3.66.11 \(\int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log (x^2)-6 x \log ^2(x^2)}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} (24 x^4-2 e^{3+x} x^4)+(-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2) \log ^2(x^2)+\log ^4(x^2)} \, dx\)

Optimal. Leaf size=31 \[ \frac {3}{12-e^{3+x}+e^{2+2 x}-\frac {\log ^2\left (x^2\right )}{x^2}} \]

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Rubi [F]  time = 4.63, 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 {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*E^(3 + x)*x^4 - 6*E^(2 + 2*x)*x^4 + 12*x*Log[x^2] - 6*x*Log[x^2]^2)/(144*x^4 - 24*E^(3 + x)*x^4 + E^(6
+ 2*x)*x^4 + E^(4 + 4*x)*x^4 + E^(2 + 2*x)*(24*x^4 - 2*E^(3 + x)*x^4) + (-24*x^2 + 2*E^(3 + x)*x^2 - 2*E^(2 +
2*x)*x^2)*Log[x^2]^2 + Log[x^2]^4),x]

[Out]

72*Defer[Int][x^4/(12*x^2 - E^(3 + x)*x^2 + E^(2 + 2*x)*x^2 - Log[x^2]^2)^2, x] - 3*Defer[Int][(E^(3 + x)*x^4)
/(12*x^2 - E^(3 + x)*x^2 + E^(2 + 2*x)*x^2 - Log[x^2]^2)^2, x] + 12*Defer[Int][(x*Log[x^2])/(12*x^2 - E^(3 + x
)*x^2 + E^(2 + 2*x)*x^2 - Log[x^2]^2)^2, x] - 6*Defer[Int][(x*Log[x^2]^2)/(12*x^2 - E^(3 + x)*x^2 + E^(2 + 2*x
)*x^2 - Log[x^2]^2)^2, x] - 6*Defer[Int][(x^2*Log[x^2]^2)/(12*x^2 - E^(3 + x)*x^2 + E^(2 + 2*x)*x^2 - Log[x^2]
^2)^2, x] - 6*Defer[Int][x^2/(12*x^2 - E^(3 + x)*x^2 + E^(2 + 2*x)*x^2 - Log[x^2]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 x \left (-e^{2+x} \left (-e+2 e^x\right ) x^3+4 \log \left (x^2\right )-2 \log ^2\left (x^2\right )\right )}{\left (\left (12-e^{3+x}+e^{2+2 x}\right ) x^2-\log ^2\left (x^2\right )\right )^2} \, dx\\ &=3 \int \frac {x \left (-e^{2+x} \left (-e+2 e^x\right ) x^3+4 \log \left (x^2\right )-2 \log ^2\left (x^2\right )\right )}{\left (\left (12-e^{3+x}+e^{2+2 x}\right ) x^2-\log ^2\left (x^2\right )\right )^2} \, dx\\ &=3 \int \left (-\frac {2 x^2}{12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )}-\frac {x \left (-24 x^3+e^{3+x} x^3-4 \log \left (x^2\right )+2 \log ^2\left (x^2\right )+2 x \log ^2\left (x^2\right )\right )}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2}\right ) \, dx\\ &=-\left (3 \int \frac {x \left (-24 x^3+e^{3+x} x^3-4 \log \left (x^2\right )+2 \log ^2\left (x^2\right )+2 x \log ^2\left (x^2\right )\right )}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2} \, dx\right )-6 \int \frac {x^2}{12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )} \, dx\\ &=-\left (3 \int \left (-\frac {24 x^4}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2}+\frac {e^{3+x} x^4}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2}-\frac {4 x \log \left (x^2\right )}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2}+\frac {2 x \log ^2\left (x^2\right )}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2}+\frac {2 x^2 \log ^2\left (x^2\right )}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2}\right ) \, dx\right )-6 \int \frac {x^2}{12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )} \, dx\\ &=-\left (3 \int \frac {e^{3+x} x^4}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2} \, dx\right )-6 \int \frac {x \log ^2\left (x^2\right )}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2} \, dx-6 \int \frac {x^2 \log ^2\left (x^2\right )}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2} \, dx-6 \int \frac {x^2}{12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )} \, dx+12 \int \frac {x \log \left (x^2\right )}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2} \, dx+72 \int \frac {x^4}{\left (12 x^2-e^{3+x} x^2+e^{2+2 x} x^2-\log ^2\left (x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.89, size = 36, normalized size = 1.16 \begin {gather*} \frac {3 x^2}{\left (12-e^{3+x}+e^{2+2 x}\right ) x^2-\log ^2\left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*E^(3 + x)*x^4 - 6*E^(2 + 2*x)*x^4 + 12*x*Log[x^2] - 6*x*Log[x^2]^2)/(144*x^4 - 24*E^(3 + x)*x^4 +
E^(6 + 2*x)*x^4 + E^(4 + 4*x)*x^4 + E^(2 + 2*x)*(24*x^4 - 2*E^(3 + x)*x^4) + (-24*x^2 + 2*E^(3 + x)*x^2 - 2*E
^(2 + 2*x)*x^2)*Log[x^2]^2 + Log[x^2]^4),x]

[Out]

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

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fricas [A]  time = 1.10, size = 46, normalized size = 1.48 \begin {gather*} \frac {3 \, x^{2} e^{4}}{12 \, x^{2} e^{4} + x^{2} e^{\left (2 \, x + 6\right )} - x^{2} e^{\left (x + 7\right )} - e^{4} \log \left (x^{2}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x*log(x^2)^2+12*x*log(x^2)-6*x^4*exp(2*x+2)+3*x^4*exp(3+x))/(log(x^2)^4+(-2*x^2*exp(2*x+2)+2*x^2
*exp(3+x)-24*x^2)*log(x^2)^2+x^4*exp(2*x+2)^2+(-2*x^4*exp(3+x)+24*x^4)*exp(2*x+2)+x^4*exp(3+x)^2-24*x^4*exp(3+
x)+144*x^4),x, algorithm="fricas")

[Out]

3*x^2*e^4/(12*x^2*e^4 + x^2*e^(2*x + 6) - x^2*e^(x + 7) - e^4*log(x^2)^2)

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giac [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((-6*x*log(x^2)^2+12*x*log(x^2)-6*x^4*exp(2*x+2)+3*x^4*exp(3+x))/(log(x^2)^4+(-2*x^2*exp(2*x+2)+2*x^2
*exp(3+x)-24*x^2)*log(x^2)^2+x^4*exp(2*x+2)^2+(-2*x^4*exp(3+x)+24*x^4)*exp(2*x+2)+x^4*exp(3+x)^2-24*x^4*exp(3+
x)+144*x^4),x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 0.50, size = 189, normalized size = 6.10

method	result	size
risch	\(\frac {12 x^{2}}{\pi ^{2} \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}-4 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}+6 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}-4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}+\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{6}+8 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-16 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+8 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 x^{2} {\mathrm e}^{2 x +2}-4 x^{2} {\mathrm e}^{3+x}+48 x^{2}-16 \ln \relax (x )^{2}}\)	\(189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x*ln(x^2)^2+12*x*ln(x^2)-6*x^4*exp(2*x+2)+3*x^4*exp(3+x))/(ln(x^2)^4+(-2*x^2*exp(2*x+2)+2*x^2*exp(3+x)
-24*x^2)*ln(x^2)^2+x^4*exp(2*x+2)^2+(-2*x^4*exp(3+x)+24*x^4)*exp(2*x+2)+x^4*exp(3+x)^2-24*x^4*exp(3+x)+144*x^4
),x,method=_RETURNVERBOSE)

[Out]

12*x^2/(Pi^2*csgn(I*x)^4*csgn(I*x^2)^2-4*Pi^2*csgn(I*x)^3*csgn(I*x^2)^3+6*Pi^2*csgn(I*x)^2*csgn(I*x^2)^4-4*Pi^
2*csgn(I*x)*csgn(I*x^2)^5+Pi^2*csgn(I*x^2)^6+8*I*ln(x)*Pi*csgn(I*x)^2*csgn(I*x^2)-16*I*ln(x)*Pi*csgn(I*x)*csgn
(I*x^2)^2+8*I*ln(x)*Pi*csgn(I*x^2)^3+4*x^2*exp(2*x+2)-4*x^2*exp(3+x)+48*x^2-16*ln(x)^2)

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maxima [A]  time = 0.49, size = 38, normalized size = 1.23 \begin {gather*} \frac {3 \, x^{2}}{x^{2} e^{\left (2 \, x + 2\right )} - x^{2} e^{\left (x + 3\right )} + 12 \, x^{2} - 4 \, \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x*log(x^2)^2+12*x*log(x^2)-6*x^4*exp(2*x+2)+3*x^4*exp(3+x))/(log(x^2)^4+(-2*x^2*exp(2*x+2)+2*x^2
*exp(3+x)-24*x^2)*log(x^2)^2+x^4*exp(2*x+2)^2+(-2*x^4*exp(3+x)+24*x^4)*exp(2*x+2)+x^4*exp(3+x)^2-24*x^4*exp(3+
x)+144*x^4),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {12\,x\,\ln \left (x^2\right )+3\,x^4\,{\mathrm {e}}^{x+3}-6\,x\,{\ln \left (x^2\right )}^2-6\,x^4\,{\mathrm {e}}^{2\,x+2}}{{\ln \left (x^2\right )}^4-24\,x^4\,{\mathrm {e}}^{x+3}-{\mathrm {e}}^{2\,x+2}\,\left (2\,x^4\,{\mathrm {e}}^{x+3}-24\,x^4\right )+x^4\,{\mathrm {e}}^{2\,x+6}+x^4\,{\mathrm {e}}^{4\,x+4}+144\,x^4-{\ln \left (x^2\right )}^2\,\left (2\,x^2\,{\mathrm {e}}^{2\,x+2}-2\,x^2\,{\mathrm {e}}^{x+3}+24\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x*log(x^2) + 3*x^4*exp(x + 3) - 6*x*log(x^2)^2 - 6*x^4*exp(2*x + 2))/(log(x^2)^4 - 24*x^4*exp(x + 3) -
exp(2*x + 2)*(2*x^4*exp(x + 3) - 24*x^4) + x^4*exp(2*x + 6) + x^4*exp(4*x + 4) + 144*x^4 - log(x^2)^2*(2*x^2*
exp(2*x + 2) - 2*x^2*exp(x + 3) + 24*x^2)),x)

[Out]

int((12*x*log(x^2) + 3*x^4*exp(x + 3) - 6*x*log(x^2)^2 - 6*x^4*exp(2*x + 2))/(log(x^2)^4 - 24*x^4*exp(x + 3) -
exp(2*x + 2)*(2*x^4*exp(x + 3) - 24*x^4) + x^4*exp(2*x + 6) + x^4*exp(4*x + 4) + 144*x^4 - log(x^2)^2*(2*x^2*
exp(2*x + 2) - 2*x^2*exp(x + 3) + 24*x^2)), x)

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sympy [A]  time = 0.43, size = 48, normalized size = 1.55 \begin {gather*} \frac {3 x^{2} e^{4}}{- x^{2} e^{4} e^{x + 3} + x^{2} e^{2 x + 6} + 12 x^{2} e^{4} - e^{4} \log {\left (x^{2} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x*ln(x**2)**2+12*x*ln(x**2)-6*x**4*exp(2*x+2)+3*x**4*exp(3+x))/(ln(x**2)**4+(-2*x**2*exp(2*x+2)+
2*x**2*exp(3+x)-24*x**2)*ln(x**2)**2+x**4*exp(2*x+2)**2+(-2*x**4*exp(3+x)+24*x**4)*exp(2*x+2)+x**4*exp(3+x)**2
-24*x**4*exp(3+x)+144*x**4),x)

[Out]

3*x**2*exp(4)/(-x**2*exp(4)*exp(x + 3) + x**2*exp(2*x + 6) + 12*x**2*exp(4) - exp(4)*log(x**2)**2)

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