3.72.94 \(\int \frac {16 x^2+e^x (8 x+8 x^2)+(6 x+e^{2 x} x+2 x^3+e^x (3+3 x^2)) \log (3+e^x x+x^2)+(64 x+e^x (32+32 x)+(24+8 e^x x+8 x^2) \log (3+e^x x+x^2)) \log ((-1+\log (9)) \log (3+e^x x+x^2))}{(3+e^x x+x^2) \log (3+e^x x+x^2)} \, dx\)

Optimal. Leaf size=26 \[ e^x+\left (x+4 \log \left ((-1+\log (9)) \log \left (3+x \left (e^x+x\right )\right )\right )\right )^2 \]

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Rubi [F]  time = 8.20, 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 {16 x^2+e^x \left (8 x+8 x^2\right )+\left (6 x+e^{2 x} x+2 x^3+e^x \left (3+3 x^2\right )\right ) \log \left (3+e^x x+x^2\right )+\left (64 x+e^x (32+32 x)+\left (24+8 e^x x+8 x^2\right ) \log \left (3+e^x x+x^2\right )\right ) \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(16*x^2 + E^x*(8*x + 8*x^2) + (6*x + E^(2*x)*x + 2*x^3 + E^x*(3 + 3*x^2))*Log[3 + E^x*x + x^2] + (64*x + E
^x*(32 + 32*x) + (24 + 8*E^x*x + 8*x^2)*Log[3 + E^x*x + x^2])*Log[(-1 + Log[9])*Log[3 + E^x*x + x^2]])/((3 + E
^x*x + x^2)*Log[3 + E^x*x + x^2]),x]

[Out]

E^x + x^2 + 8*x*Log[-((1 - Log[9])*Log[3 + E^x*x + x^2])] + 32*Defer[Int][Log[(-1 + Log[9])*Log[3 + E^x*x + x^
2]]/Log[3 + E^x*x + x^2], x] + 32*Defer[Int][Log[(-1 + Log[9])*Log[3 + E^x*x + x^2]]/(x*Log[3 + E^x*x + x^2]),
x] - 96*Defer[Int][Log[(-1 + Log[9])*Log[3 + E^x*x + x^2]]/((3 + E^x*x + x^2)*Log[3 + E^x*x + x^2]), x] - 96*
Defer[Int][Log[(-1 + Log[9])*Log[3 + E^x*x + x^2]]/(x*(3 + E^x*x + x^2)*Log[3 + E^x*x + x^2]), x] + 32*Defer[I
nt][(x*Log[(-1 + Log[9])*Log[3 + E^x*x + x^2]])/((3 + E^x*x + x^2)*Log[3 + E^x*x + x^2]), x] - 32*Defer[Int][(
x^2*Log[(-1 + Log[9])*Log[3 + E^x*x + x^2]])/((3 + E^x*x + x^2)*Log[3 + E^x*x + x^2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x-\frac {8 \left (3+3 x-x^2+x^3\right ) \left (x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )}{x \left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}+\frac {2 \left (4+4 x+x \log \left (3+e^x x+x^2\right )\right ) \left (x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )}{x \log \left (3+e^x x+x^2\right )}\right ) \, dx\\ &=2 \int \frac {\left (4+4 x+x \log \left (3+e^x x+x^2\right )\right ) \left (x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )}{x \log \left (3+e^x x+x^2\right )} \, dx-8 \int \frac {\left (3+3 x-x^2+x^3\right ) \left (x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )}{x \left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx+\int e^x \, dx\\ &=e^x+2 \int \left (\frac {4+4 x+x \log \left (3+e^x x+x^2\right )}{\log \left (3+e^x x+x^2\right )}+\frac {4 \left (4+4 x+x \log \left (3+e^x x+x^2\right )\right ) \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \log \left (3+e^x x+x^2\right )}\right ) \, dx-8 \int \left (\frac {3 \left (x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}+\frac {3 \left (x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )}{x \left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}-\frac {x \left (x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}+\frac {x^2 \left (x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}\right ) \, dx\\ &=e^x+2 \int \frac {4+4 x+x \log \left (3+e^x x+x^2\right )}{\log \left (3+e^x x+x^2\right )} \, dx+8 \int \frac {\left (4+4 x+x \log \left (3+e^x x+x^2\right )\right ) \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \log \left (3+e^x x+x^2\right )} \, dx+8 \int \frac {x \left (x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-8 \int \frac {x^2 \left (x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-24 \int \frac {x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-24 \int \frac {x+4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx\\ &=e^x+2 \int \left (x+\frac {4 (1+x)}{\log \left (3+e^x x+x^2\right )}\right ) \, dx+8 \int \left (\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )+\frac {4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\log \left (3+e^x x+x^2\right )}+\frac {4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \log \left (3+e^x x+x^2\right )}\right ) \, dx+8 \int \left (\frac {x^2}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}+\frac {4 x \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}\right ) \, dx-8 \int \left (\frac {x^3}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}+\frac {4 x^2 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}\right ) \, dx-24 \int \left (\frac {x}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}+\frac {4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}\right ) \, dx-24 \int \left (\frac {1}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}+\frac {4 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}\right ) \, dx\\ &=e^x+x^2+8 \int \frac {1+x}{\log \left (3+e^x x+x^2\right )} \, dx+8 \int \frac {x^2}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-8 \int \frac {x^3}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx+8 \int \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right ) \, dx-24 \int \frac {1}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-24 \int \frac {x}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {x \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-32 \int \frac {x^2 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx\\ &=e^x+x^2+8 x \log \left (-\left ((1-\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )+8 \int \left (\frac {1}{\log \left (3+e^x x+x^2\right )}+\frac {x}{\log \left (3+e^x x+x^2\right )}\right ) \, dx+8 \int \frac {x^2}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-8 \int \frac {x^3}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-8 \int \frac {x \left (2 x+e^x (1+x)\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-24 \int \frac {1}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-24 \int \frac {x}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {x \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-32 \int \frac {x^2 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx\\ &=e^x+x^2+8 x \log \left (-\left ((1-\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )-8 \int \left (\frac {1+x}{\log \left (3+e^x x+x^2\right )}-\frac {3+3 x-x^2+x^3}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}\right ) \, dx+8 \int \frac {1}{\log \left (3+e^x x+x^2\right )} \, dx+8 \int \frac {x}{\log \left (3+e^x x+x^2\right )} \, dx+8 \int \frac {x^2}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-8 \int \frac {x^3}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-24 \int \frac {1}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-24 \int \frac {x}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {x \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-32 \int \frac {x^2 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx\\ &=e^x+x^2+8 x \log \left (-\left ((1-\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )+8 \int \frac {1}{\log \left (3+e^x x+x^2\right )} \, dx+8 \int \frac {x}{\log \left (3+e^x x+x^2\right )} \, dx-8 \int \frac {1+x}{\log \left (3+e^x x+x^2\right )} \, dx+8 \int \frac {x^2}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-8 \int \frac {x^3}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx+8 \int \frac {3+3 x-x^2+x^3}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-24 \int \frac {1}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-24 \int \frac {x}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {x \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-32 \int \frac {x^2 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx\\ &=e^x+x^2+8 x \log \left (-\left ((1-\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )-8 \int \left (\frac {1}{\log \left (3+e^x x+x^2\right )}+\frac {x}{\log \left (3+e^x x+x^2\right )}\right ) \, dx+8 \int \left (\frac {3}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}+\frac {3 x}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}-\frac {x^2}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}+\frac {x^3}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )}\right ) \, dx+8 \int \frac {1}{\log \left (3+e^x x+x^2\right )} \, dx+8 \int \frac {x}{\log \left (3+e^x x+x^2\right )} \, dx+8 \int \frac {x^2}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-8 \int \frac {x^3}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-24 \int \frac {1}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-24 \int \frac {x}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {x \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-32 \int \frac {x^2 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx\\ &=e^x+x^2+8 x \log \left (-\left ((1-\log (9)) \log \left (3+e^x x+x^2\right )\right )\right )+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \log \left (3+e^x x+x^2\right )} \, dx+32 \int \frac {x \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-32 \int \frac {x^2 \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{\left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx-96 \int \frac {\log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )}{x \left (3+e^x x+x^2\right ) \log \left (3+e^x x+x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 48, normalized size = 1.85 \begin {gather*} e^x+x^2+8 x \log \left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right )+16 \log ^2\left ((-1+\log (9)) \log \left (3+e^x x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16*x^2 + E^x*(8*x + 8*x^2) + (6*x + E^(2*x)*x + 2*x^3 + E^x*(3 + 3*x^2))*Log[3 + E^x*x + x^2] + (64
*x + E^x*(32 + 32*x) + (24 + 8*E^x*x + 8*x^2)*Log[3 + E^x*x + x^2])*Log[(-1 + Log[9])*Log[3 + E^x*x + x^2]])/(
(3 + E^x*x + x^2)*Log[3 + E^x*x + x^2]),x]

[Out]

E^x + x^2 + 8*x*Log[(-1 + Log[9])*Log[3 + E^x*x + x^2]] + 16*Log[(-1 + Log[9])*Log[3 + E^x*x + x^2]]^2

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fricas [A]  time = 0.67, size = 49, normalized size = 1.88 \begin {gather*} x^{2} + 8 \, x \log \left ({\left (2 \, \log \relax (3) - 1\right )} \log \left (x^{2} + x e^{x} + 3\right )\right ) + 16 \, \log \left ({\left (2 \, \log \relax (3) - 1\right )} \log \left (x^{2} + x e^{x} + 3\right )\right )^{2} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*exp(x)*x+8*x^2+24)*log(exp(x)*x+x^2+3)+(32*x+32)*exp(x)+64*x)*log((2*log(3)-1)*log(exp(x)*x+x^2
+3))+(x*exp(x)^2+(3*x^2+3)*exp(x)+2*x^3+6*x)*log(exp(x)*x+x^2+3)+(8*x^2+8*x)*exp(x)+16*x^2)/(exp(x)*x+x^2+3)/l
og(exp(x)*x+x^2+3),x, algorithm="fricas")

[Out]

x^2 + 8*x*log((2*log(3) - 1)*log(x^2 + x*e^x + 3)) + 16*log((2*log(3) - 1)*log(x^2 + x*e^x + 3))^2 + e^x

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giac [B]  time = 0.19, size = 65, normalized size = 2.50 \begin {gather*} x^{2} + 8 \, x \log \left (2 \, \log \relax (3) - 1\right ) + 8 \, x \log \left (\log \left (x^{2} + x e^{x} + 3\right )\right ) + 32 \, \log \left (2 \, \log \relax (3) - 1\right ) \log \left (\log \left (x^{2} + x e^{x} + 3\right )\right ) + 16 \, \log \left (\log \left (x^{2} + x e^{x} + 3\right )\right )^{2} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*exp(x)*x+8*x^2+24)*log(exp(x)*x+x^2+3)+(32*x+32)*exp(x)+64*x)*log((2*log(3)-1)*log(exp(x)*x+x^2
+3))+(x*exp(x)^2+(3*x^2+3)*exp(x)+2*x^3+6*x)*log(exp(x)*x+x^2+3)+(8*x^2+8*x)*exp(x)+16*x^2)/(exp(x)*x+x^2+3)/l
og(exp(x)*x+x^2+3),x, algorithm="giac")

[Out]

x^2 + 8*x*log(2*log(3) - 1) + 8*x*log(log(x^2 + x*e^x + 3)) + 32*log(2*log(3) - 1)*log(log(x^2 + x*e^x + 3)) +
16*log(log(x^2 + x*e^x + 3))^2 + e^x

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maple [A]  time = 0.04, size = 50, normalized size = 1.92

method	result	size
risch	\(x^{2}+8 \ln \left (\left (2 \ln \relax (3)-1\right ) \ln \left ({\mathrm e}^{x} x +x^{2}+3\right )\right ) x +16 \ln \left (\left (2 \ln \relax (3)-1\right ) \ln \left ({\mathrm e}^{x} x +x^{2}+3\right )\right )^{2}+{\mathrm e}^{x}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8*exp(x)*x+8*x^2+24)*ln(exp(x)*x+x^2+3)+(32*x+32)*exp(x)+64*x)*ln((2*ln(3)-1)*ln(exp(x)*x+x^2+3))+(x*ex
p(x)^2+(3*x^2+3)*exp(x)+2*x^3+6*x)*ln(exp(x)*x+x^2+3)+(8*x^2+8*x)*exp(x)+16*x^2)/(exp(x)*x+x^2+3)/ln(exp(x)*x+
x^2+3),x,method=_RETURNVERBOSE)

[Out]

x^2+8*ln((2*ln(3)-1)*ln(exp(x)*x+x^2+3))*x+16*ln((2*ln(3)-1)*ln(exp(x)*x+x^2+3))^2+exp(x)

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maxima [B]  time = 0.51, size = 55, normalized size = 2.12 \begin {gather*} x^{2} + 8 \, x \log \left (2 \, \log \relax (3) - 1\right ) + 8 \, {\left (x + 4 \, \log \left (2 \, \log \relax (3) - 1\right )\right )} \log \left (\log \left (x^{2} + x e^{x} + 3\right )\right ) + 16 \, \log \left (\log \left (x^{2} + x e^{x} + 3\right )\right )^{2} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*exp(x)*x+8*x^2+24)*log(exp(x)*x+x^2+3)+(32*x+32)*exp(x)+64*x)*log((2*log(3)-1)*log(exp(x)*x+x^2
+3))+(x*exp(x)^2+(3*x^2+3)*exp(x)+2*x^3+6*x)*log(exp(x)*x+x^2+3)+(8*x^2+8*x)*exp(x)+16*x^2)/(exp(x)*x+x^2+3)/l
og(exp(x)*x+x^2+3),x, algorithm="maxima")

[Out]

x^2 + 8*x*log(2*log(3) - 1) + 8*(x + 4*log(2*log(3) - 1))*log(log(x^2 + x*e^x + 3)) + 16*log(log(x^2 + x*e^x +
3))^2 + e^x

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mupad [B]  time = 4.53, size = 49, normalized size = 1.88 \begin {gather*} {\mathrm {e}}^x+8\,x\,\ln \left (\ln \left (x\,{\mathrm {e}}^x+x^2+3\right )\,\left (2\,\ln \relax (3)-1\right )\right )+16\,{\ln \left (\ln \left (x\,{\mathrm {e}}^x+x^2+3\right )\,\left (2\,\ln \relax (3)-1\right )\right )}^2+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x*exp(x) + x^2 + 3)*(6*x + x*exp(2*x) + exp(x)*(3*x^2 + 3) + 2*x^3) + log(log(x*exp(x) + x^2 + 3)*(2*
log(3) - 1))*(64*x + log(x*exp(x) + x^2 + 3)*(8*x*exp(x) + 8*x^2 + 24) + exp(x)*(32*x + 32)) + exp(x)*(8*x + 8
*x^2) + 16*x^2)/(log(x*exp(x) + x^2 + 3)*(x*exp(x) + x^2 + 3)),x)

[Out]

exp(x) + 8*x*log(log(x*exp(x) + x^2 + 3)*(2*log(3) - 1)) + 16*log(log(x*exp(x) + x^2 + 3)*(2*log(3) - 1))^2 +
x^2

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sympy [B]  time = 1.66, size = 53, normalized size = 2.04 \begin {gather*} x^{2} + 8 x \log {\left (\left (-1 + 2 \log {\relax (3 )}\right ) \log {\left (x^{2} + x e^{x} + 3 \right )} \right )} + e^{x} + 16 \log {\left (\left (-1 + 2 \log {\relax (3 )}\right ) \log {\left (x^{2} + x e^{x} + 3 \right )} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*exp(x)*x+8*x**2+24)*ln(exp(x)*x+x**2+3)+(32*x+32)*exp(x)+64*x)*ln((2*ln(3)-1)*ln(exp(x)*x+x**2+
3))+(x*exp(x)**2+(3*x**2+3)*exp(x)+2*x**3+6*x)*ln(exp(x)*x+x**2+3)+(8*x**2+8*x)*exp(x)+16*x**2)/(exp(x)*x+x**2
+3)/ln(exp(x)*x+x**2+3),x)

[Out]

x**2 + 8*x*log((-1 + 2*log(3))*log(x**2 + x*exp(x) + 3)) + exp(x) + 16*log((-1 + 2*log(3))*log(x**2 + x*exp(x)
+ 3))**2

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