3.11.36 \(\int \frac {e^x (54+9 x-9 x^2)+e^{3 x} (-108 x^2-36 x^3+33 x^4+6 x^5-3 x^6)+e^{2 x} (108 x^2+36 x^3-33 x^4-6 x^5+3 x^6)+e^x (-54-72 x+18 x^2+9 x^3) \log (x)}{e^{4 x} (36 x^2+12 x^3-11 x^4-2 x^5+x^6)+e^{3 x} (-72 x^3-24 x^4+22 x^5+4 x^6-2 x^7)+e^{2 x} (36 x^4+12 x^5-11 x^6-2 x^7+x^8)+(e^{2 x} (-36 x-6 x^2+6 x^3)+e^x (36 x^2+6 x^3-6 x^4)) \log (x)+9 \log ^2(x)} \, dx\)
Optimal. Leaf size=36 \[ \frac {3}{e^x-x-\frac {e^{-x} \log (x)}{(2+x) \left (x-\frac {x^2}{3}\right )}} \]
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Rubi [F] time = 180.00, 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*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
Int[(E^x*(54 + 9*x - 9*x^2) + E^(3*x)*(-108*x^2 - 36*x^3 + 33*x^4 + 6*x^5 - 3*x^6) + E^(2*x)*(108*x^2 + 36*x^3
- 33*x^4 - 6*x^5 + 3*x^6) + E^x*(-54 - 72*x + 18*x^2 + 9*x^3)*Log[x])/(E^(4*x)*(36*x^2 + 12*x^3 - 11*x^4 - 2*
x^5 + x^6) + E^(3*x)*(-72*x^3 - 24*x^4 + 22*x^5 + 4*x^6 - 2*x^7) + E^(2*x)*(36*x^4 + 12*x^5 - 11*x^6 - 2*x^7 +
x^8) + (E^(2*x)*(-36*x - 6*x^2 + 6*x^3) + E^x*(36*x^2 + 6*x^3 - 6*x^4))*Log[x] + 9*Log[x]^2),x]
[Out]
$Aborted
Rubi steps
Aborted
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Mathematica [A] time = 0.11, size = 41, normalized size = 1.14 \begin {gather*} \frac {3 e^x x \left (-6-x+x^2\right )}{e^x \left (e^x-x\right ) x \left (-6-x+x^2\right )+3 \log (x)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(E^x*(54 + 9*x - 9*x^2) + E^(3*x)*(-108*x^2 - 36*x^3 + 33*x^4 + 6*x^5 - 3*x^6) + E^(2*x)*(108*x^2 +
36*x^3 - 33*x^4 - 6*x^5 + 3*x^6) + E^x*(-54 - 72*x + 18*x^2 + 9*x^3)*Log[x])/(E^(4*x)*(36*x^2 + 12*x^3 - 11*x^
4 - 2*x^5 + x^6) + E^(3*x)*(-72*x^3 - 24*x^4 + 22*x^5 + 4*x^6 - 2*x^7) + E^(2*x)*(36*x^4 + 12*x^5 - 11*x^6 - 2
*x^7 + x^8) + (E^(2*x)*(-36*x - 6*x^2 + 6*x^3) + E^x*(36*x^2 + 6*x^3 - 6*x^4))*Log[x] + 9*Log[x]^2),x]
[Out]
(3*E^x*x*(-6 - x + x^2))/(E^x*(E^x - x)*x*(-6 - x + x^2) + 3*Log[x])
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fricas [A] time = 0.65, size = 58, normalized size = 1.61 \begin {gather*} \frac {3 \, {\left (x^{3} - x^{2} - 6 \, x\right )} e^{x}}{{\left (x^{3} - x^{2} - 6 \, x\right )} e^{\left (2 \, x\right )} - {\left (x^{4} - x^{3} - 6 \, x^{2}\right )} e^{x} + 3 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((9*x^3+18*x^2-72*x-54)*exp(x)*log(x)+(-3*x^6+6*x^5+33*x^4-36*x^3-108*x^2)*exp(x)^3+(3*x^6-6*x^5-33*
x^4+36*x^3+108*x^2)*exp(x)^2+(-9*x^2+9*x+54)*exp(x))/(9*log(x)^2+((6*x^3-6*x^2-36*x)*exp(x)^2+(-6*x^4+6*x^3+36
*x^2)*exp(x))*log(x)+(x^6-2*x^5-11*x^4+12*x^3+36*x^2)*exp(x)^4+(-2*x^7+4*x^6+22*x^5-24*x^4-72*x^3)*exp(x)^3+(x
^8-2*x^7-11*x^6+12*x^5+36*x^4)*exp(x)^2),x, algorithm="fricas")
[Out]
3*(x^3 - x^2 - 6*x)*e^x/((x^3 - x^2 - 6*x)*e^(2*x) - (x^4 - x^3 - 6*x^2)*e^x + 3*log(x))
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giac [B] time = 1.66, size = 72, normalized size = 2.00 \begin {gather*} -\frac {3 \, {\left (x^{3} e^{x} - x^{2} e^{x} - 6 \, x e^{x}\right )}}{x^{4} e^{x} - x^{3} e^{\left (2 \, x\right )} - x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )} - 6 \, x^{2} e^{x} + 6 \, x e^{\left (2 \, x\right )} - 3 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((9*x^3+18*x^2-72*x-54)*exp(x)*log(x)+(-3*x^6+6*x^5+33*x^4-36*x^3-108*x^2)*exp(x)^3+(3*x^6-6*x^5-33*
x^4+36*x^3+108*x^2)*exp(x)^2+(-9*x^2+9*x+54)*exp(x))/(9*log(x)^2+((6*x^3-6*x^2-36*x)*exp(x)^2+(-6*x^4+6*x^3+36
*x^2)*exp(x))*log(x)+(x^6-2*x^5-11*x^4+12*x^3+36*x^2)*exp(x)^4+(-2*x^7+4*x^6+22*x^5-24*x^4-72*x^3)*exp(x)^3+(x
^8-2*x^7-11*x^6+12*x^5+36*x^4)*exp(x)^2),x, algorithm="giac")
[Out]
-3*(x^3*e^x - x^2*e^x - 6*x*e^x)/(x^4*e^x - x^3*e^(2*x) - x^3*e^x + x^2*e^(2*x) - 6*x^2*e^x + 6*x*e^(2*x) - 3*
log(x))
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maple [A] time = 0.04, size = 65, normalized size = 1.81
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\(-\frac {3 \left (x^{2}-x -6\right ) x \,{\mathrm e}^{x}}{{\mathrm e}^{x} x^{4}-{\mathrm e}^{2 x} x^{3}-{\mathrm e}^{x} x^{3}+{\mathrm e}^{2 x} x^{2}-6 \,{\mathrm e}^{x} x^{2}+6 x \,{\mathrm e}^{2 x}-3 \ln \relax (x )}\) |
\(65\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((9*x^3+18*x^2-72*x-54)*exp(x)*ln(x)+(-3*x^6+6*x^5+33*x^4-36*x^3-108*x^2)*exp(x)^3+(3*x^6-6*x^5-33*x^4+36*
x^3+108*x^2)*exp(x)^2+(-9*x^2+9*x+54)*exp(x))/(9*ln(x)^2+((6*x^3-6*x^2-36*x)*exp(x)^2+(-6*x^4+6*x^3+36*x^2)*ex
p(x))*ln(x)+(x^6-2*x^5-11*x^4+12*x^3+36*x^2)*exp(x)^4+(-2*x^7+4*x^6+22*x^5-24*x^4-72*x^3)*exp(x)^3+(x^8-2*x^7-
11*x^6+12*x^5+36*x^4)*exp(x)^2),x,method=_RETURNVERBOSE)
[Out]
-3*(x^2-x-6)*x*exp(x)/(exp(x)*x^4-exp(2*x)*x^3-exp(x)*x^3+exp(2*x)*x^2-6*exp(x)*x^2+6*x*exp(2*x)-3*ln(x))
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maxima [A] time = 0.53, size = 58, normalized size = 1.61 \begin {gather*} \frac {3 \, {\left (x^{3} - x^{2} - 6 \, x\right )} e^{x}}{{\left (x^{3} - x^{2} - 6 \, x\right )} e^{\left (2 \, x\right )} - {\left (x^{4} - x^{3} - 6 \, x^{2}\right )} e^{x} + 3 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((9*x^3+18*x^2-72*x-54)*exp(x)*log(x)+(-3*x^6+6*x^5+33*x^4-36*x^3-108*x^2)*exp(x)^3+(3*x^6-6*x^5-33*
x^4+36*x^3+108*x^2)*exp(x)^2+(-9*x^2+9*x+54)*exp(x))/(9*log(x)^2+((6*x^3-6*x^2-36*x)*exp(x)^2+(-6*x^4+6*x^3+36
*x^2)*exp(x))*log(x)+(x^6-2*x^5-11*x^4+12*x^3+36*x^2)*exp(x)^4+(-2*x^7+4*x^6+22*x^5-24*x^4-72*x^3)*exp(x)^3+(x
^8-2*x^7-11*x^6+12*x^5+36*x^4)*exp(x)^2),x, algorithm="maxima")
[Out]
3*(x^3 - x^2 - 6*x)*e^x/((x^3 - x^2 - 6*x)*e^(2*x) - (x^4 - x^3 - 6*x^2)*e^x + 3*log(x))
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{2\,x}\,\left (3\,x^6-6\,x^5-33\,x^4+36\,x^3+108\,x^2\right )-{\mathrm {e}}^{3\,x}\,\left (3\,x^6-6\,x^5-33\,x^4+36\,x^3+108\,x^2\right )+{\mathrm {e}}^x\,\left (-9\,x^2+9\,x+54\right )-{\mathrm {e}}^x\,\ln \relax (x)\,\left (-9\,x^3-18\,x^2+72\,x+54\right )}{9\,{\ln \relax (x)}^2+\left ({\mathrm {e}}^x\,\left (-6\,x^4+6\,x^3+36\,x^2\right )-{\mathrm {e}}^{2\,x}\,\left (-6\,x^3+6\,x^2+36\,x\right )\right )\,\ln \relax (x)-{\mathrm {e}}^{3\,x}\,\left (2\,x^7-4\,x^6-22\,x^5+24\,x^4+72\,x^3\right )+{\mathrm {e}}^{4\,x}\,\left (x^6-2\,x^5-11\,x^4+12\,x^3+36\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (x^8-2\,x^7-11\,x^6+12\,x^5+36\,x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((exp(2*x)*(108*x^2 + 36*x^3 - 33*x^4 - 6*x^5 + 3*x^6) - exp(3*x)*(108*x^2 + 36*x^3 - 33*x^4 - 6*x^5 + 3*x^
6) + exp(x)*(9*x - 9*x^2 + 54) - exp(x)*log(x)*(72*x - 18*x^2 - 9*x^3 + 54))/(9*log(x)^2 - log(x)*(exp(2*x)*(3
6*x + 6*x^2 - 6*x^3) - exp(x)*(36*x^2 + 6*x^3 - 6*x^4)) - exp(3*x)*(72*x^3 + 24*x^4 - 22*x^5 - 4*x^6 + 2*x^7)
+ exp(4*x)*(36*x^2 + 12*x^3 - 11*x^4 - 2*x^5 + x^6) + exp(2*x)*(36*x^4 + 12*x^5 - 11*x^6 - 2*x^7 + x^8)),x)
[Out]
int((exp(2*x)*(108*x^2 + 36*x^3 - 33*x^4 - 6*x^5 + 3*x^6) - exp(3*x)*(108*x^2 + 36*x^3 - 33*x^4 - 6*x^5 + 3*x^
6) + exp(x)*(9*x - 9*x^2 + 54) - exp(x)*log(x)*(72*x - 18*x^2 - 9*x^3 + 54))/(9*log(x)^2 - log(x)*(exp(2*x)*(3
6*x + 6*x^2 - 6*x^3) - exp(x)*(36*x^2 + 6*x^3 - 6*x^4)) - exp(3*x)*(72*x^3 + 24*x^4 - 22*x^5 - 4*x^6 + 2*x^7)
+ exp(4*x)*(36*x^2 + 12*x^3 - 11*x^4 - 2*x^5 + x^6) + exp(2*x)*(36*x^4 + 12*x^5 - 11*x^6 - 2*x^7 + x^8)), x)
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sympy [B] time = 0.73, size = 51, normalized size = 1.42 \begin {gather*} \frac {\left (3 x^{3} - 3 x^{2} - 18 x\right ) e^{x}}{\left (x^{3} - x^{2} - 6 x\right ) e^{2 x} + \left (- x^{4} + x^{3} + 6 x^{2}\right ) e^{x} + 3 \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((9*x**3+18*x**2-72*x-54)*exp(x)*ln(x)+(-3*x**6+6*x**5+33*x**4-36*x**3-108*x**2)*exp(x)**3+(3*x**6-6
*x**5-33*x**4+36*x**3+108*x**2)*exp(x)**2+(-9*x**2+9*x+54)*exp(x))/(9*ln(x)**2+((6*x**3-6*x**2-36*x)*exp(x)**2
+(-6*x**4+6*x**3+36*x**2)*exp(x))*ln(x)+(x**6-2*x**5-11*x**4+12*x**3+36*x**2)*exp(x)**4+(-2*x**7+4*x**6+22*x**
5-24*x**4-72*x**3)*exp(x)**3+(x**8-2*x**7-11*x**6+12*x**5+36*x**4)*exp(x)**2),x)
[Out]
(3*x**3 - 3*x**2 - 18*x)*exp(x)/((x**3 - x**2 - 6*x)*exp(2*x) + (-x**4 + x**3 + 6*x**2)*exp(x) + 3*log(x))
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