3.88.70 \(\int \frac {-140 x+130 x^3+20 x^4+10 x^6+(40 x-40 x^3) \log (x)+e^{e^x} (-70 x+10 x^4+e^x (-30 x^2-10 x^5)+(20 x+10 e^x x^2) \log (x))}{9+6 x^3+x^6+(-6-2 x^3) \log (x)+\log ^2(x)} \, dx\)

Optimal. Leaf size=28 \[ \frac {10 x^2 \left (2+e^{e^x}-x^2\right )}{-3-x^3+\log (x)} \]

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Rubi [F]  time = 2.96, 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 {-140 x+130 x^3+20 x^4+10 x^6+\left (40 x-40 x^3\right ) \log (x)+e^{e^x} \left (-70 x+10 x^4+e^x \left (-30 x^2-10 x^5\right )+\left (20 x+10 e^x x^2\right ) \log (x)\right )}{9+6 x^3+x^6+\left (-6-2 x^3\right ) \log (x)+\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-140*x + 130*x^3 + 20*x^4 + 10*x^6 + (40*x - 40*x^3)*Log[x] + E^E^x*(-70*x + 10*x^4 + E^x*(-30*x^2 - 10*x
^5) + (20*x + 10*E^x*x^2)*Log[x]))/(9 + 6*x^3 + x^6 + (-6 - 2*x^3)*Log[x] + Log[x]^2),x]

[Out]

-20*Defer[Int][x/(3 + x^3 - Log[x])^2, x] - 10*Defer[Int][(E^E^x*x)/(3 + x^3 - Log[x])^2, x] + 10*Defer[Int][x
^3/(3 + x^3 - Log[x])^2, x] + 60*Defer[Int][x^4/(3 + x^3 - Log[x])^2, x] + 30*Defer[Int][(E^E^x*x^4)/(3 + x^3
- Log[x])^2, x] - 30*Defer[Int][x^6/(3 + x^3 - Log[x])^2, x] - 40*Defer[Int][x/(3 + x^3 - Log[x]), x] - 20*Def
er[Int][(E^E^x*x)/(3 + x^3 - Log[x]), x] - 10*Defer[Int][(E^(E^x + x)*x^2)/(3 + x^3 - Log[x]), x] + 40*Defer[I
nt][x^3/(3 + x^3 - Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-140 x+130 x^3+20 x^4+10 x^6+\left (40 x-40 x^3\right ) \log (x)+e^{e^x} \left (-70 x+10 x^4+e^x \left (-30 x^2-10 x^5\right )+\left (20 x+10 e^x x^2\right ) \log (x)\right )}{\left (3+x^3-\log (x)\right )^2} \, dx\\ &=\int \left (-\frac {140 x}{\left (3+x^3-\log (x)\right )^2}-\frac {70 e^{e^x} x}{\left (3+x^3-\log (x)\right )^2}+\frac {130 x^3}{\left (3+x^3-\log (x)\right )^2}+\frac {20 x^4}{\left (3+x^3-\log (x)\right )^2}+\frac {10 e^{e^x} x^4}{\left (3+x^3-\log (x)\right )^2}+\frac {10 x^6}{\left (3+x^3-\log (x)\right )^2}-\frac {10 e^{e^x+x} x^2}{3+x^3-\log (x)}+\frac {20 e^{e^x} x \log (x)}{\left (3+x^3-\log (x)\right )^2}-\frac {40 (-1+x) x (1+x) \log (x)}{\left (3+x^3-\log (x)\right )^2}\right ) \, dx\\ &=10 \int \frac {e^{e^x} x^4}{\left (3+x^3-\log (x)\right )^2} \, dx+10 \int \frac {x^6}{\left (3+x^3-\log (x)\right )^2} \, dx-10 \int \frac {e^{e^x+x} x^2}{3+x^3-\log (x)} \, dx+20 \int \frac {x^4}{\left (3+x^3-\log (x)\right )^2} \, dx+20 \int \frac {e^{e^x} x \log (x)}{\left (3+x^3-\log (x)\right )^2} \, dx-40 \int \frac {(-1+x) x (1+x) \log (x)}{\left (3+x^3-\log (x)\right )^2} \, dx-70 \int \frac {e^{e^x} x}{\left (3+x^3-\log (x)\right )^2} \, dx+130 \int \frac {x^3}{\left (3+x^3-\log (x)\right )^2} \, dx-140 \int \frac {x}{\left (3+x^3-\log (x)\right )^2} \, dx\\ &=10 \int \frac {e^{e^x} x^4}{\left (3+x^3-\log (x)\right )^2} \, dx+10 \int \frac {x^6}{\left (3+x^3-\log (x)\right )^2} \, dx-10 \int \frac {e^{e^x+x} x^2}{3+x^3-\log (x)} \, dx+20 \int \left (\frac {e^{e^x} x \left (3+x^3\right )}{\left (3+x^3-\log (x)\right )^2}-\frac {e^{e^x} x}{3+x^3-\log (x)}\right ) \, dx+20 \int \frac {x^4}{\left (3+x^3-\log (x)\right )^2} \, dx-40 \int \left (\frac {x \left (-3+3 x^2-x^3+x^5\right )}{\left (3+x^3-\log (x)\right )^2}+\frac {x \left (1-x^2\right )}{3+x^3-\log (x)}\right ) \, dx-70 \int \frac {e^{e^x} x}{\left (3+x^3-\log (x)\right )^2} \, dx+130 \int \frac {x^3}{\left (3+x^3-\log (x)\right )^2} \, dx-140 \int \frac {x}{\left (3+x^3-\log (x)\right )^2} \, dx\\ &=10 \int \frac {e^{e^x} x^4}{\left (3+x^3-\log (x)\right )^2} \, dx+10 \int \frac {x^6}{\left (3+x^3-\log (x)\right )^2} \, dx-10 \int \frac {e^{e^x+x} x^2}{3+x^3-\log (x)} \, dx+20 \int \frac {x^4}{\left (3+x^3-\log (x)\right )^2} \, dx+20 \int \frac {e^{e^x} x \left (3+x^3\right )}{\left (3+x^3-\log (x)\right )^2} \, dx-20 \int \frac {e^{e^x} x}{3+x^3-\log (x)} \, dx-40 \int \frac {x \left (-3+3 x^2-x^3+x^5\right )}{\left (3+x^3-\log (x)\right )^2} \, dx-40 \int \frac {x \left (1-x^2\right )}{3+x^3-\log (x)} \, dx-70 \int \frac {e^{e^x} x}{\left (3+x^3-\log (x)\right )^2} \, dx+130 \int \frac {x^3}{\left (3+x^3-\log (x)\right )^2} \, dx-140 \int \frac {x}{\left (3+x^3-\log (x)\right )^2} \, dx\\ &=10 \int \frac {e^{e^x} x^4}{\left (3+x^3-\log (x)\right )^2} \, dx+10 \int \frac {x^6}{\left (3+x^3-\log (x)\right )^2} \, dx-10 \int \frac {e^{e^x+x} x^2}{3+x^3-\log (x)} \, dx+20 \int \left (\frac {3 e^{e^x} x}{\left (3+x^3-\log (x)\right )^2}+\frac {e^{e^x} x^4}{\left (3+x^3-\log (x)\right )^2}\right ) \, dx+20 \int \frac {x^4}{\left (3+x^3-\log (x)\right )^2} \, dx-20 \int \frac {e^{e^x} x}{3+x^3-\log (x)} \, dx-40 \int \left (-\frac {3 x}{\left (3+x^3-\log (x)\right )^2}+\frac {3 x^3}{\left (3+x^3-\log (x)\right )^2}-\frac {x^4}{\left (3+x^3-\log (x)\right )^2}+\frac {x^6}{\left (3+x^3-\log (x)\right )^2}\right ) \, dx-40 \int \left (\frac {x}{3+x^3-\log (x)}-\frac {x^3}{3+x^3-\log (x)}\right ) \, dx-70 \int \frac {e^{e^x} x}{\left (3+x^3-\log (x)\right )^2} \, dx+130 \int \frac {x^3}{\left (3+x^3-\log (x)\right )^2} \, dx-140 \int \frac {x}{\left (3+x^3-\log (x)\right )^2} \, dx\\ &=10 \int \frac {e^{e^x} x^4}{\left (3+x^3-\log (x)\right )^2} \, dx+10 \int \frac {x^6}{\left (3+x^3-\log (x)\right )^2} \, dx-10 \int \frac {e^{e^x+x} x^2}{3+x^3-\log (x)} \, dx+20 \int \frac {x^4}{\left (3+x^3-\log (x)\right )^2} \, dx+20 \int \frac {e^{e^x} x^4}{\left (3+x^3-\log (x)\right )^2} \, dx-20 \int \frac {e^{e^x} x}{3+x^3-\log (x)} \, dx+40 \int \frac {x^4}{\left (3+x^3-\log (x)\right )^2} \, dx-40 \int \frac {x^6}{\left (3+x^3-\log (x)\right )^2} \, dx-40 \int \frac {x}{3+x^3-\log (x)} \, dx+40 \int \frac {x^3}{3+x^3-\log (x)} \, dx+60 \int \frac {e^{e^x} x}{\left (3+x^3-\log (x)\right )^2} \, dx-70 \int \frac {e^{e^x} x}{\left (3+x^3-\log (x)\right )^2} \, dx+120 \int \frac {x}{\left (3+x^3-\log (x)\right )^2} \, dx-120 \int \frac {x^3}{\left (3+x^3-\log (x)\right )^2} \, dx+130 \int \frac {x^3}{\left (3+x^3-\log (x)\right )^2} \, dx-140 \int \frac {x}{\left (3+x^3-\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 28, normalized size = 1.00 \begin {gather*} \frac {10 x^2 \left (-2-e^{e^x}+x^2\right )}{3+x^3-\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-140*x + 130*x^3 + 20*x^4 + 10*x^6 + (40*x - 40*x^3)*Log[x] + E^E^x*(-70*x + 10*x^4 + E^x*(-30*x^2
- 10*x^5) + (20*x + 10*E^x*x^2)*Log[x]))/(9 + 6*x^3 + x^6 + (-6 - 2*x^3)*Log[x] + Log[x]^2),x]

[Out]

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

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fricas [A]  time = 0.62, size = 30, normalized size = 1.07 \begin {gather*} \frac {10 \, {\left (x^{4} - x^{2} e^{\left (e^{x}\right )} - 2 \, x^{2}\right )}}{x^{3} - \log \relax (x) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*exp(x)*x^2+20*x)*log(x)+(-10*x^5-30*x^2)*exp(x)+10*x^4-70*x)*exp(exp(x))+(-40*x^3+40*x)*log(x)
+10*x^6+20*x^4+130*x^3-140*x)/(log(x)^2+(-2*x^3-6)*log(x)+x^6+6*x^3+9),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.20, size = 45, normalized size = 1.61 \begin {gather*} \frac {10 \, {\left (x^{4} e^{x} - x^{2} e^{\left (x + e^{x}\right )} - 2 \, x^{2} e^{x}\right )}}{x^{3} e^{x} - e^{x} \log \relax (x) + 3 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*exp(x)*x^2+20*x)*log(x)+(-10*x^5-30*x^2)*exp(x)+10*x^4-70*x)*exp(exp(x))+(-40*x^3+40*x)*log(x)
+10*x^6+20*x^4+130*x^3-140*x)/(log(x)^2+(-2*x^3-6)*log(x)+x^6+6*x^3+9),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (10 \,{\mathrm e}^{x} x^{2}+20 x \right ) \ln \relax (x )+\left (-10 x^{5}-30 x^{2}\right ) {\mathrm e}^{x}+10 x^{4}-70 x \right ) {\mathrm e}^{{\mathrm e}^{x}}+\left (-40 x^{3}+40 x \right ) \ln \relax (x )+10 x^{6}+20 x^{4}+130 x^{3}-140 x}{\ln \relax (x )^{2}+\left (-2 x^{3}-6\right ) \ln \relax (x )+x^{6}+6 x^{3}+9}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((10*exp(x)*x^2+20*x)*ln(x)+(-10*x^5-30*x^2)*exp(x)+10*x^4-70*x)*exp(exp(x))+(-40*x^3+40*x)*ln(x)+10*x^6+
20*x^4+130*x^3-140*x)/(ln(x)^2+(-2*x^3-6)*ln(x)+x^6+6*x^3+9),x)

[Out]

int((((10*exp(x)*x^2+20*x)*ln(x)+(-10*x^5-30*x^2)*exp(x)+10*x^4-70*x)*exp(exp(x))+(-40*x^3+40*x)*ln(x)+10*x^6+
20*x^4+130*x^3-140*x)/(ln(x)^2+(-2*x^3-6)*ln(x)+x^6+6*x^3+9),x)

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maxima [A]  time = 0.38, size = 30, normalized size = 1.07 \begin {gather*} \frac {10 \, {\left (x^{4} - x^{2} e^{\left (e^{x}\right )} - 2 \, x^{2}\right )}}{x^{3} - \log \relax (x) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*exp(x)*x^2+20*x)*log(x)+(-10*x^5-30*x^2)*exp(x)+10*x^4-70*x)*exp(exp(x))+(-40*x^3+40*x)*log(x)
+10*x^6+20*x^4+130*x^3-140*x)/(log(x)^2+(-2*x^3-6)*log(x)+x^6+6*x^3+9),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.77, size = 104, normalized size = 3.71 \begin {gather*} \frac {40\,x}{3}+\frac {\frac {40\,x}{9}-\frac {40\,x^2}{3}}{x^3-\frac {1}{3}}-\frac {\frac {10\,x^2\,\left (x^5+2\,x^3+13\,x^2-14\right )}{3\,x^3-1}-\frac {40\,x^2\,\ln \relax (x)\,\left (x^2-1\right )}{3\,x^3-1}}{x^3-\ln \relax (x)+3}-\frac {10\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^x}}{x^3-\ln \relax (x)+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(40*x - 40*x^3) - exp(exp(x))*(70*x + exp(x)*(30*x^2 + 10*x^5) - log(x)*(20*x + 10*x^2*exp(x)) - 1
0*x^4) - 140*x + 130*x^3 + 20*x^4 + 10*x^6)/(log(x)^2 + 6*x^3 + x^6 - log(x)*(2*x^3 + 6) + 9),x)

[Out]

(40*x)/3 + ((40*x)/9 - (40*x^2)/3)/(x^3 - 1/3) - ((10*x^2*(13*x^2 + 2*x^3 + x^5 - 14))/(3*x^3 - 1) - (40*x^2*l
og(x)*(x^2 - 1))/(3*x^3 - 1))/(x^3 - log(x) + 3) - (10*x^2*exp(exp(x)))/(x^3 - log(x) + 3)

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sympy [A]  time = 0.41, size = 36, normalized size = 1.29 \begin {gather*} - \frac {10 x^{2} e^{e^{x}}}{x^{3} - \log {\relax (x )} + 3} + \frac {- 10 x^{4} + 20 x^{2}}{- x^{3} + \log {\relax (x )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*exp(x)*x**2+20*x)*ln(x)+(-10*x**5-30*x**2)*exp(x)+10*x**4-70*x)*exp(exp(x))+(-40*x**3+40*x)*ln
(x)+10*x**6+20*x**4+130*x**3-140*x)/(ln(x)**2+(-2*x**3-6)*ln(x)+x**6+6*x**3+9),x)

[Out]

-10*x**2*exp(exp(x))/(x**3 - log(x) + 3) + (-10*x**4 + 20*x**2)/(-x**3 + log(x) - 3)

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