3.19.100 \(\int \frac {e^{\frac {2 (-x+e \log (x))}{e}} (-2 x^2+e (4 x+x^2))+e (8 x+4 x^2) \log (5)+(e^{\frac {2 (-x+e \log (x))}{e}} (-2 x+e (2+x))+4 e x \log (5)) \log (\frac {1}{4} (e^{\frac {2 (-x+e \log (x))}{e}}+4 \log (5)))}{2 e^{1+\frac {2 (-x+e \log (x))}{e}} x+8 e x \log (5)} \, dx\)

Optimal. Leaf size=29 \[ x+\frac {1}{4} \left (x+\log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )\right )^2 \]

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Rubi [F]  time = 5.93, 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 {e^{\frac {2 (-x+e \log (x))}{e}} \left (-2 x^2+e \left (4 x+x^2\right )\right )+e \left (8 x+4 x^2\right ) \log (5)+\left (e^{\frac {2 (-x+e \log (x))}{e}} (-2 x+e (2+x))+4 e x \log (5)\right ) \log \left (\frac {1}{4} \left (e^{\frac {2 (-x+e \log (x))}{e}}+4 \log (5)\right )\right )}{2 e^{1+\frac {2 (-x+e \log (x))}{e}} x+8 e x \log (5)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*(-x + E*Log[x]))/E)*(-2*x^2 + E*(4*x + x^2)) + E*(8*x + 4*x^2)*Log[5] + (E^((2*(-x + E*Log[x]))/E)*
(-2*x + E*(2 + x)) + 4*E*x*Log[5])*Log[(E^((2*(-x + E*Log[x]))/E) + 4*Log[5])/4])/(2*E^(1 + (2*(-x + E*Log[x])
)/E)*x + 8*E*x*Log[5]),x]

[Out]

(4*x*Log[5])/Log[625] + (x^2*Log[5])/Log[625] + (2*x*Log[5]*Log[x^2/(4*E^((2*x)/E)) + Log[5]])/Log[625] - (4*L
og[5]*Defer[Int][x^2/(x^2 + 4*E^((2*x)/E)*Log[5]), x])/Log[625] + (4*Log[5]*Defer[Int][x^3/(x^2 + 4*E^((2*x)/E
)*Log[5]), x])/(E*Log[625]) - (Log[4]*Log[390625]*Defer[Int][x/(x^2 + E^((2*x)/E)*Log[625]), x])/(2*Log[625])
+ (Log[390625]*Log[x^2/E^((2*x)/E) + Log[625]]*Defer[Int][x/(x^2 + E^((2*x)/E)*Log[625]), x])/(2*Log[625]) + (
Log[390625]*Defer[Int][x^2/(x^2 + E^((2*x)/E)*Log[625]), x])/(2*Log[625]) + (Log[4]*Log[390625]*Defer[Int][x^2
/(x^2 + E^((2*x)/E)*Log[625]), x])/(2*E*Log[625]) - (Log[390625]*Log[x^2/E^((2*x)/E) + Log[625]]*Defer[Int][x^
2/(x^2 + E^((2*x)/E)*Log[625]), x])/(2*E*Log[625]) - (Log[390625]*Defer[Int][x^3/(x^2 + E^((2*x)/E)*Log[625]),
x])/(2*E*Log[625]) - (Log[390625]*Defer[Int][(x*Defer[Int][x/(x^2 + E^((2*x)/E)*Log[625]), x])/(x^2 + E^((2*x
)/E)*Log[625]), x])/Log[625] + (Log[390625]*Defer[Int][(x^2*Defer[Int][x/(x^2 + E^((2*x)/E)*Log[625]), x])/(x^
2 + E^((2*x)/E)*Log[625]), x])/(E*Log[625]) + (Log[390625]*Defer[Int][(x*Defer[Int][x^2/(x^2 + E^((2*x)/E)*Log
[625]), x])/(x^2 + E^((2*x)/E)*Log[625]), x])/(E*Log[625]) - (Log[390625]*Defer[Int][(x^2*Defer[Int][x^2/(x^2
+ E^((2*x)/E)*Log[625]), x])/(x^2 + E^((2*x)/E)*Log[625]), x])/(E^2*Log[625])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-1+\frac {2 x}{e}} \left (e^{\frac {2 (-x+e \log (x))}{e}} \left (-2 x^2+e \left (4 x+x^2\right )\right )+e \left (8 x+4 x^2\right ) \log (5)+\left (e^{\frac {2 (-x+e \log (x))}{e}} (-2 x+e (2+x))+4 e x \log (5)\right ) \log \left (\frac {1}{4} \left (e^{\frac {2 (-x+e \log (x))}{e}}+4 \log (5)\right )\right )\right )}{2 x \left (x^2+e^{\frac {2 x}{e}} \log (625)\right )} \, dx\\ &=\frac {1}{2} \int \frac {e^{-1+\frac {2 x}{e}} \left (e^{\frac {2 (-x+e \log (x))}{e}} \left (-2 x^2+e \left (4 x+x^2\right )\right )+e \left (8 x+4 x^2\right ) \log (5)+\left (e^{\frac {2 (-x+e \log (x))}{e}} (-2 x+e (2+x))+4 e x \log (5)\right ) \log \left (\frac {1}{4} \left (e^{\frac {2 (-x+e \log (x))}{e}}+4 \log (5)\right )\right )\right )}{x \left (x^2+e^{\frac {2 x}{e}} \log (625)\right )} \, dx\\ &=\frac {1}{2} \int \frac {-2 x^3+e x^2 (4+x)+4 e^{1+\frac {2 x}{e}} (2+x) \log (5)+\left (-2 x^2+e x (2+x)+4 e^{1+\frac {2 x}{e}} \log (5)\right ) \log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )}{e \left (x^2+e^{\frac {2 x}{e}} \log (625)\right )} \, dx\\ &=\frac {\int \frac {-2 x^3+e x^2 (4+x)+4 e^{1+\frac {2 x}{e}} (2+x) \log (5)+\left (-2 x^2+e x (2+x)+4 e^{1+\frac {2 x}{e}} \log (5)\right ) \log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e}\\ &=\frac {\int \left (\frac {4 e \log (5) \left (2+x+\log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )\right )}{\log (625)}+\frac {(e-x) x \log (390625) \left (x-\log (4)+\log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right )}{\log (625) \left (x^2+e^{\frac {2 x}{e}} \log (625)\right )}\right ) \, dx}{2 e}\\ &=\frac {(2 \log (5)) \int \left (2+x+\log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )\right ) \, dx}{\log (625)}+\frac {\log (390625) \int \frac {(e-x) x \left (x-\log (4)+\log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right )}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}\\ &=\frac {4 x \log (5)}{\log (625)}+\frac {x^2 \log (5)}{\log (625)}+\frac {(2 \log (5)) \int \log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right ) \, dx}{\log (625)}+\frac {\log (390625) \int \left (\frac {e x \left (x-\log (4)+\log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right )}{x^2+e^{\frac {2 x}{e}} \log (625)}-\frac {x^2 \left (x-\log (4)+\log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right )}{x^2+e^{\frac {2 x}{e}} \log (625)}\right ) \, dx}{2 e \log (625)}\\ &=\frac {4 x \log (5)}{\log (625)}+\frac {x^2 \log (5)}{\log (625)}+\frac {2 x \log (5) \log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )}{\log (625)}-\frac {(2 \log (5)) \int \frac {2 (e-x) x^2}{e \left (x^2+4 e^{\frac {2 x}{e}} \log (5)\right )} \, dx}{\log (625)}+\frac {\log (390625) \int \frac {x \left (x-\log (4)+\log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right )}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}-\frac {\log (390625) \int \frac {x^2 \left (x-\log (4)+\log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right )}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}\\ &=\frac {4 x \log (5)}{\log (625)}+\frac {x^2 \log (5)}{\log (625)}+\frac {2 x \log (5) \log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )}{\log (625)}-\frac {(4 \log (5)) \int \frac {(e-x) x^2}{x^2+4 e^{\frac {2 x}{e}} \log (5)} \, dx}{e \log (625)}+\frac {\log (390625) \int \left (\frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)}-\frac {x \log (4)}{x^2+e^{\frac {2 x}{e}} \log (625)}+\frac {x \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )}{x^2+e^{\frac {2 x}{e}} \log (625)}\right ) \, dx}{2 \log (625)}-\frac {\log (390625) \int \left (\frac {x^3}{x^2+e^{\frac {2 x}{e}} \log (625)}-\frac {x^2 \log (4)}{x^2+e^{\frac {2 x}{e}} \log (625)}+\frac {x^2 \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )}{x^2+e^{\frac {2 x}{e}} \log (625)}\right ) \, dx}{2 e \log (625)}\\ &=\frac {4 x \log (5)}{\log (625)}+\frac {x^2 \log (5)}{\log (625)}+\frac {2 x \log (5) \log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )}{\log (625)}-\frac {(4 \log (5)) \int \left (\frac {e x^2}{x^2+4 e^{\frac {2 x}{e}} \log (5)}-\frac {x^3}{x^2+4 e^{\frac {2 x}{e}} \log (5)}\right ) \, dx}{e \log (625)}+\frac {\log (390625) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}+\frac {\log (390625) \int \frac {x \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}-\frac {\log (390625) \int \frac {x^3}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}-\frac {\log (390625) \int \frac {x^2 \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}-\frac {(\log (4) \log (390625)) \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}+\frac {(\log (4) \log (390625)) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}\\ &=\frac {4 x \log (5)}{\log (625)}+\frac {x^2 \log (5)}{\log (625)}+\frac {2 x \log (5) \log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )}{\log (625)}-\frac {(4 \log (5)) \int \frac {x^2}{x^2+4 e^{\frac {2 x}{e}} \log (5)} \, dx}{\log (625)}+\frac {(4 \log (5)) \int \frac {x^3}{x^2+4 e^{\frac {2 x}{e}} \log (5)} \, dx}{e \log (625)}+\frac {\log (390625) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}-\frac {\log (390625) \int \frac {2 (e-x) x \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{e \left (x^2+e^{\frac {2 x}{e}} \log (625)\right )} \, dx}{2 \log (625)}-\frac {\log (390625) \int \frac {x^3}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}+\frac {\log (390625) \int \frac {2 (e-x) x \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{e \left (x^2+e^{\frac {2 x}{e}} \log (625)\right )} \, dx}{2 e \log (625)}-\frac {(\log (4) \log (390625)) \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}+\frac {(\log (4) \log (390625)) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}+\frac {\left (\log (390625) \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right ) \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}-\frac {\left (\log (390625) \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right ) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}\\ &=\frac {4 x \log (5)}{\log (625)}+\frac {x^2 \log (5)}{\log (625)}+\frac {2 x \log (5) \log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )}{\log (625)}-\frac {(4 \log (5)) \int \frac {x^2}{x^2+4 e^{\frac {2 x}{e}} \log (5)} \, dx}{\log (625)}+\frac {(4 \log (5)) \int \frac {x^3}{x^2+4 e^{\frac {2 x}{e}} \log (5)} \, dx}{e \log (625)}+\frac {\log (390625) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}+\frac {\log (390625) \int \frac {(e-x) x \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{e^2 \log (625)}-\frac {\log (390625) \int \frac {x^3}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}-\frac {\log (390625) \int \frac {(e-x) x \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{e \log (625)}-\frac {(\log (4) \log (390625)) \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}+\frac {(\log (4) \log (390625)) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}+\frac {\left (\log (390625) \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right ) \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}-\frac {\left (\log (390625) \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right ) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}\\ &=\frac {4 x \log (5)}{\log (625)}+\frac {x^2 \log (5)}{\log (625)}+\frac {2 x \log (5) \log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )}{\log (625)}-\frac {(4 \log (5)) \int \frac {x^2}{x^2+4 e^{\frac {2 x}{e}} \log (5)} \, dx}{\log (625)}+\frac {(4 \log (5)) \int \frac {x^3}{x^2+4 e^{\frac {2 x}{e}} \log (5)} \, dx}{e \log (625)}+\frac {\log (390625) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}+\frac {\log (390625) \int \left (\frac {e x \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{x^2+e^{\frac {2 x}{e}} \log (625)}-\frac {x^2 \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{x^2+e^{\frac {2 x}{e}} \log (625)}\right ) \, dx}{e^2 \log (625)}-\frac {\log (390625) \int \frac {x^3}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}-\frac {\log (390625) \int \left (\frac {e x \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{x^2+e^{\frac {2 x}{e}} \log (625)}-\frac {x^2 \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{x^2+e^{\frac {2 x}{e}} \log (625)}\right ) \, dx}{e \log (625)}-\frac {(\log (4) \log (390625)) \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}+\frac {(\log (4) \log (390625)) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}+\frac {\left (\log (390625) \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right ) \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}-\frac {\left (\log (390625) \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right ) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}\\ &=\frac {4 x \log (5)}{\log (625)}+\frac {x^2 \log (5)}{\log (625)}+\frac {2 x \log (5) \log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right )}{\log (625)}-\frac {(4 \log (5)) \int \frac {x^2}{x^2+4 e^{\frac {2 x}{e}} \log (5)} \, dx}{\log (625)}+\frac {(4 \log (5)) \int \frac {x^3}{x^2+4 e^{\frac {2 x}{e}} \log (5)} \, dx}{e \log (625)}+\frac {\log (390625) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}-\frac {\log (390625) \int \frac {x \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{\log (625)}-\frac {\log (390625) \int \frac {x^2 \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{e^2 \log (625)}-\frac {\log (390625) \int \frac {x^3}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}+\frac {\log (390625) \int \frac {x^2 \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{e \log (625)}+\frac {\log (390625) \int \frac {x \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{e \log (625)}-\frac {(\log (4) \log (390625)) \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}+\frac {(\log (4) \log (390625)) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}+\frac {\left (\log (390625) \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right ) \int \frac {x}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 \log (625)}-\frac {\left (\log (390625) \log \left (e^{-\frac {2 x}{e}} x^2+\log (625)\right )\right ) \int \frac {x^2}{x^2+e^{\frac {2 x}{e}} \log (625)} \, dx}{2 e \log (625)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.18, size = 114, normalized size = 3.93 \begin {gather*} \frac {-\frac {2 x^2}{e}+\frac {1}{2} e x (4+x)+2 x \log \left (x^2+4 e^{\frac {2 x}{e}} \log (5)\right )-\frac {1}{2} e \log ^2\left (x^2+4 e^{\frac {2 x}{e}} \log (5)\right )+\log \left (\frac {1}{4} e^{-\frac {2 x}{e}} x^2+\log (5)\right ) \left ((-2+e) x+e \log \left (x^2+4 e^{\frac {2 x}{e}} \log (5)\right )\right )}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(-x + E*Log[x]))/E)*(-2*x^2 + E*(4*x + x^2)) + E*(8*x + 4*x^2)*Log[5] + (E^((2*(-x + E*Log[x]
))/E)*(-2*x + E*(2 + x)) + 4*E*x*Log[5])*Log[(E^((2*(-x + E*Log[x]))/E) + 4*Log[5])/4])/(2*E^(1 + (2*(-x + E*L
og[x]))/E)*x + 8*E*x*Log[5]),x]

[Out]

((-2*x^2)/E + (E*x*(4 + x))/2 + 2*x*Log[x^2 + 4*E^((2*x)/E)*Log[5]] - (E*Log[x^2 + 4*E^((2*x)/E)*Log[5]]^2)/2
+ Log[x^2/(4*E^((2*x)/E)) + Log[5]]*((-2 + E)*x + E*Log[x^2 + 4*E^((2*x)/E)*Log[5]]))/(2*E)

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fricas [B]  time = 0.93, size = 70, normalized size = 2.41 \begin {gather*} \frac {1}{4} \, x^{2} + \frac {1}{2} \, x \log \left (\frac {1}{4} \, {\left (4 \, e \log \relax (5) + e^{\left ({\left (2 \, e \log \relax (x) - 2 \, x + e\right )} e^{\left (-1\right )}\right )}\right )} e^{\left (-1\right )}\right ) + \frac {1}{4} \, \log \left (\frac {1}{4} \, {\left (4 \, e \log \relax (5) + e^{\left ({\left (2 \, e \log \relax (x) - 2 \, x + e\right )} e^{\left (-1\right )}\right )}\right )} e^{\left (-1\right )}\right )^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2+x)*exp(1)-2*x)*exp((exp(1)*log(x)-x)/exp(1))^2+4*x*exp(1)*log(5))*log(1/4*exp((exp(1)*log(x)-x
)/exp(1))^2+log(5))+((x^2+4*x)*exp(1)-2*x^2)*exp((exp(1)*log(x)-x)/exp(1))^2+(4*x^2+8*x)*exp(1)*log(5))/(2*x*e
xp(1)*exp((exp(1)*log(x)-x)/exp(1))^2+8*x*exp(1)*log(5)),x, algorithm="fricas")

[Out]

1/4*x^2 + 1/2*x*log(1/4*(4*e*log(5) + e^((2*e*log(x) - 2*x + e)*e^(-1)))*e^(-1)) + 1/4*log(1/4*(4*e*log(5) + e
^((2*e*log(x) - 2*x + e)*e^(-1)))*e^(-1))^2 + x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, {\left (x^{2} + 2 \, x\right )} e \log \relax (5) - {\left (2 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e\right )} e^{\left (2 \, {\left (e \log \relax (x) - x\right )} e^{\left (-1\right )}\right )} + {\left (4 \, x e \log \relax (5) + {\left ({\left (x + 2\right )} e - 2 \, x\right )} e^{\left (2 \, {\left (e \log \relax (x) - x\right )} e^{\left (-1\right )}\right )}\right )} \log \left (\frac {1}{4} \, e^{\left (2 \, {\left (e \log \relax (x) - x\right )} e^{\left (-1\right )}\right )} + \log \relax (5)\right )}{2 \, {\left (4 \, x e \log \relax (5) + x e^{\left (2 \, {\left (e \log \relax (x) - x\right )} e^{\left (-1\right )} + 1\right )}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2+x)*exp(1)-2*x)*exp((exp(1)*log(x)-x)/exp(1))^2+4*x*exp(1)*log(5))*log(1/4*exp((exp(1)*log(x)-x
)/exp(1))^2+log(5))+((x^2+4*x)*exp(1)-2*x^2)*exp((exp(1)*log(x)-x)/exp(1))^2+(4*x^2+8*x)*exp(1)*log(5))/(2*x*e
xp(1)*exp((exp(1)*log(x)-x)/exp(1))^2+8*x*exp(1)*log(5)),x, algorithm="giac")

[Out]

integrate(1/2*(4*(x^2 + 2*x)*e*log(5) - (2*x^2 - (x^2 + 4*x)*e)*e^(2*(e*log(x) - x)*e^(-1)) + (4*x*e*log(5) +
((x + 2)*e - 2*x)*e^(2*(e*log(x) - x)*e^(-1)))*log(1/4*e^(2*(e*log(x) - x)*e^(-1)) + log(5)))/(4*x*e*log(5) +
x*e^(2*(e*log(x) - x)*e^(-1) + 1)), x)

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maple [A]  time = 0.25, size = 45, normalized size = 1.55

method	result	size
risch	\(\frac {\ln \left (\frac {x^{2} {\mathrm e}^{-2 \,{\mathrm e}^{-1} x}}{4}+\ln \relax (5)\right )^{2}}{4}+\frac {x \ln \left (\frac {x^{2} {\mathrm e}^{-2 \,{\mathrm e}^{-1} x}}{4}+\ln \relax (5)\right )}{2}+\frac {x^{2}}{4}+x\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((2+x)*exp(1)-2*x)*exp((exp(1)*ln(x)-x)/exp(1))^2+4*x*exp(1)*ln(5))*ln(1/4*exp((exp(1)*ln(x)-x)/exp(1))^
2+ln(5))+((x^2+4*x)*exp(1)-2*x^2)*exp((exp(1)*ln(x)-x)/exp(1))^2+(4*x^2+8*x)*exp(1)*ln(5))/(2*x*exp(1)*exp((ex
p(1)*ln(x)-x)/exp(1))^2+8*x*exp(1)*ln(5)),x,method=_RETURNVERBOSE)

[Out]

1/4*ln(1/4*x^2*exp(-2*exp(-1)*x)+ln(5))^2+1/2*x*ln(1/4*x^2*exp(-2*exp(-1)*x)+ln(5))+1/4*x^2+x

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maxima [B]  time = 0.66, size = 87, normalized size = 3.00 \begin {gather*} \frac {1}{4} \, {\left (x^{2} {\left (e^{2} - 4 \, e + 4\right )} + e^{2} \log \left (x^{2} + 4 \, e^{\left (2 \, x e^{\left (-1\right )}\right )} \log \relax (5)\right )^{2} - 4 \, {\left ({\left (\log \relax (2) - 1\right )} e^{2} - 2 \, e \log \relax (2)\right )} x + 2 \, {\left (x {\left (e^{2} - 2 \, e\right )} - 2 \, e^{2} \log \relax (2)\right )} \log \left (x^{2} + 4 \, e^{\left (2 \, x e^{\left (-1\right )}\right )} \log \relax (5)\right )\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2+x)*exp(1)-2*x)*exp((exp(1)*log(x)-x)/exp(1))^2+4*x*exp(1)*log(5))*log(1/4*exp((exp(1)*log(x)-x
)/exp(1))^2+log(5))+((x^2+4*x)*exp(1)-2*x^2)*exp((exp(1)*log(x)-x)/exp(1))^2+(4*x^2+8*x)*exp(1)*log(5))/(2*x*e
xp(1)*exp((exp(1)*log(x)-x)/exp(1))^2+8*x*exp(1)*log(5)),x, algorithm="maxima")

[Out]

1/4*(x^2*(e^2 - 4*e + 4) + e^2*log(x^2 + 4*e^(2*x*e^(-1))*log(5))^2 - 4*((log(2) - 1)*e^2 - 2*e*log(2))*x + 2*
(x*(e^2 - 2*e) - 2*e^2*log(2))*log(x^2 + 4*e^(2*x*e^(-1))*log(5)))*e^(-2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{-2\,{\mathrm {e}}^{-1}\,\left (x-\mathrm {e}\,\ln \relax (x)\right )}\,\left (\mathrm {e}\,\left (x^2+4\,x\right )-2\,x^2\right )-\ln \left (\ln \relax (5)+\frac {{\mathrm {e}}^{-2\,{\mathrm {e}}^{-1}\,\left (x-\mathrm {e}\,\ln \relax (x)\right )}}{4}\right )\,\left ({\mathrm {e}}^{-2\,{\mathrm {e}}^{-1}\,\left (x-\mathrm {e}\,\ln \relax (x)\right )}\,\left (2\,x-\mathrm {e}\,\left (x+2\right )\right )-4\,x\,\mathrm {e}\,\ln \relax (5)\right )+\mathrm {e}\,\ln \relax (5)\,\left (4\,x^2+8\,x\right )}{8\,x\,\mathrm {e}\,\ln \relax (5)+2\,x\,\mathrm {e}\,{\mathrm {e}}^{-2\,{\mathrm {e}}^{-1}\,\left (x-\mathrm {e}\,\ln \relax (x)\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-2*exp(-1)*(x - exp(1)*log(x)))*(exp(1)*(4*x + x^2) - 2*x^2) - log(log(5) + exp(-2*exp(-1)*(x - exp(1
)*log(x)))/4)*(exp(-2*exp(-1)*(x - exp(1)*log(x)))*(2*x - exp(1)*(x + 2)) - 4*x*exp(1)*log(5)) + exp(1)*log(5)
*(8*x + 4*x^2))/(8*x*exp(1)*log(5) + 2*x*exp(1)*exp(-2*exp(-1)*(x - exp(1)*log(x)))),x)

[Out]

int((exp(-2*exp(-1)*(x - exp(1)*log(x)))*(exp(1)*(4*x + x^2) - 2*x^2) - log(log(5) + exp(-2*exp(-1)*(x - exp(1
)*log(x)))/4)*(exp(-2*exp(-1)*(x - exp(1)*log(x)))*(2*x - exp(1)*(x + 2)) - 4*x*exp(1)*log(5)) + exp(1)*log(5)
*(8*x + 4*x^2))/(8*x*exp(1)*log(5) + 2*x*exp(1)*exp(-2*exp(-1)*(x - exp(1)*log(x)))), x)

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sympy [B]  time = 0.66, size = 60, normalized size = 2.07 \begin {gather*} \frac {x^{2}}{4} + \frac {x \log {\left (\frac {e^{\frac {2 \left (- x + e \log {\relax (x )}\right )}{e}}}{4} + \log {\relax (5 )} \right )}}{2} + x + \frac {\log {\left (\frac {e^{\frac {2 \left (- x + e \log {\relax (x )}\right )}{e}}}{4} + \log {\relax (5 )} \right )}^{2}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2+x)*exp(1)-2*x)*exp((exp(1)*ln(x)-x)/exp(1))**2+4*x*exp(1)*ln(5))*ln(1/4*exp((exp(1)*ln(x)-x)/e
xp(1))**2+ln(5))+((x**2+4*x)*exp(1)-2*x**2)*exp((exp(1)*ln(x)-x)/exp(1))**2+(4*x**2+8*x)*exp(1)*ln(5))/(2*x*ex
p(1)*exp((exp(1)*ln(x)-x)/exp(1))**2+8*x*exp(1)*ln(5)),x)

[Out]

x**2/4 + x*log(exp(2*(-x + E*log(x))*exp(-1))/4 + log(5))/2 + x + log(exp(2*(-x + E*log(x))*exp(-1))/4 + log(5
))**2/4

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