3.47.83 \(\int \frac {-3 x^2+6 x^3+e^x (2 x^2+6 x^3-2 x^4)+(-6 x^2+e^x (-8 x-4 x^2+4 x^3)) \log (\frac {1}{3} e^{-2 x} (-3 x+e^x (-4-2 x+2 x^2))) \log (\log (\frac {1}{3} e^{-2 x} (-3 x+e^x (-4-2 x+2 x^2))))}{(-3 x+e^x (-4-2 x+2 x^2)) \log (\frac {1}{3} e^{-2 x} (-3 x+e^x (-4-2 x+2 x^2)))} \, dx\)

Optimal. Leaf size=34 \[ x^2 \log \left (\log \left (\frac {1}{3} e^{-x} \left (-4-\left (2+3 e^{-x}\right ) x+2 x^2\right )\right )\right ) \]

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Rubi [F]  time = 5.41, 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 x^2+6 x^3+e^x \left (2 x^2+6 x^3-2 x^4\right )+\left (-6 x^2+e^x \left (-8 x-4 x^2+4 x^3\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )\right )}{\left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+e^x \left (-4-2 x+2 x^2\right )\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*x^2 + 6*x^3 + E^x*(2*x^2 + 6*x^3 - 2*x^4) + (-6*x^2 + E^x*(-8*x - 4*x^2 + 4*x^3))*Log[(-3*x + E^x*(-4
- 2*x + 2*x^2))/(3*E^(2*x))]*Log[Log[(-3*x + E^x*(-4 - 2*x + 2*x^2))/(3*E^(2*x))]])/((-3*x + E^x*(-4 - 2*x + 2
*x^2))*Log[(-3*x + E^x*(-4 - 2*x + 2*x^2))/(3*E^(2*x))]),x]

[Out]

Defer[Int][Log[(-3*x + 2*E^x*(-2 - x + x^2))/(3*E^(2*x))]^(-1), x] + 4*Defer[Int][1/((-2 + x)*Log[(-3*x + 2*E^
x*(-2 - x + x^2))/(3*E^(2*x))]), x] + 2*Defer[Int][x/Log[(-3*x + 2*E^x*(-2 - x + x^2))/(3*E^(2*x))], x] - Defe
r[Int][x^2/Log[(-3*x + 2*E^x*(-2 - x + x^2))/(3*E^(2*x))], x] + Defer[Int][1/((1 + x)*Log[(-3*x + 2*E^x*(-2 -
x + x^2))/(3*E^(2*x))]), x] + 15*Defer[Int][1/((-4*E^x - 3*x - 2*E^x*x + 2*E^x*x^2)*Log[(-3*x + 2*E^x*(-2 - x
+ x^2))/(3*E^(2*x))]), x] + 24*Defer[Int][1/((-2 + x)*(-4*E^x - 3*x - 2*E^x*x + 2*E^x*x^2)*Log[(-3*x + 2*E^x*(
-2 - x + x^2))/(3*E^(2*x))]), x] + 3*Defer[Int][x/((-4*E^x - 3*x - 2*E^x*x + 2*E^x*x^2)*Log[(-3*x + 2*E^x*(-2
- x + x^2))/(3*E^(2*x))]), x] + 3*Defer[Int][x^2/((-4*E^x - 3*x - 2*E^x*x + 2*E^x*x^2)*Log[(-3*x + 2*E^x*(-2 -
x + x^2))/(3*E^(2*x))]), x] + 3*Defer[Int][x^3/((-4*E^x - 3*x - 2*E^x*x + 2*E^x*x^2)*Log[(-3*x + 2*E^x*(-2 -
x + x^2))/(3*E^(2*x))]), x] - 3*Defer[Int][1/((1 + x)*(-4*E^x - 3*x - 2*E^x*x + 2*E^x*x^2)*Log[(-3*x + 2*E^x*(
-2 - x + x^2))/(3*E^(2*x))]), x] + 2*Defer[Int][x*Log[Log[(-3*x + 2*E^x*(-2 - x + x^2))/(3*E^(2*x))]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3 x^2 \left (2-2 x+x^3\right )}{\left (-2-x+x^2\right ) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}-\frac {x \left (-x-3 x^2+x^3+4 \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right )+2 x \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right )-2 x^2 \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right )\right )}{\left (-2-x+x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}\right ) \, dx\\ &=3 \int \frac {x^2 \left (2-2 x+x^3\right )}{\left (-2-x+x^2\right ) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx-\int \frac {x \left (-x-3 x^2+x^3+4 \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right )+2 x \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right )-2 x^2 \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right )\right )}{\left (-2-x+x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx\\ &=3 \int \left (\frac {5}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}+\frac {8}{(-2+x) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}+\frac {x}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}+\frac {x^2}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}+\frac {x^3}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}-\frac {1}{(1+x) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}\right ) \, dx-\int \frac {x \left (-\frac {x \left (-1-3 x+x^2\right )}{\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}-2 \left (2+x-x^2\right ) \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right )\right )}{2+x-x^2} \, dx\\ &=3 \int \frac {x}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+3 \int \frac {x^2}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+3 \int \frac {x^3}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx-3 \int \frac {1}{(1+x) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+15 \int \frac {1}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+24 \int \frac {1}{(-2+x) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx-\int \left (\frac {x^2 \left (-1-3 x+x^2\right )}{(-2+x) (1+x) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}-2 x \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right )\right ) \, dx\\ &=2 \int x \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right ) \, dx+3 \int \frac {x}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+3 \int \frac {x^2}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+3 \int \frac {x^3}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx-3 \int \frac {1}{(1+x) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+15 \int \frac {1}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+24 \int \frac {1}{(-2+x) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx-\int \frac {x^2 \left (-1-3 x+x^2\right )}{(-2+x) (1+x) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx\\ &=2 \int x \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right ) \, dx+3 \int \frac {x}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+3 \int \frac {x^2}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+3 \int \frac {x^3}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx-3 \int \frac {1}{(1+x) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+15 \int \frac {1}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+24 \int \frac {1}{(-2+x) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx-\int \left (-\frac {1}{\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}-\frac {4}{(-2+x) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}-\frac {2 x}{\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}+\frac {x^2}{\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}-\frac {1}{(1+x) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )}\right ) \, dx\\ &=2 \int \frac {x}{\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+2 \int x \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right ) \, dx+3 \int \frac {x}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+3 \int \frac {x^2}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+3 \int \frac {x^3}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx-3 \int \frac {1}{(1+x) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+4 \int \frac {1}{(-2+x) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+15 \int \frac {1}{\left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+24 \int \frac {1}{(-2+x) \left (-4 e^x-3 x-2 e^x x+2 e^x x^2\right ) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+\int \frac {1}{\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx-\int \frac {x^2}{\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx+\int \frac {1}{(1+x) \log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 32, normalized size = 0.94 \begin {gather*} x^2 \log \left (\log \left (\frac {1}{3} e^{-2 x} \left (-3 x+2 e^x \left (-2-x+x^2\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x^2 + 6*x^3 + E^x*(2*x^2 + 6*x^3 - 2*x^4) + (-6*x^2 + E^x*(-8*x - 4*x^2 + 4*x^3))*Log[(-3*x + E^
x*(-4 - 2*x + 2*x^2))/(3*E^(2*x))]*Log[Log[(-3*x + E^x*(-4 - 2*x + 2*x^2))/(3*E^(2*x))]])/((-3*x + E^x*(-4 - 2
*x + 2*x^2))*Log[(-3*x + E^x*(-4 - 2*x + 2*x^2))/(3*E^(2*x))]),x]

[Out]

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

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fricas [A]  time = 0.77, size = 28, normalized size = 0.82 \begin {gather*} x^{2} \log \left (\log \left (\frac {1}{3} \, {\left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )} e^{\left (-2 \, x\right )}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3-4*x^2-8*x)*exp(x)-6*x^2)*log(1/3*((2*x^2-2*x-4)*exp(x)-3*x)/exp(x)^2)*log(log(1/3*((2*x^2-2
*x-4)*exp(x)-3*x)/exp(x)^2))+(-2*x^4+6*x^3+2*x^2)*exp(x)+6*x^3-3*x^2)/((2*x^2-2*x-4)*exp(x)-3*x)/log(1/3*((2*x
^2-2*x-4)*exp(x)-3*x)/exp(x)^2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {6 \, x^{3} - 2 \, {\left (3 \, x^{2} - 2 \, {\left (x^{3} - x^{2} - 2 \, x\right )} e^{x}\right )} \log \left (\frac {1}{3} \, {\left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )} e^{\left (-2 \, x\right )}\right ) \log \left (\log \left (\frac {1}{3} \, {\left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )} e^{\left (-2 \, x\right )}\right )\right ) - 3 \, x^{2} - 2 \, {\left (x^{4} - 3 \, x^{3} - x^{2}\right )} e^{x}}{{\left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )} \log \left (\frac {1}{3} \, {\left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )} e^{\left (-2 \, x\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3-4*x^2-8*x)*exp(x)-6*x^2)*log(1/3*((2*x^2-2*x-4)*exp(x)-3*x)/exp(x)^2)*log(log(1/3*((2*x^2-2
*x-4)*exp(x)-3*x)/exp(x)^2))+(-2*x^4+6*x^3+2*x^2)*exp(x)+6*x^3-3*x^2)/((2*x^2-2*x-4)*exp(x)-3*x)/log(1/3*((2*x
^2-2*x-4)*exp(x)-3*x)/exp(x)^2),x, algorithm="giac")

[Out]

integrate((6*x^3 - 2*(3*x^2 - 2*(x^3 - x^2 - 2*x)*e^x)*log(1/3*(2*(x^2 - x - 2)*e^x - 3*x)*e^(-2*x))*log(log(1
/3*(2*(x^2 - x - 2)*e^x - 3*x)*e^(-2*x))) - 3*x^2 - 2*(x^4 - 3*x^3 - x^2)*e^x)/((2*(x^2 - x - 2)*e^x - 3*x)*lo
g(1/3*(2*(x^2 - x - 2)*e^x - 3*x)*e^(-2*x))), x)

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maple [C]  time = 0.25, size = 191, normalized size = 5.62

method	result	size
risch	\(x^{2} \ln \left (\ln \relax (2)-\ln \relax (3)-2 \ln \left ({\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x} x^{2}+\left (-{\mathrm e}^{x}-\frac {3}{2}\right ) x -2 \,{\mathrm e}^{x}\right )+\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left ({\mathrm e}^{x} x^{2}+\left (-{\mathrm e}^{x}-\frac {3}{2}\right ) x -2 \,{\mathrm e}^{x}\right )\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left ({\mathrm e}^{x} x^{2}+\left (-{\mathrm e}^{x}-\frac {3}{2}\right ) x -2 \,{\mathrm e}^{x}\right )\right )+\mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right )\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left ({\mathrm e}^{x} x^{2}+\left (-{\mathrm e}^{x}-\frac {3}{2}\right ) x -2 \,{\mathrm e}^{x}\right )\right )+\mathrm {csgn}\left (i \left ({\mathrm e}^{x} x^{2}+\left (-{\mathrm e}^{x}-\frac {3}{2}\right ) x -2 \,{\mathrm e}^{x}\right )\right )\right )}{2}\right )\)	\(191\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*ln(ln(2)-ln(3)-2*ln(exp(x))+ln(exp(x)*x^2+(-exp(x)-3/2)*x-2*exp(x))+1/2*I*Pi*csgn(I*exp(2*x))*(-csgn(I*exp
(2*x))+csgn(I*exp(x)))^2-1/2*I*Pi*csgn(I*exp(-2*x)*(exp(x)*x^2+(-exp(x)-3/2)*x-2*exp(x)))*(-csgn(I*exp(-2*x)*(
exp(x)*x^2+(-exp(x)-3/2)*x-2*exp(x)))+csgn(I*exp(-2*x)))*(-csgn(I*exp(-2*x)*(exp(x)*x^2+(-exp(x)-3/2)*x-2*exp(
x)))+csgn(I*(exp(x)*x^2+(-exp(x)-3/2)*x-2*exp(x)))))

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maxima [A]  time = 0.62, size = 30, normalized size = 0.88 \begin {gather*} x^{2} \log \left (-2 \, x - \log \relax (3) + \log \left (2 \, {\left (x^{2} - x - 2\right )} e^{x} - 3 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3-4*x^2-8*x)*exp(x)-6*x^2)*log(1/3*((2*x^2-2*x-4)*exp(x)-3*x)/exp(x)^2)*log(log(1/3*((2*x^2-2
*x-4)*exp(x)-3*x)/exp(x)^2))+(-2*x^4+6*x^3+2*x^2)*exp(x)+6*x^3-3*x^2)/((2*x^2-2*x-4)*exp(x)-3*x)/log(1/3*((2*x
^2-2*x-4)*exp(x)-3*x)/exp(x)^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.59, size = 28, normalized size = 0.82 \begin {gather*} x^2\,\ln \left (\ln \left (-{\mathrm {e}}^{-2\,x}\,\left (x+\frac {{\mathrm {e}}^x\,\left (-2\,x^2+2\,x+4\right )}{3}\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(2*x^2 + 6*x^3 - 2*x^4) - 3*x^2 + 6*x^3 - log(-exp(-2*x)*(x + (exp(x)*(2*x - 2*x^2 + 4))/3))*log(
log(-exp(-2*x)*(x + (exp(x)*(2*x - 2*x^2 + 4))/3)))*(6*x^2 + exp(x)*(8*x + 4*x^2 - 4*x^3)))/(log(-exp(-2*x)*(x
+ (exp(x)*(2*x - 2*x^2 + 4))/3))*(3*x + exp(x)*(2*x - 2*x^2 + 4))),x)

[Out]

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

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sympy [A]  time = 3.27, size = 27, normalized size = 0.79 \begin {gather*} x^{2} \log {\left (\log {\left (\left (- x + \frac {\left (2 x^{2} - 2 x - 4\right ) e^{x}}{3}\right ) e^{- 2 x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x**3-4*x**2-8*x)*exp(x)-6*x**2)*ln(1/3*((2*x**2-2*x-4)*exp(x)-3*x)/exp(x)**2)*ln(ln(1/3*((2*x**
2-2*x-4)*exp(x)-3*x)/exp(x)**2))+(-2*x**4+6*x**3+2*x**2)*exp(x)+6*x**3-3*x**2)/((2*x**2-2*x-4)*exp(x)-3*x)/ln(
1/3*((2*x**2-2*x-4)*exp(x)-3*x)/exp(x)**2),x)

[Out]

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

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