3.95.44 \(\int \frac {-972-231 x+3 x^2+e^{3 e^3} (-3 x+3 x^2)+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} (-72 x+20 x^2+2 x^3)+(-288+152 x-12 x^2-2 x^3+e^{3 e^3} (8 x-2 x^2)) \log (x)+(16-8 x+x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=33 \[ 2-\frac {3 x}{4-x-\frac {\left (5+e^{3 e^3}\right ) x}{9+x-\log (x)}} \]

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Rubi [F]  time = 3.25, 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 {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-972 - 231*x + 3*x^2 + E^(3*E^3)*(-3*x + 3*x^2) + (216 + 24*x)*Log[x] - 12*Log[x]^2)/(1296 - 720*x + 28*x
^2 + E^(6*E^3)*x^2 + 20*x^3 + x^4 + E^(3*E^3)*(-72*x + 20*x^2 + 2*x^3) + (-288 + 152*x - 12*x^2 - 2*x^3 + E^(3
*E^3)*(8*x - 2*x^2))*Log[x] + (16 - 8*x + x^2)*Log[x]^2),x]

[Out]

-12/(4 - x) - 12*(5 + E^(3*E^3))^2*Defer[Int][(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])^(-2),
x] - 192*(5 + E^(3*E^3))^2*Defer[Int][1/((4 - x)^2*(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])
^2), x] + 96*(5 + E^(3*E^3))^2*Defer[Int][1/((4 - x)*(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x]
)^2), x] - 3*(5 + E^(3*E^3))*Defer[Int][x/(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])^2, x] + 3
*(5 + E^(3*E^3))*Defer[Int][x^2/(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])^2, x] - 96*(5 + E^(
3*E^3))*Defer[Int][1/((4 - x)^2*(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])), x] - 24*(5 + E^(3
*E^3))*Defer[Int][1/((-4 + x)*(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+\left (28+e^{6 e^3}\right ) x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx\\ &=\int \frac {3 \left (-324-\left (77+e^{3 e^3}\right ) x+\left (1+e^{3 e^3}\right ) x^2+8 (9+x) \log (x)-4 \log ^2(x)\right )}{\left (36-\left (10+e^{3 e^3}\right ) x-x^2+(-4+x) \log (x)\right )^2} \, dx\\ &=3 \int \frac {-324-\left (77+e^{3 e^3}\right ) x+\left (1+e^{3 e^3}\right ) x^2+8 (9+x) \log (x)-4 \log ^2(x)}{\left (36-\left (10+e^{3 e^3}\right ) x-x^2+(-4+x) \log (x)\right )^2} \, dx\\ &=3 \int \left (-\frac {4}{(-4+x)^2}+\frac {x \left (-16 \left (5+e^{3 e^3}\right )+4 \left (5-4 e^{3 e^3}-e^{6 e^3}\right ) x-9 \left (5+e^{3 e^3}\right ) x^2+\left (5+e^{3 e^3}\right ) x^3\right )}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}+\frac {8 \left (-5-e^{3 e^3}\right ) x}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )}\right ) \, dx\\ &=-\frac {12}{4-x}+3 \int \frac {x \left (-16 \left (5+e^{3 e^3}\right )+4 \left (5-4 e^{3 e^3}-e^{6 e^3}\right ) x-9 \left (5+e^{3 e^3}\right ) x^2+\left (5+e^{3 e^3}\right ) x^3\right )}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx-\left (24 \left (5+e^{3 e^3}\right )\right ) \int \frac {x}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )} \, dx\\ &=-\frac {12}{4-x}+3 \int \left (-\frac {4 \left (5+e^{3 e^3}\right )^2}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}-\frac {64 \left (5+e^{3 e^3}\right )^2}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}+\frac {32 \left (5+e^{3 e^3}\right )^2}{(4-x) \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}+\frac {\left (-5-e^{3 e^3}\right ) x}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}+\frac {\left (5+e^{3 e^3}\right ) x^2}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}\right ) \, dx-\left (24 \left (5+e^{3 e^3}\right )\right ) \int \left (\frac {4}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )}+\frac {1}{(-4+x) \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )}\right ) \, dx\\ &=-\frac {12}{4-x}-\left (3 \left (5+e^{3 e^3}\right )\right ) \int \frac {x}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx+\left (3 \left (5+e^{3 e^3}\right )\right ) \int \frac {x^2}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx-\left (24 \left (5+e^{3 e^3}\right )\right ) \int \frac {1}{(-4+x) \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )} \, dx-\left (96 \left (5+e^{3 e^3}\right )\right ) \int \frac {1}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )} \, dx-\left (12 \left (5+e^{3 e^3}\right )^2\right ) \int \frac {1}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx+\left (96 \left (5+e^{3 e^3}\right )^2\right ) \int \frac {1}{(4-x) \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx-\left (192 \left (5+e^{3 e^3}\right )^2\right ) \int \frac {1}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.73, size = 45, normalized size = 1.36 \begin {gather*} \frac {3 \left (36-\left (1+e^{3 e^3}\right ) x-4 \log (x)\right )}{-36+\left (10+e^{3 e^3}\right ) x+x^2-(-4+x) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-972 - 231*x + 3*x^2 + E^(3*E^3)*(-3*x + 3*x^2) + (216 + 24*x)*Log[x] - 12*Log[x]^2)/(1296 - 720*x
+ 28*x^2 + E^(6*E^3)*x^2 + 20*x^3 + x^4 + E^(3*E^3)*(-72*x + 20*x^2 + 2*x^3) + (-288 + 152*x - 12*x^2 - 2*x^3
+ E^(3*E^3)*(8*x - 2*x^2))*Log[x] + (16 - 8*x + x^2)*Log[x]^2),x]

[Out]

(3*(36 - (1 + E^(3*E^3))*x - 4*Log[x]))/(-36 + (10 + E^(3*E^3))*x + x^2 - (-4 + x)*Log[x])

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fricas [A]  time = 0.52, size = 40, normalized size = 1.21 \begin {gather*} -\frac {3 \, {\left (x e^{\left (3 \, e^{3}\right )} + x + 4 \, \log \relax (x) - 36\right )}}{x^{2} + x e^{\left (3 \, e^{3}\right )} - {\left (x - 4\right )} \log \relax (x) + 10 \, x - 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*log(x)^2+(24*x+216)*log(x)+(3*x^2-3*x)*exp(3*exp(3))+3*x^2-231*x-972)/((x^2-8*x+16)*log(x)^2+((
-2*x^2+8*x)*exp(3*exp(3))-2*x^3-12*x^2+152*x-288)*log(x)+x^2*exp(3*exp(3))^2+(2*x^3+20*x^2-72*x)*exp(3*exp(3))
+x^4+20*x^3+28*x^2-720*x+1296),x, algorithm="fricas")

[Out]

-3*(x*e^(3*e^3) + x + 4*log(x) - 36)/(x^2 + x*e^(3*e^3) - (x - 4)*log(x) + 10*x - 36)

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giac [A]  time = 0.16, size = 42, normalized size = 1.27 \begin {gather*} -\frac {3 \, {\left (x e^{\left (3 \, e^{3}\right )} + x + 4 \, \log \relax (x) - 36\right )}}{x^{2} + x e^{\left (3 \, e^{3}\right )} - x \log \relax (x) + 10 \, x + 4 \, \log \relax (x) - 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*log(x)^2+(24*x+216)*log(x)+(3*x^2-3*x)*exp(3*exp(3))+3*x^2-231*x-972)/((x^2-8*x+16)*log(x)^2+((
-2*x^2+8*x)*exp(3*exp(3))-2*x^3-12*x^2+152*x-288)*log(x)+x^2*exp(3*exp(3))^2+(2*x^3+20*x^2-72*x)*exp(3*exp(3))
+x^4+20*x^3+28*x^2-720*x+1296),x, algorithm="giac")

[Out]

-3*(x*e^(3*e^3) + x + 4*log(x) - 36)/(x^2 + x*e^(3*e^3) - x*log(x) + 10*x + 4*log(x) - 36)

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maple [A]  time = 0.22, size = 45, normalized size = 1.36

method	result	size
norman	\(\frac {-12 \ln \relax (x )+\left (-3 \,{\mathrm e}^{3 \,{\mathrm e}^{3}}-3\right ) x +108}{{\mathrm e}^{3 \,{\mathrm e}^{3}} x +x^{2}-x \ln \relax (x )+10 x +4 \ln \relax (x )-36}\)	\(45\)
risch	\(\frac {12}{x -4}-\frac {3 \left (5+{\mathrm e}^{3 \,{\mathrm e}^{3}}\right ) x^{2}}{\left (x -4\right ) \left ({\mathrm e}^{3 \,{\mathrm e}^{3}} x +x^{2}-x \ln \relax (x )+10 x +4 \ln \relax (x )-36\right )}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12*ln(x)^2+(24*x+216)*ln(x)+(3*x^2-3*x)*exp(3*exp(3))+3*x^2-231*x-972)/((x^2-8*x+16)*ln(x)^2+((-2*x^2+8*
x)*exp(3*exp(3))-2*x^3-12*x^2+152*x-288)*ln(x)+x^2*exp(3*exp(3))^2+(2*x^3+20*x^2-72*x)*exp(3*exp(3))+x^4+20*x^
3+28*x^2-720*x+1296),x,method=_RETURNVERBOSE)

[Out]

(-12*ln(x)+(-3*exp(exp(3))^3-3)*x+108)/(exp(3*exp(3))*x+x^2-x*ln(x)+10*x+4*ln(x)-36)

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maxima [A]  time = 0.39, size = 40, normalized size = 1.21 \begin {gather*} -\frac {3 \, {\left (x {\left (e^{\left (3 \, e^{3}\right )} + 1\right )} + 4 \, \log \relax (x) - 36\right )}}{x^{2} + x {\left (e^{\left (3 \, e^{3}\right )} + 10\right )} - {\left (x - 4\right )} \log \relax (x) - 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*log(x)^2+(24*x+216)*log(x)+(3*x^2-3*x)*exp(3*exp(3))+3*x^2-231*x-972)/((x^2-8*x+16)*log(x)^2+((
-2*x^2+8*x)*exp(3*exp(3))-2*x^3-12*x^2+152*x-288)*log(x)+x^2*exp(3*exp(3))^2+(2*x^3+20*x^2-72*x)*exp(3*exp(3))
+x^4+20*x^3+28*x^2-720*x+1296),x, algorithm="maxima")

[Out]

-3*(x*(e^(3*e^3) + 1) + 4*log(x) - 36)/(x^2 + x*(e^(3*e^3) + 10) - (x - 4)*log(x) - 36)

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mupad [B]  time = 8.23, size = 45, normalized size = 1.36 \begin {gather*} -\frac {12\,\ln \relax (x)+x\,\left (3\,{\mathrm {e}}^{3\,{\mathrm {e}}^3}+3\right )-108}{10\,x+4\,\ln \relax (x)+x\,{\mathrm {e}}^{3\,{\mathrm {e}}^3}-x\,\ln \relax (x)+x^2-36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(231*x + 12*log(x)^2 - log(x)*(24*x + 216) - 3*x^2 + exp(3*exp(3))*(3*x - 3*x^2) + 972)/(exp(3*exp(3))*(2
0*x^2 - 72*x + 2*x^3) - 720*x + x^2*exp(6*exp(3)) + log(x)^2*(x^2 - 8*x + 16) + 28*x^2 + 20*x^3 + x^4 - log(x)
*(12*x^2 - 152*x + 2*x^3 - exp(3*exp(3))*(8*x - 2*x^2) + 288) + 1296),x)

[Out]

-(12*log(x) + x*(3*exp(3*exp(3)) + 3) - 108)/(10*x + 4*log(x) + x*exp(3*exp(3)) - x*log(x) + x^2 - 36)

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sympy [B]  time = 0.38, size = 66, normalized size = 2.00 \begin {gather*} \frac {15 x^{2} + 3 x^{2} e^{3 e^{3}}}{- x^{3} - x^{2} e^{3 e^{3}} - 6 x^{2} + 76 x + 4 x e^{3 e^{3}} + \left (x^{2} - 8 x + 16\right ) \log {\relax (x )} - 144} + \frac {12}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*ln(x)**2+(24*x+216)*ln(x)+(3*x**2-3*x)*exp(3*exp(3))+3*x**2-231*x-972)/((x**2-8*x+16)*ln(x)**2+
((-2*x**2+8*x)*exp(3*exp(3))-2*x**3-12*x**2+152*x-288)*ln(x)+x**2*exp(3*exp(3))**2+(2*x**3+20*x**2-72*x)*exp(3
*exp(3))+x**4+20*x**3+28*x**2-720*x+1296),x)

[Out]

(15*x**2 + 3*x**2*exp(3*exp(3)))/(-x**3 - x**2*exp(3*exp(3)) - 6*x**2 + 76*x + 4*x*exp(3*exp(3)) + (x**2 - 8*x
+ 16)*log(x) - 144) + 12/(x - 4)

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