3.92.99 \(\int \frac {\frac {e^{4-2 e^x} (2 x+(2-2 x^2) \log (4)+e^x (2 x^2+(2 x-2 x^3) \log (4)))}{x^2}+(-2 x+4 x^2 \log (4)) \log (-x+(-1+x^2) \log (4))+\frac {e^{2-e^x} (2 x-4 x^2 \log (4)+(-2 x+(-2+2 x^2) \log (4)+e^x (-2 x^2+(-2 x+2 x^3) \log (4))) \log (-x+(-1+x^2) \log (4)))}{x}}{-9 x^2+(-9 x+9 x^3) \log (4)} \, dx\)

Optimal. Leaf size=35 \[ \frac {1}{9} \left (\frac {e^{2-e^x}}{x}-\log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )^2 \]

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Rubi [F]  time = 16.00, 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*} \int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((E^(4 - 2*E^x)*(2*x + (2 - 2*x^2)*Log[4] + E^x*(2*x^2 + (2*x - 2*x^3)*Log[4])))/x^2 + (-2*x + 4*x^2*Log[4
])*Log[-x + (-1 + x^2)*Log[4]] + (E^(2 - E^x)*(2*x - 4*x^2*Log[4] + (-2*x + (-2 + 2*x^2)*Log[4] + E^x*(-2*x^2
+ (-2*x + 2*x^3)*Log[4]))*Log[-x + (-1 + x^2)*Log[4]]))/x)/(-9*x^2 + (-9*x + 9*x^3)*Log[4]),x]

[Out]

(Log[16]*Log[-x - Log[4] + x^2*Log[4]]*Log[1 - 2*x*Log[4] - Sqrt[1 + 4*Log[4]^2]])/(9*Log[4]) - (Log[16]*Log[1
- 2*x*Log[4] - Sqrt[1 + 4*Log[4]^2]]^2)/(18*Log[4]) + (Log[16]*Log[-x - Log[4] + x^2*Log[4]]*Log[1 - 2*x*Log[
4] + Sqrt[1 + 4*Log[4]^2]])/(9*Log[4]) - (Log[16]*Log[-1/2*(1 - 2*x*Log[4] - Sqrt[1 + 4*Log[4]^2])/Sqrt[1 + 4*
Log[4]^2]]*Log[1 - 2*x*Log[4] + Sqrt[1 + 4*Log[4]^2]])/(9*Log[4]) - (Log[16]*Log[1 - 2*x*Log[4] + Sqrt[1 + 4*L
og[4]^2]]^2)/(18*Log[4]) - (Log[16]*Log[1 - 2*x*Log[4] - Sqrt[1 + 4*Log[4]^2]]*Log[(1 - 2*x*Log[4] + Sqrt[1 +
4*Log[4]^2])/(2*Sqrt[1 + 4*Log[4]^2])])/(9*Log[4]) - (Log[16]*PolyLog[2, (1 - 2*x*Log[4] + Sqrt[1 + 4*Log[4]^2
])/(2*Sqrt[1 + 4*Log[4]^2])])/(9*Log[4]) - (Log[16]*PolyLog[2, -1/2*(1 - Sqrt[1 + 4*Log[4]^2] - x*Log[16])/Sqr
t[1 + 4*Log[4]^2]])/(9*Log[4]) - (2*Defer[Int][1/(E^(2*(-2 + E^x))*x^3), x])/9 + (2*Log[-x - Log[4] + x^2*Log[
4]]*Defer[Int][E^(2 - E^x)/x^2, x])/9 - (2*Defer[Int][E^(4 - 2*E^x + x)/x^2, x])/9 - (2*Defer[Int][E^(2 - E^x)
/x, x])/(9*Log[4]) + (2*Log[-x - Log[4] + x^2*Log[4]]*Defer[Int][E^(2 - E^x + x)/x, x])/9 - ((Sqrt[1 + 4*Log[4
]^2]*Log[16]^2 - Log[4]*Log[256])*Defer[Int][E^(2 - E^x)/(-1 + 2*x*Log[4] - Sqrt[1 + 4*Log[4]^2]), x])/(18*Log
[4]^2) + ((Sqrt[1 + 4*Log[4]^2]*Log[16]^2 + Log[4]*Log[256])*Defer[Int][E^(2 - E^x)/(-1 + 2*x*Log[4] + Sqrt[1
+ 4*Log[4]^2]), x])/(18*Log[4]^2) - (2*(1 + 1/Sqrt[1 + 4*Log[4]^2])*Log[16]*Defer[Int][Defer[Int][E^(2 - E^x)/
x^2, x]/(-1 + 2*x*Log[4] - Sqrt[1 + 4*Log[4]^2]), x])/9 - (4*Log[4]*Defer[Int][Defer[Int][E^(2 - E^x)/x^2, x]/
(1 - 2*x*Log[4] + Sqrt[1 + 4*Log[4]^2]), x])/(9*Sqrt[1 + 4*Log[4]^2]) - (2*(1 - 1/Sqrt[1 + 4*Log[4]^2])*Log[16
]*Defer[Int][Defer[Int][E^(2 - E^x)/x^2, x]/(-1 + 2*x*Log[4] + Sqrt[1 + 4*Log[4]^2]), x])/9 - (4*Log[4]*Defer[
Int][Defer[Int][E^(2 - E^x)/x^2, x]/(-1 + Sqrt[1 + 4*Log[4]^2] + x*Log[16]), x])/(9*Sqrt[1 + 4*Log[4]^2]) - (2
*(1 + 1/Sqrt[1 + 4*Log[4]^2])*Log[16]*Defer[Int][Defer[Int][E^(2 - E^x + x)/x, x]/(-1 + 2*x*Log[4] - Sqrt[1 +
4*Log[4]^2]), x])/9 - (4*Log[4]*Defer[Int][Defer[Int][E^(2 - E^x + x)/x, x]/(1 - 2*x*Log[4] + Sqrt[1 + 4*Log[4
]^2]), x])/(9*Sqrt[1 + 4*Log[4]^2]) - (2*(1 - 1/Sqrt[1 + 4*Log[4]^2])*Log[16]*Defer[Int][Defer[Int][E^(2 - E^x
+ x)/x, x]/(-1 + 2*x*Log[4] + Sqrt[1 + 4*Log[4]^2]), x])/9 - (4*Log[4]*Defer[Int][Defer[Int][E^(2 - E^x + x)/
x, x]/(-1 + Sqrt[1 + 4*Log[4]^2] + x*Log[16]), x])/(9*Sqrt[1 + 4*Log[4]^2])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-2 e^x} \left (e^2 \left (-x-\log (4)+x^2 \log (4)\right )+e^{2+x} x \left (-x-\log (4)+x^2 \log (4)\right )+e^{e^x} x^2 (-1+x \log (16))\right ) \left (e^2-e^{e^x} x \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{9 x^3 \left (x+\log (4)-x^2 \log (4)\right )} \, dx\\ &=\frac {2}{9} \int \frac {e^{-2 e^x} \left (e^2 \left (-x-\log (4)+x^2 \log (4)\right )+e^{2+x} x \left (-x-\log (4)+x^2 \log (4)\right )+e^{e^x} x^2 (-1+x \log (16))\right ) \left (e^2-e^{e^x} x \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x^3 \left (x+\log (4)-x^2 \log (4)\right )} \, dx\\ &=\frac {2}{9} \int \left (-\frac {e^{2-2 e^x+x} \left (e^2-e^{e^x} x \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x^2}+\frac {e^{-2 e^x} \left (-e^2 x-e^{e^x} x^2-e^2 \log (4)+e^2 x^2 \log (4)+e^{e^x} x^3 \log (16)\right ) \left (-e^2+e^{e^x} x \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x^3 \left (-x-\log (4)+x^2 \log (4)\right )}\right ) \, dx\\ &=-\left (\frac {2}{9} \int \frac {e^{2-2 e^x+x} \left (e^2-e^{e^x} x \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x^2} \, dx\right )+\frac {2}{9} \int \frac {e^{-2 e^x} \left (-e^2 x-e^{e^x} x^2-e^2 \log (4)+e^2 x^2 \log (4)+e^{e^x} x^3 \log (16)\right ) \left (-e^2+e^{e^x} x \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x^3 \left (-x-\log (4)+x^2 \log (4)\right )} \, dx\\ &=-\left (\frac {2}{9} \int \left (\frac {e^{4-2 e^x+x}}{x^2}-\frac {e^{2-e^x+x} \log \left (-x-\log (4)+x^2 \log (4)\right )}{x}\right ) \, dx\right )+\frac {2}{9} \int \left (-\frac {e^{4-2 e^x}}{x^3}+\frac {(1-x \log (16)) \log \left (-x-\log (4)+x^2 \log (4)\right )}{x+\log (4)-x^2 \log (4)}+\frac {e^{2-e^x} \left (x-x^2 \log (16)-x \log \left (-x+\left (-1+x^2\right ) \log (4)\right )-\log (4) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+x^2 \log (4) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x^2 \left (-x-\log (4)+x^2 \log (4)\right )}\right ) \, dx\\ &=-\left (\frac {2}{9} \int \frac {e^{4-2 e^x}}{x^3} \, dx\right )-\frac {2}{9} \int \frac {e^{4-2 e^x+x}}{x^2} \, dx+\frac {2}{9} \int \frac {e^{2-e^x+x} \log \left (-x-\log (4)+x^2 \log (4)\right )}{x} \, dx+\frac {2}{9} \int \frac {(1-x \log (16)) \log \left (-x-\log (4)+x^2 \log (4)\right )}{x+\log (4)-x^2 \log (4)} \, dx+\frac {2}{9} \int \frac {e^{2-e^x} \left (x-x^2 \log (16)-x \log \left (-x+\left (-1+x^2\right ) \log (4)\right )-\log (4) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+x^2 \log (4) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x^2 \left (-x-\log (4)+x^2 \log (4)\right )} \, dx\\ &=-\left (\frac {2}{9} \int \frac {e^{-2 \left (-2+e^x\right )}}{x^3} \, dx\right )-\frac {2}{9} \int \frac {e^{4-2 e^x+x}}{x^2} \, dx+\frac {2}{9} \int \left (-\frac {\log (16) \log \left (-x-\log (4)+x^2 \log (4)\right )}{1-2 x \log (4)-\sqrt {1+4 \log ^2(4)}}-\frac {\log (16) \log \left (-x-\log (4)+x^2 \log (4)\right )}{1-2 x \log (4)+\sqrt {1+4 \log ^2(4)}}\right ) \, dx+\frac {2}{9} \int \frac {e^{2-e^x} \left (-x+x^2 \log (16)-\left (-x-\log (4)+x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x^2 \left (x+\log (4)-x^2 \log (4)\right )} \, dx-\frac {2}{9} \int \frac {(1-x \log (16)) \int \frac {e^{2-e^x+x}}{x} \, dx}{x+\log (4)-x^2 \log (4)} \, dx+\frac {1}{9} \left (2 \log \left (-x-\log (4)+x^2 \log (4)\right )\right ) \int \frac {e^{2-e^x+x}}{x} \, dx\\ &=-\left (\frac {2}{9} \int \frac {e^{-2 \left (-2+e^x\right )}}{x^3} \, dx\right )-\frac {2}{9} \int \frac {e^{4-2 e^x+x}}{x^2} \, dx+\frac {2}{9} \int \left (\frac {e^{2-e^x} (1-x \log (16))}{x \left (-x-\log (4)+x^2 \log (4)\right )}+\frac {e^{2-e^x} \log \left (-x-\log (4)+x^2 \log (4)\right )}{x^2}\right ) \, dx-\frac {2}{9} \int \left (-\frac {\int \frac {e^{2-e^x+x}}{x} \, dx}{-x-\log (4)+x^2 \log (4)}+\frac {x \log (16) \int \frac {e^{2-e^x+x}}{x} \, dx}{-x-\log (4)+x^2 \log (4)}\right ) \, dx-\frac {1}{9} (2 \log (16)) \int \frac {\log \left (-x-\log (4)+x^2 \log (4)\right )}{1-2 x \log (4)-\sqrt {1+4 \log ^2(4)}} \, dx-\frac {1}{9} (2 \log (16)) \int \frac {\log \left (-x-\log (4)+x^2 \log (4)\right )}{1-2 x \log (4)+\sqrt {1+4 \log ^2(4)}} \, dx+\frac {1}{9} \left (2 \log \left (-x-\log (4)+x^2 \log (4)\right )\right ) \int \frac {e^{2-e^x+x}}{x} \, dx\\ &=\frac {\log (16) \log \left (-x-\log (4)+x^2 \log (4)\right ) \log \left (1-2 x \log (4)-\sqrt {1+4 \log ^2(4)}\right )}{9 \log (4)}+\frac {\log (16) \log \left (-x-\log (4)+x^2 \log (4)\right ) \log \left (1-2 x \log (4)+\sqrt {1+4 \log ^2(4)}\right )}{9 \log (4)}-\frac {2}{9} \int \frac {e^{-2 \left (-2+e^x\right )}}{x^3} \, dx-\frac {2}{9} \int \frac {e^{4-2 e^x+x}}{x^2} \, dx+\frac {2}{9} \int \frac {e^{2-e^x} (1-x \log (16))}{x \left (-x-\log (4)+x^2 \log (4)\right )} \, dx+\frac {2}{9} \int \frac {e^{2-e^x} \log \left (-x-\log (4)+x^2 \log (4)\right )}{x^2} \, dx+\frac {2}{9} \int \frac {\int \frac {e^{2-e^x+x}}{x} \, dx}{-x-\log (4)+x^2 \log (4)} \, dx-\frac {1}{9} (2 \log (16)) \int \frac {x \int \frac {e^{2-e^x+x}}{x} \, dx}{-x-\log (4)+x^2 \log (4)} \, dx-\frac {\log (16) \int \frac {(-1+2 x \log (4)) \log \left (1-2 x \log (4)-\sqrt {1+4 \log ^2(4)}\right )}{-x-\log (4)+x^2 \log (4)} \, dx}{9 \log (4)}-\frac {\log (16) \int \frac {(-1+2 x \log (4)) \log \left (1-2 x \log (4)+\sqrt {1+4 \log ^2(4)}\right )}{-x-\log (4)+x^2 \log (4)} \, dx}{9 \log (4)}+\frac {1}{9} \left (2 \log \left (-x-\log (4)+x^2 \log (4)\right )\right ) \int \frac {e^{2-e^x+x}}{x} \, dx\\ &=\frac {\log (16) \log \left (-x-\log (4)+x^2 \log (4)\right ) \log \left (1-2 x \log (4)-\sqrt {1+4 \log ^2(4)}\right )}{9 \log (4)}+\frac {\log (16) \log \left (-x-\log (4)+x^2 \log (4)\right ) \log \left (1-2 x \log (4)+\sqrt {1+4 \log ^2(4)}\right )}{9 \log (4)}-\frac {2}{9} \int \frac {e^{-2 \left (-2+e^x\right )}}{x^3} \, dx-\frac {2}{9} \int \frac {e^{4-2 e^x+x}}{x^2} \, dx+\frac {2}{9} \int \left (-\frac {e^{2-e^x}}{x \log (4)}+\frac {e^{2-e^x} (-1+x \log (4)-\log (4) \log (16))}{\log (4) \left (-x-\log (4)+x^2 \log (4)\right )}\right ) \, dx-\frac {2}{9} \int \frac {(1-x \log (16)) \int \frac {e^{2-e^x}}{x^2} \, dx}{x+\log (4)-x^2 \log (4)} \, dx+\frac {2}{9} \int \left (-\frac {2 \log (4) \int \frac {e^{2-e^x+x}}{x} \, dx}{\sqrt {1+4 \log ^2(4)} \left (1-2 x \log (4)+\sqrt {1+4 \log ^2(4)}\right )}-\frac {2 \log (4) \int \frac {e^{2-e^x+x}}{x} \, dx}{\sqrt {1+4 \log ^2(4)} \left (-1+\sqrt {1+4 \log ^2(4)}+x \log (16)\right )}\right ) \, dx-\frac {1}{9} (2 \log (16)) \int \left (\frac {\left (1+\frac {1}{\sqrt {1+4 \log ^2(4)}}\right ) \int \frac {e^{2-e^x+x}}{x} \, dx}{-1+2 x \log (4)-\sqrt {1+4 \log ^2(4)}}+\frac {\left (1-\frac {1}{\sqrt {1+4 \log ^2(4)}}\right ) \int \frac {e^{2-e^x+x}}{x} \, dx}{-1+2 x \log (4)+\sqrt {1+4 \log ^2(4)}}\right ) \, dx-\frac {\log (16) \int \left (\frac {2 \log (4) \log \left (1-2 x \log (4)-\sqrt {1+4 \log ^2(4)}\right )}{-1+2 x \log (4)-\sqrt {1+4 \log ^2(4)}}+\frac {2 \log (4) \log \left (1-2 x \log (4)-\sqrt {1+4 \log ^2(4)}\right )}{-1+2 x \log (4)+\sqrt {1+4 \log ^2(4)}}\right ) \, dx}{9 \log (4)}-\frac {\log (16) \int \left (\frac {2 \log (4) \log \left (1-2 x \log (4)+\sqrt {1+4 \log ^2(4)}\right )}{-1+2 x \log (4)-\sqrt {1+4 \log ^2(4)}}+\frac {2 \log (4) \log \left (1-2 x \log (4)+\sqrt {1+4 \log ^2(4)}\right )}{-1+2 x \log (4)+\sqrt {1+4 \log ^2(4)}}\right ) \, dx}{9 \log (4)}+\frac {1}{9} \left (2 \log \left (-x-\log (4)+x^2 \log (4)\right )\right ) \int \frac {e^{2-e^x}}{x^2} \, dx+\frac {1}{9} \left (2 \log \left (-x-\log (4)+x^2 \log (4)\right )\right ) \int \frac {e^{2-e^x+x}}{x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((E^(4 - 2*E^x)*(2*x + (2 - 2*x^2)*Log[4] + E^x*(2*x^2 + (2*x - 2*x^3)*Log[4])))/x^2 + (-2*x + 4*x^2
*Log[4])*Log[-x + (-1 + x^2)*Log[4]] + (E^(2 - E^x)*(2*x - 4*x^2*Log[4] + (-2*x + (-2 + 2*x^2)*Log[4] + E^x*(-
2*x^2 + (-2*x + 2*x^3)*Log[4]))*Log[-x + (-1 + x^2)*Log[4]]))/x)/(-9*x^2 + (-9*x + 9*x^3)*Log[4]),x]

[Out]

Integrate[((E^(4 - 2*E^x)*(2*x + (2 - 2*x^2)*Log[4] + E^x*(2*x^2 + (2*x - 2*x^3)*Log[4])))/x^2 + (-2*x + 4*x^2
*Log[4])*Log[-x + (-1 + x^2)*Log[4]] + (E^(2 - E^x)*(2*x - 4*x^2*Log[4] + (-2*x + (-2 + 2*x^2)*Log[4] + E^x*(-
2*x^2 + (-2*x + 2*x^3)*Log[4]))*Log[-x + (-1 + x^2)*Log[4]]))/x)/(-9*x^2 + (-9*x + 9*x^3)*Log[4]), x]

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fricas [A]  time = 0.80, size = 59, normalized size = 1.69 \begin {gather*} -\frac {2}{9} \, e^{\left (-e^{x} - \log \relax (x) + 2\right )} \log \left (2 \, {\left (x^{2} - 1\right )} \log \relax (2) - x\right ) + \frac {1}{9} \, \log \left (2 \, {\left (x^{2} - 1\right )} \log \relax (2) - x\right )^{2} + \frac {1}{9} \, e^{\left (-2 \, e^{x} - 2 \, \log \relax (x) + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*(-2*x^3+2*x)*log(2)+2*x^2)*exp(x)+2*(-2*x^2+2)*log(2)+2*x)*exp(-log(x)-exp(x)+2)^2+(((2*(2*x^3-
2*x)*log(2)-2*x^2)*exp(x)+2*(2*x^2-2)*log(2)-2*x)*log(2*(x^2-1)*log(2)-x)-8*x^2*log(2)+2*x)*exp(-log(x)-exp(x)
+2)+(8*x^2*log(2)-2*x)*log(2*(x^2-1)*log(2)-x))/(2*(9*x^3-9*x)*log(2)-9*x^2),x, algorithm="fricas")

[Out]

-2/9*e^(-e^x - log(x) + 2)*log(2*(x^2 - 1)*log(2) - x) + 1/9*log(2*(x^2 - 1)*log(2) - x)^2 + 1/9*e^(-2*e^x - 2
*log(x) + 4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left ({\left (4 \, x^{2} \log \relax (2) + {\left ({\left (x^{2} - 2 \, {\left (x^{3} - x\right )} \log \relax (2)\right )} e^{x} - 2 \, {\left (x^{2} - 1\right )} \log \relax (2) + x\right )} \log \left (2 \, {\left (x^{2} - 1\right )} \log \relax (2) - x\right ) - x\right )} e^{\left (-e^{x} - \log \relax (x) + 2\right )} - {\left ({\left (x^{2} - 2 \, {\left (x^{3} - x\right )} \log \relax (2)\right )} e^{x} - 2 \, {\left (x^{2} - 1\right )} \log \relax (2) + x\right )} e^{\left (-2 \, e^{x} - 2 \, \log \relax (x) + 4\right )} - {\left (4 \, x^{2} \log \relax (2) - x\right )} \log \left (2 \, {\left (x^{2} - 1\right )} \log \relax (2) - x\right )\right )}}{9 \, {\left (x^{2} - 2 \, {\left (x^{3} - x\right )} \log \relax (2)\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*(-2*x^3+2*x)*log(2)+2*x^2)*exp(x)+2*(-2*x^2+2)*log(2)+2*x)*exp(-log(x)-exp(x)+2)^2+(((2*(2*x^3-
2*x)*log(2)-2*x^2)*exp(x)+2*(2*x^2-2)*log(2)-2*x)*log(2*(x^2-1)*log(2)-x)-8*x^2*log(2)+2*x)*exp(-log(x)-exp(x)
+2)+(8*x^2*log(2)-2*x)*log(2*(x^2-1)*log(2)-x))/(2*(9*x^3-9*x)*log(2)-9*x^2),x, algorithm="giac")

[Out]

integrate(2/9*((4*x^2*log(2) + ((x^2 - 2*(x^3 - x)*log(2))*e^x - 2*(x^2 - 1)*log(2) + x)*log(2*(x^2 - 1)*log(2
) - x) - x)*e^(-e^x - log(x) + 2) - ((x^2 - 2*(x^3 - x)*log(2))*e^x - 2*(x^2 - 1)*log(2) + x)*e^(-2*e^x - 2*lo
g(x) + 4) - (4*x^2*log(2) - x)*log(2*(x^2 - 1)*log(2) - x))/(x^2 - 2*(x^3 - x)*log(2)), x)

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maple [A]  time = 0.11, size = 58, normalized size = 1.66

method	result	size
risch	\(\frac {\ln \left (2 \left (x^{2}-1\right ) \ln \relax (2)-x \right )^{2}}{9}+\frac {{\mathrm e}^{-2 \,{\mathrm e}^{x}+4}}{9 x^{2}}-\frac {2 \ln \left (2 \left (x^{2}-1\right ) \ln \relax (2)-x \right ) {\mathrm e}^{-{\mathrm e}^{x}+2}}{9 x}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*(-2*x^3+2*x)*ln(2)+2*x^2)*exp(x)+2*(-2*x^2+2)*ln(2)+2*x)*exp(-ln(x)-exp(x)+2)^2+(((2*(2*x^3-2*x)*ln(2
)-2*x^2)*exp(x)+2*(2*x^2-2)*ln(2)-2*x)*ln(2*(x^2-1)*ln(2)-x)-8*x^2*ln(2)+2*x)*exp(-ln(x)-exp(x)+2)+(8*x^2*ln(2
)-2*x)*ln(2*(x^2-1)*ln(2)-x))/(2*(9*x^3-9*x)*ln(2)-9*x^2),x,method=_RETURNVERBOSE)

[Out]

1/9*ln(2*(x^2-1)*ln(2)-x)^2+1/9/x^2*exp(-2*exp(x)+4)-2/9*ln(2*(x^2-1)*ln(2)-x)/x*exp(-exp(x)+2)

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maxima [A]  time = 0.53, size = 61, normalized size = 1.74 \begin {gather*} \frac {x^{2} \log \left (2 \, x^{2} \log \relax (2) - x - 2 \, \log \relax (2)\right )^{2} - 2 \, x e^{\left (-e^{x} + 2\right )} \log \left (2 \, x^{2} \log \relax (2) - x - 2 \, \log \relax (2)\right ) + e^{\left (-2 \, e^{x} + 4\right )}}{9 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*(-2*x^3+2*x)*log(2)+2*x^2)*exp(x)+2*(-2*x^2+2)*log(2)+2*x)*exp(-log(x)-exp(x)+2)^2+(((2*(2*x^3-
2*x)*log(2)-2*x^2)*exp(x)+2*(2*x^2-2)*log(2)-2*x)*log(2*(x^2-1)*log(2)-x)-8*x^2*log(2)+2*x)*exp(-log(x)-exp(x)
+2)+(8*x^2*log(2)-2*x)*log(2*(x^2-1)*log(2)-x))/(2*(9*x^3-9*x)*log(2)-9*x^2),x, algorithm="maxima")

[Out]

1/9*(x^2*log(2*x^2*log(2) - x - 2*log(2))^2 - 2*x*e^(-e^x + 2)*log(2*x^2*log(2) - x - 2*log(2)) + e^(-2*e^x +
4))/x^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{2-\ln \relax (x)-{\mathrm {e}}^x}\,\left (8\,x^2\,\ln \relax (2)-2\,x+\ln \left (2\,\ln \relax (2)\,\left (x^2-1\right )-x\right )\,\left (2\,x+{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left (2\,x-2\,x^3\right )+2\,x^2\right )-2\,\ln \relax (2)\,\left (2\,x^2-2\right )\right )\right )-{\mathrm {e}}^{4-2\,\ln \relax (x)-2\,{\mathrm {e}}^x}\,\left (2\,x+{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left (2\,x-2\,x^3\right )+2\,x^2\right )-2\,\ln \relax (2)\,\left (2\,x^2-2\right )\right )+\ln \left (2\,\ln \relax (2)\,\left (x^2-1\right )-x\right )\,\left (2\,x-8\,x^2\,\ln \relax (2)\right )}{2\,\ln \relax (2)\,\left (9\,x-9\,x^3\right )+9\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2 - log(x) - exp(x))*(8*x^2*log(2) - 2*x + log(2*log(2)*(x^2 - 1) - x)*(2*x + exp(x)*(2*log(2)*(2*x -
2*x^3) + 2*x^2) - 2*log(2)*(2*x^2 - 2))) - exp(4 - 2*log(x) - 2*exp(x))*(2*x + exp(x)*(2*log(2)*(2*x - 2*x^3)
+ 2*x^2) - 2*log(2)*(2*x^2 - 2)) + log(2*log(2)*(x^2 - 1) - x)*(2*x - 8*x^2*log(2)))/(2*log(2)*(9*x - 9*x^3)
+ 9*x^2),x)

[Out]

int((exp(2 - log(x) - exp(x))*(8*x^2*log(2) - 2*x + log(2*log(2)*(x^2 - 1) - x)*(2*x + exp(x)*(2*log(2)*(2*x -
2*x^3) + 2*x^2) - 2*log(2)*(2*x^2 - 2))) - exp(4 - 2*log(x) - 2*exp(x))*(2*x + exp(x)*(2*log(2)*(2*x - 2*x^3)
+ 2*x^2) - 2*log(2)*(2*x^2 - 2)) + log(2*log(2)*(x^2 - 1) - x)*(2*x - 8*x^2*log(2)))/(2*log(2)*(9*x - 9*x^3)
+ 9*x^2), x)

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sympy [B]  time = 1.14, size = 58, normalized size = 1.66 \begin {gather*} \frac {\log {\left (- x + \left (2 x^{2} - 2\right ) \log {\relax (2 )} \right )}^{2}}{9} + \frac {- 18 x^{2} e^{2 - e^{x}} \log {\left (- x + \left (2 x^{2} - 2\right ) \log {\relax (2 )} \right )} + 9 x e^{4 - 2 e^{x}}}{81 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*(-2*x**3+2*x)*ln(2)+2*x**2)*exp(x)+2*(-2*x**2+2)*ln(2)+2*x)*exp(-ln(x)-exp(x)+2)**2+(((2*(2*x**
3-2*x)*ln(2)-2*x**2)*exp(x)+2*(2*x**2-2)*ln(2)-2*x)*ln(2*(x**2-1)*ln(2)-x)-8*x**2*ln(2)+2*x)*exp(-ln(x)-exp(x)
+2)+(8*x**2*ln(2)-2*x)*ln(2*(x**2-1)*ln(2)-x))/(2*(9*x**3-9*x)*ln(2)-9*x**2),x)

[Out]

log(-x + (2*x**2 - 2)*log(2))**2/9 + (-18*x**2*exp(2 - exp(x))*log(-x + (2*x**2 - 2)*log(2)) + 9*x*exp(4 - 2*e
xp(x)))/(81*x**3)

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