3.95.89 \(\int \frac {-3 e^3 x^2+3 x^3+e^{-2-2 x} (-18 e^3+18 x)+e^{-1-x} (-15 e^3 x+12 x^2-3 x^3)+(e^{-2-2 x} (-18+18 e^3)-3 x^2+3 e^3 x^2+e^{-1-x} (-12 x+15 e^3 x+3 x^2)) \log (x)+(18 e^{-2-2 x}+15 e^{-1-x} x+3 x^2+(-18 e^{-2-2 x}-15 e^{-1-x} x-3 x^2) \log (x)) \log (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x})}{6 e^{4-2 x} x^2+5 e^{5-x} x^3+e^6 x^4+(-12 e^{1-2 x} x^2-10 e^{2-x} x^3-2 e^3 x^4) \log (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x})+(6 e^{-2-2 x} x^2+5 e^{-1-x} x^3+x^4) \log ^2(\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x})} \, dx\)

Optimal. Leaf size=35 \[ \frac {3 (-x+\log (x))}{x \left (-e^3+\log \left (x+\frac {x}{2+e^{1+x} x}\right )\right )} \]

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Rubi [F]  time = 11.31, 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^2+3 x^3+e^{-2-2 x} \left (-18 e^3+18 x\right )+e^{-1-x} \left (-15 e^3 x+12 x^2-3 x^3\right )+\left (e^{-2-2 x} \left (-18+18 e^3\right )-3 x^2+3 e^3 x^2+e^{-1-x} \left (-12 x+15 e^3 x+3 x^2\right )\right ) \log (x)+\left (18 e^{-2-2 x}+15 e^{-1-x} x+3 x^2+\left (-18 e^{-2-2 x}-15 e^{-1-x} x-3 x^2\right ) \log (x)\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )}{6 e^{4-2 x} x^2+5 e^{5-x} x^3+e^6 x^4+\left (-12 e^{1-2 x} x^2-10 e^{2-x} x^3-2 e^3 x^4\right ) \log \left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )+\left (6 e^{-2-2 x} x^2+5 e^{-1-x} x^3+x^4\right ) \log ^2\left (\frac {3 e^{-1-x} x+x^2}{2 e^{-1-x}+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*E^3*x^2 + 3*x^3 + E^(-2 - 2*x)*(-18*E^3 + 18*x) + E^(-1 - x)*(-15*E^3*x + 12*x^2 - 3*x^3) + (E^(-2 - 2
*x)*(-18 + 18*E^3) - 3*x^2 + 3*E^3*x^2 + E^(-1 - x)*(-12*x + 15*E^3*x + 3*x^2))*Log[x] + (18*E^(-2 - 2*x) + 15
*E^(-1 - x)*x + 3*x^2 + (-18*E^(-2 - 2*x) - 15*E^(-1 - x)*x - 3*x^2)*Log[x])*Log[(3*E^(-1 - x)*x + x^2)/(2*E^(
-1 - x) + x)])/(6*E^(4 - 2*x)*x^2 + 5*E^(5 - x)*x^3 + E^6*x^4 + (-12*E^(1 - 2*x)*x^2 - 10*E^(2 - x)*x^3 - 2*E^
3*x^4)*Log[(3*E^(-1 - x)*x + x^2)/(2*E^(-1 - x) + x)] + (6*E^(-2 - 2*x)*x^2 + 5*E^(-1 - x)*x^3 + x^4)*Log[(3*E
^(-1 - x)*x + x^2)/(2*E^(-1 - x) + x)]^2),x]

[Out]

3*Defer[Int][1/(x*(E^3 - Log[(x*(3 + E^(1 + x)*x))/(2 + E^(1 + x)*x)])^2), x] + 6*Defer[Int][1/((2 + E^(1 + x)
*x)*(E^3 - Log[(x*(3 + E^(1 + x)*x))/(2 + E^(1 + x)*x)])^2), x] + 6*Defer[Int][1/(x*(2 + E^(1 + x)*x)*(E^3 - L
og[(x*(3 + E^(1 + x)*x))/(2 + E^(1 + x)*x)])^2), x] - 9*Defer[Int][1/((3 + E^(1 + x)*x)*(E^3 - Log[(x*(3 + E^(
1 + x)*x))/(2 + E^(1 + x)*x)])^2), x] - 9*Defer[Int][1/(x*(3 + E^(1 + x)*x)*(E^3 - Log[(x*(3 + E^(1 + x)*x))/(
2 + E^(1 + x)*x)])^2), x] - 3*Defer[Int][Log[x]/(x^2*(E^3 - Log[(x*(3 + E^(1 + x)*x))/(2 + E^(1 + x)*x)])^2),
x] - 6*Defer[Int][Log[x]/(x^2*(2 + E^(1 + x)*x)*(E^3 - Log[(x*(3 + E^(1 + x)*x))/(2 + E^(1 + x)*x)])^2), x] -
6*Defer[Int][Log[x]/(x*(2 + E^(1 + x)*x)*(E^3 - Log[(x*(3 + E^(1 + x)*x))/(2 + E^(1 + x)*x)])^2), x] + 9*Defer
[Int][Log[x]/(x^2*(3 + E^(1 + x)*x)*(E^3 - Log[(x*(3 + E^(1 + x)*x))/(2 + E^(1 + x)*x)])^2), x] + 9*Defer[Int]
[Log[x]/(x*(3 + E^(1 + x)*x)*(E^3 - Log[(x*(3 + E^(1 + x)*x))/(2 + E^(1 + x)*x)])^2), x] - 3*Defer[Int][1/(x^2
*(E^3 - Log[(x*(3 + E^(1 + x)*x))/(2 + E^(1 + x)*x)])), x] + 3*Defer[Int][Log[x]/(x^2*(E^3 - Log[(x*(3 + E^(1
+ x)*x))/(2 + E^(1 + x)*x)])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (-6 e^3+6 x-5 e^{4+x} x+4 e^{1+x} x^2-e^{5+2 x} x^2-e^{1+x} x^3+e^{2+2 x} x^3+\left (6+5 e^{1+x} x+e^{2+2 x} x^2\right ) \log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )+\log (x) \left (-6+6 e^3+5 e^{4+x} x+e^{1+x} (-4+x) x-e^{2+2 x} x^2+e^{5+2 x} x^2-\left (6+5 e^{1+x} x+e^{2+2 x} x^2\right ) \log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )\right )}{x^2 \left (6+5 e^{1+x} x+e^{2+2 x} x^2\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx\\ &=3 \int \frac {-6 e^3+6 x-5 e^{4+x} x+4 e^{1+x} x^2-e^{5+2 x} x^2-e^{1+x} x^3+e^{2+2 x} x^3+\left (6+5 e^{1+x} x+e^{2+2 x} x^2\right ) \log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )+\log (x) \left (-6+6 e^3+5 e^{4+x} x+e^{1+x} (-4+x) x-e^{2+2 x} x^2+e^{5+2 x} x^2-\left (6+5 e^{1+x} x+e^{2+2 x} x^2\right ) \log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )}{x^2 \left (6+5 e^{1+x} x+e^{2+2 x} x^2\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx\\ &=3 \int \left (\frac {2 (1+x) (x-\log (x))}{x^2 \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}-\frac {3 (1+x) (x-\log (x))}{x^2 \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}+\frac {-e^3+x-\left (1-e^3\right ) \log (x)+\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )-\log (x) \log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}\right ) \, dx\\ &=3 \int \frac {-e^3+x-\left (1-e^3\right ) \log (x)+\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )-\log (x) \log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx+6 \int \frac {(1+x) (x-\log (x))}{x^2 \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx-9 \int \frac {(1+x) (x-\log (x))}{x^2 \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx\\ &=3 \int \left (\frac {x-\log (x)}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}+\frac {-1+\log (x)}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )}\right ) \, dx+6 \int \left (\frac {x-\log (x)}{x^2 \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}+\frac {x-\log (x)}{x \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}\right ) \, dx-9 \int \left (\frac {x-\log (x)}{x^2 \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}+\frac {x-\log (x)}{x \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}\right ) \, dx\\ &=3 \int \frac {x-\log (x)}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx+3 \int \frac {-1+\log (x)}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )} \, dx+6 \int \frac {x-\log (x)}{x^2 \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx+6 \int \frac {x-\log (x)}{x \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx-9 \int \frac {x-\log (x)}{x^2 \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx-9 \int \frac {x-\log (x)}{x \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx\\ &=3 \int \left (\frac {1}{x \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}-\frac {\log (x)}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}\right ) \, dx+3 \int \left (-\frac {1}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )}+\frac {\log (x)}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )}\right ) \, dx+6 \int \left (\frac {1}{x \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}-\frac {\log (x)}{x^2 \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}\right ) \, dx+6 \int \left (\frac {1}{\left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}-\frac {\log (x)}{x \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}\right ) \, dx-9 \int \left (\frac {1}{x \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}-\frac {\log (x)}{x^2 \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}\right ) \, dx-9 \int \left (\frac {1}{\left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}-\frac {\log (x)}{x \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2}\right ) \, dx\\ &=3 \int \frac {1}{x \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx-3 \int \frac {\log (x)}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx-3 \int \frac {1}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )} \, dx+3 \int \frac {\log (x)}{x^2 \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )} \, dx+6 \int \frac {1}{\left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx+6 \int \frac {1}{x \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx-6 \int \frac {\log (x)}{x^2 \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx-6 \int \frac {\log (x)}{x \left (2+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx-9 \int \frac {1}{\left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx-9 \int \frac {1}{x \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx+9 \int \frac {\log (x)}{x^2 \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx+9 \int \frac {\log (x)}{x \left (3+e^{1+x} x\right ) \left (e^3-\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 42, normalized size = 1.20 \begin {gather*} \frac {3 (-x+\log (x))}{x \left (-e^3+\log \left (\frac {x \left (3+e^{1+x} x\right )}{2+e^{1+x} x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*E^3*x^2 + 3*x^3 + E^(-2 - 2*x)*(-18*E^3 + 18*x) + E^(-1 - x)*(-15*E^3*x + 12*x^2 - 3*x^3) + (E^(
-2 - 2*x)*(-18 + 18*E^3) - 3*x^2 + 3*E^3*x^2 + E^(-1 - x)*(-12*x + 15*E^3*x + 3*x^2))*Log[x] + (18*E^(-2 - 2*x
) + 15*E^(-1 - x)*x + 3*x^2 + (-18*E^(-2 - 2*x) - 15*E^(-1 - x)*x - 3*x^2)*Log[x])*Log[(3*E^(-1 - x)*x + x^2)/
(2*E^(-1 - x) + x)])/(6*E^(4 - 2*x)*x^2 + 5*E^(5 - x)*x^3 + E^6*x^4 + (-12*E^(1 - 2*x)*x^2 - 10*E^(2 - x)*x^3
- 2*E^3*x^4)*Log[(3*E^(-1 - x)*x + x^2)/(2*E^(-1 - x) + x)] + (6*E^(-2 - 2*x)*x^2 + 5*E^(-1 - x)*x^3 + x^4)*Lo
g[(3*E^(-1 - x)*x + x^2)/(2*E^(-1 - x) + x)]^2),x]

[Out]

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

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fricas [A]  time = 0.62, size = 51, normalized size = 1.46 \begin {gather*} \frac {3 \, {\left (x - \log \relax (x)\right )}}{x e^{3} - x \log \left (\frac {x^{2} e^{6} + 3 \, x e^{\left (-x + 5\right )}}{x e^{6} + 2 \, e^{\left (-x + 5\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-18*exp(-x-1)^2-15*x*exp(-x-1)-3*x^2)*log(x)+18*exp(-x-1)^2+15*x*exp(-x-1)+3*x^2)*log((3*x*exp(-x
-1)+x^2)/(2*exp(-x-1)+x))+((18*exp(3)-18)*exp(-x-1)^2+(15*x*exp(3)+3*x^2-12*x)*exp(-x-1)+3*x^2*exp(3)-3*x^2)*l
og(x)+(-18*exp(3)+18*x)*exp(-x-1)^2+(-15*x*exp(3)-3*x^3+12*x^2)*exp(-x-1)-3*x^2*exp(3)+3*x^3)/((6*x^2*exp(-x-1
)^2+5*x^3*exp(-x-1)+x^4)*log((3*x*exp(-x-1)+x^2)/(2*exp(-x-1)+x))^2+(-12*x^2*exp(3)*exp(-x-1)^2-10*x^3*exp(3)*
exp(-x-1)-2*x^4*exp(3))*log((3*x*exp(-x-1)+x^2)/(2*exp(-x-1)+x))+6*x^2*exp(3)^2*exp(-x-1)^2+5*x^3*exp(3)^2*exp
(-x-1)+x^4*exp(3)^2),x, algorithm="fricas")

[Out]

3*(x - log(x))/(x*e^3 - x*log((x^2*e^6 + 3*x*e^(-x + 5))/(x*e^6 + 2*e^(-x + 5))))

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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((((-18*exp(-x-1)^2-15*x*exp(-x-1)-3*x^2)*log(x)+18*exp(-x-1)^2+15*x*exp(-x-1)+3*x^2)*log((3*x*exp(-x
-1)+x^2)/(2*exp(-x-1)+x))+((18*exp(3)-18)*exp(-x-1)^2+(15*x*exp(3)+3*x^2-12*x)*exp(-x-1)+3*x^2*exp(3)-3*x^2)*l
og(x)+(-18*exp(3)+18*x)*exp(-x-1)^2+(-15*x*exp(3)-3*x^3+12*x^2)*exp(-x-1)-3*x^2*exp(3)+3*x^3)/((6*x^2*exp(-x-1
)^2+5*x^3*exp(-x-1)+x^4)*log((3*x*exp(-x-1)+x^2)/(2*exp(-x-1)+x))^2+(-12*x^2*exp(3)*exp(-x-1)^2-10*x^3*exp(3)*
exp(-x-1)-2*x^4*exp(3))*log((3*x*exp(-x-1)+x^2)/(2*exp(-x-1)+x))+6*x^2*exp(3)^2*exp(-x-1)^2+5*x^3*exp(3)^2*exp
(-x-1)+x^4*exp(3)^2),x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 0.25, size = 427, normalized size = 12.20

method	result	size
risch	\(-\frac {6 \left (x -\ln \relax (x )\right )}{x \left (-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-x -1}+x \right )}{2 \,{\mathrm e}^{-x -1}+x}\right ) \mathrm {csgn}\left (\frac {i x \left (3 \,{\mathrm e}^{-x -1}+x \right )}{2 \,{\mathrm e}^{-x -1}+x}\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \left (3 \,{\mathrm e}^{-x -1}+x \right )}{2 \,{\mathrm e}^{-x -1}+x}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{2 \,{\mathrm e}^{-x -1}+x}\right ) \mathrm {csgn}\left (i \left (3 \,{\mathrm e}^{-x -1}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-x -1}+x \right )}{2 \,{\mathrm e}^{-x -1}+x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{2 \,{\mathrm e}^{-x -1}+x}\right ) \mathrm {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-x -1}+x \right )}{2 \,{\mathrm e}^{-x -1}+x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (3 \,{\mathrm e}^{-x -1}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-x -1}+x \right )}{2 \,{\mathrm e}^{-x -1}+x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-x -1}+x \right )}{2 \,{\mathrm e}^{-x -1}+x}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i \left (3 \,{\mathrm e}^{-x -1}+x \right )}{2 \,{\mathrm e}^{-x -1}+x}\right ) \mathrm {csgn}\left (\frac {i x \left (3 \,{\mathrm e}^{-x -1}+x \right )}{2 \,{\mathrm e}^{-x -1}+x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i x \left (3 \,{\mathrm e}^{-x -1}+x \right )}{2 \,{\mathrm e}^{-x -1}+x}\right )^{3}-2 \,{\mathrm e}^{3}+2 \ln \relax (x )-2 \ln \left (2 \,{\mathrm e}^{-x -1}+x \right )+2 \ln \left (3 \,{\mathrm e}^{-x -1}+x \right )\right )}\)	\(427\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-18*exp(-x-1)^2-15*x*exp(-x-1)-3*x^2)*ln(x)+18*exp(-x-1)^2+15*x*exp(-x-1)+3*x^2)*ln((3*x*exp(-x-1)+x^2)
/(2*exp(-x-1)+x))+((18*exp(3)-18)*exp(-x-1)^2+(15*x*exp(3)+3*x^2-12*x)*exp(-x-1)+3*x^2*exp(3)-3*x^2)*ln(x)+(-1
8*exp(3)+18*x)*exp(-x-1)^2+(-15*x*exp(3)-3*x^3+12*x^2)*exp(-x-1)-3*x^2*exp(3)+3*x^3)/((6*x^2*exp(-x-1)^2+5*x^3
*exp(-x-1)+x^4)*ln((3*x*exp(-x-1)+x^2)/(2*exp(-x-1)+x))^2+(-12*x^2*exp(3)*exp(-x-1)^2-10*x^3*exp(3)*exp(-x-1)-
2*x^4*exp(3))*ln((3*x*exp(-x-1)+x^2)/(2*exp(-x-1)+x))+6*x^2*exp(3)^2*exp(-x-1)^2+5*x^3*exp(3)^2*exp(-x-1)+x^4*
exp(3)^2),x,method=_RETURNVERBOSE)

[Out]

-6*(x-ln(x))/x/(-I*Pi*csgn(I*x)*csgn(I*(3*exp(-x-1)+x)/(2*exp(-x-1)+x))*csgn(I*x/(2*exp(-x-1)+x)*(3*exp(-x-1)+
x))+I*Pi*csgn(I*x)*csgn(I*x/(2*exp(-x-1)+x)*(3*exp(-x-1)+x))^2-I*Pi*csgn(I/(2*exp(-x-1)+x))*csgn(I*(3*exp(-x-1
)+x))*csgn(I*(3*exp(-x-1)+x)/(2*exp(-x-1)+x))+I*Pi*csgn(I/(2*exp(-x-1)+x))*csgn(I*(3*exp(-x-1)+x)/(2*exp(-x-1)
+x))^2+I*Pi*csgn(I*(3*exp(-x-1)+x))*csgn(I*(3*exp(-x-1)+x)/(2*exp(-x-1)+x))^2-I*Pi*csgn(I*(3*exp(-x-1)+x)/(2*e
xp(-x-1)+x))^3+I*Pi*csgn(I*(3*exp(-x-1)+x)/(2*exp(-x-1)+x))*csgn(I*x/(2*exp(-x-1)+x)*(3*exp(-x-1)+x))^2-I*Pi*c
sgn(I*x/(2*exp(-x-1)+x)*(3*exp(-x-1)+x))^3-2*exp(3)+2*ln(x)-2*ln(2*exp(-x-1)+x)+2*ln(3*exp(-x-1)+x))

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maxima [A]  time = 0.77, size = 43, normalized size = 1.23 \begin {gather*} \frac {3 \, {\left (x - \log \relax (x)\right )}}{x e^{3} - x \log \left (x e^{\left (x + 1\right )} + 3\right ) + x \log \left (x e^{\left (x + 1\right )} + 2\right ) - x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-18*exp(-x-1)^2-15*x*exp(-x-1)-3*x^2)*log(x)+18*exp(-x-1)^2+15*x*exp(-x-1)+3*x^2)*log((3*x*exp(-x
-1)+x^2)/(2*exp(-x-1)+x))+((18*exp(3)-18)*exp(-x-1)^2+(15*x*exp(3)+3*x^2-12*x)*exp(-x-1)+3*x^2*exp(3)-3*x^2)*l
og(x)+(-18*exp(3)+18*x)*exp(-x-1)^2+(-15*x*exp(3)-3*x^3+12*x^2)*exp(-x-1)-3*x^2*exp(3)+3*x^3)/((6*x^2*exp(-x-1
)^2+5*x^3*exp(-x-1)+x^4)*log((3*x*exp(-x-1)+x^2)/(2*exp(-x-1)+x))^2+(-12*x^2*exp(3)*exp(-x-1)^2-10*x^3*exp(3)*
exp(-x-1)-2*x^4*exp(3))*log((3*x*exp(-x-1)+x^2)/(2*exp(-x-1)+x))+6*x^2*exp(3)^2*exp(-x-1)^2+5*x^3*exp(3)^2*exp
(-x-1)+x^4*exp(3)^2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \relax (x)\,\left ({\mathrm {e}}^{-2\,x-2}\,\left (18\,{\mathrm {e}}^3-18\right )+3\,x^2\,{\mathrm {e}}^3+{\mathrm {e}}^{-x-1}\,\left (15\,x\,{\mathrm {e}}^3-12\,x+3\,x^2\right )-3\,x^2\right )-{\mathrm {e}}^{-x-1}\,\left (3\,x^3-12\,x^2+15\,{\mathrm {e}}^3\,x\right )+\ln \left (\frac {3\,x\,{\mathrm {e}}^{-x-1}+x^2}{x+2\,{\mathrm {e}}^{-x-1}}\right )\,\left (18\,{\mathrm {e}}^{-2\,x-2}-\ln \relax (x)\,\left (18\,{\mathrm {e}}^{-2\,x-2}+15\,x\,{\mathrm {e}}^{-x-1}+3\,x^2\right )+15\,x\,{\mathrm {e}}^{-x-1}+3\,x^2\right )-3\,x^2\,{\mathrm {e}}^3+3\,x^3+{\mathrm {e}}^{-2\,x-2}\,\left (18\,x-18\,{\mathrm {e}}^3\right )}{{\ln \left (\frac {3\,x\,{\mathrm {e}}^{-x-1}+x^2}{x+2\,{\mathrm {e}}^{-x-1}}\right )}^2\,\left (5\,x^3\,{\mathrm {e}}^{-x-1}+6\,x^2\,{\mathrm {e}}^{-2\,x-2}+x^4\right )+x^4\,{\mathrm {e}}^6-\ln \left (\frac {3\,x\,{\mathrm {e}}^{-x-1}+x^2}{x+2\,{\mathrm {e}}^{-x-1}}\right )\,\left (2\,x^4\,{\mathrm {e}}^3+12\,x^2\,{\mathrm {e}}^{1-2\,x}+10\,x^3\,{\mathrm {e}}^{2-x}\right )+6\,x^2\,{\mathrm {e}}^{4-2\,x}+5\,x^3\,{\mathrm {e}}^{5-x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(exp(- 2*x - 2)*(18*exp(3) - 18) + 3*x^2*exp(3) + exp(- x - 1)*(15*x*exp(3) - 12*x + 3*x^2) - 3*x^
2) - exp(- x - 1)*(15*x*exp(3) - 12*x^2 + 3*x^3) + log((3*x*exp(- x - 1) + x^2)/(x + 2*exp(- x - 1)))*(18*exp(
- 2*x - 2) - log(x)*(18*exp(- 2*x - 2) + 15*x*exp(- x - 1) + 3*x^2) + 15*x*exp(- x - 1) + 3*x^2) - 3*x^2*exp(3
) + 3*x^3 + exp(- 2*x - 2)*(18*x - 18*exp(3)))/(log((3*x*exp(- x - 1) + x^2)/(x + 2*exp(- x - 1)))^2*(5*x^3*ex
p(- x - 1) + 6*x^2*exp(- 2*x - 2) + x^4) - log((3*x*exp(- x - 1) + x^2)/(x + 2*exp(- x - 1)))*(2*x^4*exp(3) +
10*x^3*exp(3)*exp(- x - 1) + 12*x^2*exp(3)*exp(- 2*x - 2)) + x^4*exp(6) + 5*x^3*exp(6)*exp(- x - 1) + 6*x^2*ex
p(6)*exp(- 2*x - 2)),x)

[Out]

int((log(x)*(exp(- 2*x - 2)*(18*exp(3) - 18) + 3*x^2*exp(3) + exp(- x - 1)*(15*x*exp(3) - 12*x + 3*x^2) - 3*x^
2) - exp(- x - 1)*(15*x*exp(3) - 12*x^2 + 3*x^3) + log((3*x*exp(- x - 1) + x^2)/(x + 2*exp(- x - 1)))*(18*exp(
- 2*x - 2) - log(x)*(18*exp(- 2*x - 2) + 15*x*exp(- x - 1) + 3*x^2) + 15*x*exp(- x - 1) + 3*x^2) - 3*x^2*exp(3
) + 3*x^3 + exp(- 2*x - 2)*(18*x - 18*exp(3)))/(log((3*x*exp(- x - 1) + x^2)/(x + 2*exp(- x - 1)))^2*(5*x^3*ex
p(- x - 1) + 6*x^2*exp(- 2*x - 2) + x^4) + x^4*exp(6) - log((3*x*exp(- x - 1) + x^2)/(x + 2*exp(- x - 1)))*(2*
x^4*exp(3) + 12*x^2*exp(1 - 2*x) + 10*x^3*exp(2 - x)) + 6*x^2*exp(4 - 2*x) + 5*x^3*exp(5 - x)), x)

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sympy [A]  time = 0.81, size = 39, normalized size = 1.11 \begin {gather*} \frac {- 3 x + 3 \log {\relax (x )}}{x \log {\left (\frac {x^{2} + 3 x e^{- x - 1}}{x + 2 e^{- x - 1}} \right )} - x e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-18*exp(-x-1)**2-15*x*exp(-x-1)-3*x**2)*ln(x)+18*exp(-x-1)**2+15*x*exp(-x-1)+3*x**2)*ln((3*x*exp(
-x-1)+x**2)/(2*exp(-x-1)+x))+((18*exp(3)-18)*exp(-x-1)**2+(15*x*exp(3)+3*x**2-12*x)*exp(-x-1)+3*x**2*exp(3)-3*
x**2)*ln(x)+(-18*exp(3)+18*x)*exp(-x-1)**2+(-15*x*exp(3)-3*x**3+12*x**2)*exp(-x-1)-3*x**2*exp(3)+3*x**3)/((6*x
**2*exp(-x-1)**2+5*x**3*exp(-x-1)+x**4)*ln((3*x*exp(-x-1)+x**2)/(2*exp(-x-1)+x))**2+(-12*x**2*exp(3)*exp(-x-1)
**2-10*x**3*exp(3)*exp(-x-1)-2*x**4*exp(3))*ln((3*x*exp(-x-1)+x**2)/(2*exp(-x-1)+x))+6*x**2*exp(3)**2*exp(-x-1
)**2+5*x**3*exp(3)**2*exp(-x-1)+x**4*exp(3)**2),x)

[Out]

(-3*x + 3*log(x))/(x*log((x**2 + 3*x*exp(-x - 1))/(x + 2*exp(-x - 1))) - x*exp(3))

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