + (40*log(3)^2 + 21)*x - 48*log(3)^2 +
4*(x^3*log(3) - 11*x^2*log(3) + 40*x*log(3) - 48*log(3))*log(x) - 36)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (25) = 50\).
time = 0.36, size = 75, normalized size = 3.00 \begin {gather*} -\frac {x \log \left (3\right )^{2} \log \left (x\right ) + 4 \, x \log \left (3\right ) \log \left (x\right )^{2} + 4 \, x \log \left (x\right )^{3}}{{\left (x^{2} - 7 \, x + 12\right )} \log \left (3\right )^{2} + 4 \, {\left (x^{2} - 7 \, x + 12\right )} \log \left (3\right ) \log \left (x\right ) + 4 \, {\left (x^{2} - 7 \, x + 12\right )} \log \left (x\right )^{2} - 3 \, x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-12)*log(x)-x^2+7*x-12)*log(3*x^2)^4+(-9*log(x)+3*x-9)*log(3*x^2)^2+(12*x-36)*log(x)*log(3*x^2
))/((x^4-14*x^3+73*x^2-168*x+144)*log(3*x^2)^4+(-6*x^3+60*x^2-198*x+216)*log(3*x^2)^2+9*x^2-54*x+81),x, algori
thm="fricas")

[Out]

-(x*log(3)^2*log(x) + 4*x*log(3)*log(x)^2 + 4*x*log(x)^3)/((x^2 - 7*x + 12)*log(3)^2 + 4*(x^2 - 7*x + 12)*log(
3)*log(x) + 4*(x^2 - 7*x + 12)*log(x)^2 - 3*x + 9)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (22) = 44\).
time = 0.38, size = 117, normalized size = 4.68 \begin {gather*} - \frac {3 x \log {\left (x \right )}}{x^{3} \log {\left (3 \right )}^{2} - 11 x^{2} \log {\left (3 \right )}^{2} - 3 x^{2} + 21 x + 40 x \log {\left (3 \right )}^{2} + \left (4 x^{3} - 44 x^{2} + 160 x - 192\right ) \log {\left (x \right )}^{2} + \left (4 x^{3} \log {\left (3 \right )} - 44 x^{2} \log {\left (3 \right )} + 160 x \log {\left (3 \right )} - 192 \log {\left (3 \right )}\right ) \log {\left (x \right )} - 48 \log {\left (3 \right )}^{2} - 36} - \frac {x \log {\left (x \right )}}{x^{2} - 7 x + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-12)*ln(x)-x**2+7*x-12)*ln(3*x**2)**4+(-9*ln(x)+3*x-9)*ln(3*x**2)**2+(12*x-36)*ln(x)*ln(3*x**
2))/((x**4-14*x**3+73*x**2-168*x+144)*ln(3*x**2)**4+(-6*x**3+60*x**2-198*x+216)*ln(3*x**2)**2+9*x**2-54*x+81),
x)

[Out]

-3*x*log(x)/(x**3*log(3)**2 - 11*x**2*log(3)**2 - 3*x**2 + 21*x + 40*x*log(3)**2 + (4*x**3 - 44*x**2 + 160*x -
192)*log(x)**2 + (4*x**3*log(3) - 44*x**2*log(3) + 160*x*log(3) - 192*log(3))*log(x) - 48*log(3)**2 - 36) - x
*log(x)/(x**2 - 7*x + 12)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (25) = 50\).
time = 0.59, size = 125, normalized size = 5.00 \begin {gather*} -\frac {3 \, x \log \left (x\right )}{x^{3} \log \left (3\right )^{2} + 4 \, x^{3} \log \left (3\right ) \log \left (x\right ) + 4 \, x^{3} \log \left (x\right )^{2} - 11 \, x^{2} \log \left (3\right )^{2} - 44 \, x^{2} \log \left (3\right ) \log \left (x\right ) - 44 \, x^{2} \log \left (x\right )^{2} + 40 \, x \log \left (3\right )^{2} + 160 \, x \log \left (3\right ) \log \left (x\right ) + 160 \, x \log \left (x\right )^{2} - 3 \, x^{2} - 48 \, \log \left (3\right )^{2} - 192 \, \log \left (3\right ) \log \left (x\right ) - 192 \, \log \left (x\right )^{2} + 21 \, x - 36} - \frac {x \log \left (x\right )}{x^{2} - 7 \, x + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-12)*log(x)-x^2+7*x-12)*log(3*x^2)^4+(-9*log(x)+3*x-9)*log(3*x^2)^2+(12*x-36)*log(x)*log(3*x^2
))/((x^4-14*x^3+73*x^2-168*x+144)*log(3*x^2)^4+(-6*x^3+60*x^2-198*x+216)*log(3*x^2)^2+9*x^2-54*x+81),x, algori
thm="giac")

[Out]

-3*x*log(x)/(x^3*log(3)^2 + 4*x^3*log(3)*log(x) + 4*x^3*log(x)^2 - 11*x^2*log(3)^2 - 44*x^2*log(3)*log(x) - 44
*x^2*log(x)^2 + 40*x*log(3)^2 + 160*x*log(3)*log(x) + 160*x*log(x)^2 - 3*x^2 - 48*log(3)^2 - 192*log(3)*log(x)
- 192*log(x)^2 + 21*x - 36) - x*log(x)/(x^2 - 7*x + 12)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\left (7\,x+\ln \left (x\right )\,\left (x^2-12\right )-x^2-12\right )\,{\ln \left (3\,x^2\right )}^4+\left (3\,x-9\,\ln \left (x\right )-9\right )\,{\ln \left (3\,x^2\right )}^2+\ln \left (x\right )\,\left (12\,x-36\right )\,\ln \left (3\,x^2\right )}{9\,x^2-{\ln \left (3\,x^2\right )}^2\,\left (6\,x^3-60\,x^2+198\,x-216\right )-54\,x+{\ln \left (3\,x^2\right )}^4\,\left (x^4-14\,x^3+73\,x^2-168\,x+144\right )+81} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3*x^2)^4*(7*x + log(x)*(x^2 - 12) - x^2 - 12) - log(3*x^2)^2*(9*log(x) - 3*x + 9) + log(3*x^2)*log(x)
*(12*x - 36))/(9*x^2 - log(3*x^2)^2*(198*x - 60*x^2 + 6*x^3 - 216) - 54*x + log(3*x^2)^4*(73*x^2 - 168*x - 14*
x^3 + x^4 + 144) + 81),x)

[Out]

int((log(3*x^2)^4*(7*x + log(x)*(x^2 - 12) - x^2 - 12) - log(3*x^2)^2*(9*log(x) - 3*x + 9) + log(3*x^2)*log(x)
*(12*x - 36))/(9*x^2 - log(3*x^2)^2*(198*x - 60*x^2 + 6*x^3 - 216) - 54*x + log(3*x^2)^4*(73*x^2 - 168*x - 14*
x^3 + x^4 + 144) + 81), x)

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