3.17.11 \(\int \frac {-2 x^2 \log (x) \log (\frac {5 e^x \log (x)}{x})+e^{10} \log ^2(\frac {5 e^x \log (x)}{x}) (-32-16 x-2 x^2+(32-16 x-14 x^2-2 x^3) \log (x)+(-8 x-2 x^2) \log (x) \log (\frac {5 e^x \log (x)}{x}))+e^5 \log (\frac {5 e^x \log (x)}{x}) (8 x+2 x^2+(-8 x+6 x^2+2 x^3) \log (x)+(8 x+4 x^2) \log (x) \log (\frac {5 e^x \log (x)}{x}))}{x \log (x) \log (\frac {5 e^x \log (x)}{x})} \, dx\) [1611]

Optimal. Leaf size=28 \[ 3-\left (-x+e^5 (4+x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )^2 \]

[Out]

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

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Rubi [F]
time = 1.61, 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 {-2 x^2 \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )+e^{10} \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \left (-32-16 x-2 x^2+\left (32-16 x-14 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )+e^5 \log \left (\frac {5 e^x \log (x)}{x}\right ) \left (8 x+2 x^2+\left (-8 x+6 x^2+2 x^3\right ) \log (x)+\left (8 x+4 x^2\right ) \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right )}{x \log (x) \log \left (\frac {5 e^x \log (x)}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x^2*Log[x]*Log[(5*E^x*Log[x])/x] + E^10*Log[(5*E^x*Log[x])/x]^2*(-32 - 16*x - 2*x^2 + (32 - 16*x - 14*
x^2 - 2*x^3)*Log[x] + (-8*x - 2*x^2)*Log[x]*Log[(5*E^x*Log[x])/x]) + E^5*Log[(5*E^x*Log[x])/x]*(8*x + 2*x^2 +
(-8*x + 6*x^2 + 2*x^3)*Log[x] + (8*x + 4*x^2)*Log[x]*Log[(5*E^x*Log[x])/x]))/(x*Log[x]*Log[(5*E^x*Log[x])/x]),
x]

[Out]

-8*E^5*x + 16*E^10*x + 8*E^5*(1 - 2*E^5)*x - x^2 + 3*E^5*x^2 + (E^5*(2 - 7*E^5)*x^2)/2 - 4*E^5*(1 - 2*E^5)*x^2
+ (2*E^5*x^3)/3 - (2*E^10*x^3)/9 - (E^5*(2 - 7*E^5)*x^3)/3 + (E^10*x^4)/6 + 2*E^5*ExpIntegralEi[2*Log[x]] - 1
6*E^10*ExpIntegralEi[2*Log[x]] - E^5*(2 - 7*E^5)*ExpIntegralEi[2*Log[x]] + (2*E^10*ExpIntegralEi[3*Log[x]])/3
+ 8*E^5*(1 - 2*E^5)*x*Log[(5*E^x*Log[x])/x] + E^5*(2 - 7*E^5)*x^2*Log[(5*E^x*Log[x])/x] - (2*E^10*x^3*Log[(5*E
^x*Log[x])/x])/3 + 8*E^5*LogIntegral[x] - 8*E^5*(1 - 2*E^5)*LogIntegral[x] + 16*E^10*x*LogIntegral[x] - 16*E^1
0*Log[x]*LogIntegral[x] - 16*E^10*Log[(5*E^x*Log[x])/x]*LogIntegral[x] + 32*E^10*Defer[Int][Log[(5*E^x*Log[x])
/x]/x, x] - 32*E^10*Defer[Int][Log[(5*E^x*Log[x])/x]/(x*Log[x]), x] - 2*E^10*Defer[Int][(x*Log[(5*E^x*Log[x])/
x])/Log[x], x] - 8*E^10*Defer[Int][Log[(5*E^x*Log[x])/x]^2, x] - 2*E^10*Defer[Int][x*Log[(5*E^x*Log[x])/x]^2,
x] + 16*E^10*Defer[Int][LogIntegral[x]/(x*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (x-e^5 (4+x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right ) \left (e^5 (4+x)+\log (x) \left (-x+e^5 \left (-4+3 x+x^2\right )+e^5 x \log \left (\frac {5 e^x \log (x)}{x}\right )\right )\right )}{x \log (x)} \, dx\\ &=2 \int \frac {\left (x-e^5 (4+x) \log \left (\frac {5 e^x \log (x)}{x}\right )\right ) \left (e^5 (4+x)+\log (x) \left (-x+e^5 \left (-4+3 x+x^2\right )+e^5 x \log \left (\frac {5 e^x \log (x)}{x}\right )\right )\right )}{x \log (x)} \, dx\\ &=2 \int \left (\frac {4 e^5+e^5 x-4 e^5 \log (x)-\left (1-3 e^5\right ) x \log (x)+e^5 x^2 \log (x)}{\log (x)}+\frac {e^5 \left (-16 e^5-8 e^5 x-e^5 x^2+16 e^5 \log (x)+4 \left (1-2 e^5\right ) x \log (x)+2 \left (1-\frac {7 e^5}{2}\right ) x^2 \log (x)-e^5 x^3 \log (x)\right ) \log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)}-e^{10} (4+x) \log ^2\left (\frac {5 e^x \log (x)}{x}\right )\right ) \, dx\\ &=2 \int \frac {4 e^5+e^5 x-4 e^5 \log (x)-\left (1-3 e^5\right ) x \log (x)+e^5 x^2 \log (x)}{\log (x)} \, dx+\left (2 e^5\right ) \int \frac {\left (-16 e^5-8 e^5 x-e^5 x^2+16 e^5 \log (x)+4 \left (1-2 e^5\right ) x \log (x)+2 \left (1-\frac {7 e^5}{2}\right ) x^2 \log (x)-e^5 x^3 \log (x)\right ) \log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)} \, dx-\left (2 e^{10}\right ) \int (4+x) \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx\\ &=2 \int \left (-x+e^5 \left (-4+3 x+x^2\right )+\frac {e^5 (4+x)}{\log (x)}\right ) \, dx+\left (2 e^5\right ) \int \frac {\left (-e^5 (4+x)^2-\left (-2 x (2+x)+e^5 (-1+x) (4+x)^2\right ) \log (x)\right ) \log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)} \, dx-\left (2 e^{10}\right ) \int \left (4 \log ^2\left (\frac {5 e^x \log (x)}{x}\right )+x \log ^2\left (\frac {5 e^x \log (x)}{x}\right )\right ) \, dx\\ &=-x^2+\left (2 e^5\right ) \int \left (-4+3 x+x^2\right ) \, dx+\left (2 e^5\right ) \int \frac {4+x}{\log (x)} \, dx+\left (2 e^5\right ) \int \left (4 \left (1-2 e^5\right ) \log \left (\frac {5 e^x \log (x)}{x}\right )+\frac {16 e^5 \log \left (\frac {5 e^x \log (x)}{x}\right )}{x}+2 \left (1-\frac {7 e^5}{2}\right ) x \log \left (\frac {5 e^x \log (x)}{x}\right )-e^5 x^2 \log \left (\frac {5 e^x \log (x)}{x}\right )-\frac {8 e^5 \log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)}-\frac {16 e^5 \log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)}-\frac {e^5 x \log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)}\right ) \, dx-\left (2 e^{10}\right ) \int x \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx-\left (8 e^{10}\right ) \int \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx\\ &=-8 e^5 x-x^2+3 e^5 x^2+\frac {2 e^5 x^3}{3}+\left (2 e^5\right ) \int \left (\frac {4}{\log (x)}+\frac {x}{\log (x)}\right ) \, dx-\left (2 e^{10}\right ) \int x^2 \log \left (\frac {5 e^x \log (x)}{x}\right ) \, dx-\left (2 e^{10}\right ) \int \frac {x \log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)} \, dx-\left (2 e^{10}\right ) \int x \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx-\left (8 e^{10}\right ) \int \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx-\left (16 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)} \, dx+\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x} \, dx-\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)} \, dx+\left (2 e^5 \left (2-7 e^5\right )\right ) \int x \log \left (\frac {5 e^x \log (x)}{x}\right ) \, dx+\left (8 e^5 \left (1-2 e^5\right )\right ) \int \log \left (\frac {5 e^x \log (x)}{x}\right ) \, dx\\ &=-8 e^5 x-x^2+3 e^5 x^2+\frac {2 e^5 x^3}{3}+8 e^5 \left (1-2 e^5\right ) x \log \left (\frac {5 e^x \log (x)}{x}\right )+e^5 \left (2-7 e^5\right ) x^2 \log \left (\frac {5 e^x \log (x)}{x}\right )-\frac {2}{3} e^{10} x^3 \log \left (\frac {5 e^x \log (x)}{x}\right )-16 e^{10} \log \left (\frac {5 e^x \log (x)}{x}\right ) \text {li}(x)+\left (2 e^5\right ) \int \frac {x}{\log (x)} \, dx+\left (8 e^5\right ) \int \frac {1}{\log (x)} \, dx+\left (2 e^{10}\right ) \int \frac {x^2 (1+(-1+x) \log (x))}{3 \log (x)} \, dx-\left (2 e^{10}\right ) \int \frac {x \log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)} \, dx-\left (2 e^{10}\right ) \int x \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx-\left (8 e^{10}\right ) \int \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx+\left (16 e^{10}\right ) \int \frac {(1+(-1+x) \log (x)) \text {li}(x)}{x \log (x)} \, dx+\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x} \, dx-\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)} \, dx-\left (2 e^5 \left (2-7 e^5\right )\right ) \int \frac {x (1+(-1+x) \log (x))}{2 \log (x)} \, dx-\left (8 e^5 \left (1-2 e^5\right )\right ) \int \left (-1+x+\frac {1}{\log (x)}\right ) \, dx\\ &=-8 e^5 x+8 e^5 \left (1-2 e^5\right ) x-x^2+3 e^5 x^2-4 e^5 \left (1-2 e^5\right ) x^2+\frac {2 e^5 x^3}{3}+8 e^5 \left (1-2 e^5\right ) x \log \left (\frac {5 e^x \log (x)}{x}\right )+e^5 \left (2-7 e^5\right ) x^2 \log \left (\frac {5 e^x \log (x)}{x}\right )-\frac {2}{3} e^{10} x^3 \log \left (\frac {5 e^x \log (x)}{x}\right )+8 e^5 \text {li}(x)-16 e^{10} \log \left (\frac {5 e^x \log (x)}{x}\right ) \text {li}(x)+\left (2 e^5\right ) \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )+\frac {1}{3} \left (2 e^{10}\right ) \int \frac {x^2 (1+(-1+x) \log (x))}{\log (x)} \, dx-\left (2 e^{10}\right ) \int \frac {x \log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)} \, dx-\left (2 e^{10}\right ) \int x \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx-\left (8 e^{10}\right ) \int \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx+\left (16 e^{10}\right ) \int \left (\text {li}(x)-\frac {\text {li}(x)}{x}+\frac {\text {li}(x)}{x \log (x)}\right ) \, dx+\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x} \, dx-\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)} \, dx-\left (e^5 \left (2-7 e^5\right )\right ) \int \frac {x (1+(-1+x) \log (x))}{\log (x)} \, dx-\left (8 e^5 \left (1-2 e^5\right )\right ) \int \frac {1}{\log (x)} \, dx\\ &=-8 e^5 x+8 e^5 \left (1-2 e^5\right ) x-x^2+3 e^5 x^2-4 e^5 \left (1-2 e^5\right ) x^2+\frac {2 e^5 x^3}{3}+2 e^5 \text {Ei}(2 \log (x))+8 e^5 \left (1-2 e^5\right ) x \log \left (\frac {5 e^x \log (x)}{x}\right )+e^5 \left (2-7 e^5\right ) x^2 \log \left (\frac {5 e^x \log (x)}{x}\right )-\frac {2}{3} e^{10} x^3 \log \left (\frac {5 e^x \log (x)}{x}\right )+8 e^5 \text {li}(x)-8 e^5 \left (1-2 e^5\right ) \text {li}(x)-16 e^{10} \log \left (\frac {5 e^x \log (x)}{x}\right ) \text {li}(x)+\frac {1}{3} \left (2 e^{10}\right ) \int \left ((-1+x) x^2+\frac {x^2}{\log (x)}\right ) \, dx-\left (2 e^{10}\right ) \int \frac {x \log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)} \, dx-\left (2 e^{10}\right ) \int x \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx-\left (8 e^{10}\right ) \int \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx+\left (16 e^{10}\right ) \int \text {li}(x) \, dx-\left (16 e^{10}\right ) \int \frac {\text {li}(x)}{x} \, dx+\left (16 e^{10}\right ) \int \frac {\text {li}(x)}{x \log (x)} \, dx+\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x} \, dx-\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)} \, dx-\left (e^5 \left (2-7 e^5\right )\right ) \int x \left (-1+x+\frac {1}{\log (x)}\right ) \, dx\\ &=-8 e^5 x+16 e^{10} x+8 e^5 \left (1-2 e^5\right ) x-x^2+3 e^5 x^2-4 e^5 \left (1-2 e^5\right ) x^2+\frac {2 e^5 x^3}{3}+2 e^5 \text {Ei}(2 \log (x))-16 e^{10} \text {Ei}(2 \log (x))+8 e^5 \left (1-2 e^5\right ) x \log \left (\frac {5 e^x \log (x)}{x}\right )+e^5 \left (2-7 e^5\right ) x^2 \log \left (\frac {5 e^x \log (x)}{x}\right )-\frac {2}{3} e^{10} x^3 \log \left (\frac {5 e^x \log (x)}{x}\right )+8 e^5 \text {li}(x)-8 e^5 \left (1-2 e^5\right ) \text {li}(x)+16 e^{10} x \text {li}(x)-16 e^{10} \log (x) \text {li}(x)-16 e^{10} \log \left (\frac {5 e^x \log (x)}{x}\right ) \text {li}(x)+\frac {1}{3} \left (2 e^{10}\right ) \int (-1+x) x^2 \, dx+\frac {1}{3} \left (2 e^{10}\right ) \int \frac {x^2}{\log (x)} \, dx-\left (2 e^{10}\right ) \int \frac {x \log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)} \, dx-\left (2 e^{10}\right ) \int x \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx-\left (8 e^{10}\right ) \int \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx+\left (16 e^{10}\right ) \int \frac {\text {li}(x)}{x \log (x)} \, dx+\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x} \, dx-\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)} \, dx-\left (e^5 \left (2-7 e^5\right )\right ) \int \left ((-1+x) x+\frac {x}{\log (x)}\right ) \, dx\\ &=-8 e^5 x+16 e^{10} x+8 e^5 \left (1-2 e^5\right ) x-x^2+3 e^5 x^2-4 e^5 \left (1-2 e^5\right ) x^2+\frac {2 e^5 x^3}{3}+2 e^5 \text {Ei}(2 \log (x))-16 e^{10} \text {Ei}(2 \log (x))+8 e^5 \left (1-2 e^5\right ) x \log \left (\frac {5 e^x \log (x)}{x}\right )+e^5 \left (2-7 e^5\right ) x^2 \log \left (\frac {5 e^x \log (x)}{x}\right )-\frac {2}{3} e^{10} x^3 \log \left (\frac {5 e^x \log (x)}{x}\right )+8 e^5 \text {li}(x)-8 e^5 \left (1-2 e^5\right ) \text {li}(x)+16 e^{10} x \text {li}(x)-16 e^{10} \log (x) \text {li}(x)-16 e^{10} \log \left (\frac {5 e^x \log (x)}{x}\right ) \text {li}(x)+\frac {1}{3} \left (2 e^{10}\right ) \int \left (-x^2+x^3\right ) \, dx+\frac {1}{3} \left (2 e^{10}\right ) \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )-\left (2 e^{10}\right ) \int \frac {x \log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)} \, dx-\left (2 e^{10}\right ) \int x \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx-\left (8 e^{10}\right ) \int \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx+\left (16 e^{10}\right ) \int \frac {\text {li}(x)}{x \log (x)} \, dx+\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x} \, dx-\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)} \, dx-\left (e^5 \left (2-7 e^5\right )\right ) \int (-1+x) x \, dx-\left (e^5 \left (2-7 e^5\right )\right ) \int \frac {x}{\log (x)} \, dx\\ &=-8 e^5 x+16 e^{10} x+8 e^5 \left (1-2 e^5\right ) x-x^2+3 e^5 x^2-4 e^5 \left (1-2 e^5\right ) x^2+\frac {2 e^5 x^3}{3}-\frac {2 e^{10} x^3}{9}+\frac {e^{10} x^4}{6}+2 e^5 \text {Ei}(2 \log (x))-16 e^{10} \text {Ei}(2 \log (x))+\frac {2}{3} e^{10} \text {Ei}(3 \log (x))+8 e^5 \left (1-2 e^5\right ) x \log \left (\frac {5 e^x \log (x)}{x}\right )+e^5 \left (2-7 e^5\right ) x^2 \log \left (\frac {5 e^x \log (x)}{x}\right )-\frac {2}{3} e^{10} x^3 \log \left (\frac {5 e^x \log (x)}{x}\right )+8 e^5 \text {li}(x)-8 e^5 \left (1-2 e^5\right ) \text {li}(x)+16 e^{10} x \text {li}(x)-16 e^{10} \log (x) \text {li}(x)-16 e^{10} \log \left (\frac {5 e^x \log (x)}{x}\right ) \text {li}(x)-\left (2 e^{10}\right ) \int \frac {x \log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)} \, dx-\left (2 e^{10}\right ) \int x \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx-\left (8 e^{10}\right ) \int \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx+\left (16 e^{10}\right ) \int \frac {\text {li}(x)}{x \log (x)} \, dx+\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x} \, dx-\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)} \, dx-\left (e^5 \left (2-7 e^5\right )\right ) \int \left (-x+x^2\right ) \, dx-\left (e^5 \left (2-7 e^5\right )\right ) \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=-8 e^5 x+16 e^{10} x+8 e^5 \left (1-2 e^5\right ) x-x^2+3 e^5 x^2+\frac {1}{2} e^5 \left (2-7 e^5\right ) x^2-4 e^5 \left (1-2 e^5\right ) x^2+\frac {2 e^5 x^3}{3}-\frac {2 e^{10} x^3}{9}-\frac {1}{3} e^5 \left (2-7 e^5\right ) x^3+\frac {e^{10} x^4}{6}+2 e^5 \text {Ei}(2 \log (x))-16 e^{10} \text {Ei}(2 \log (x))-e^5 \left (2-7 e^5\right ) \text {Ei}(2 \log (x))+\frac {2}{3} e^{10} \text {Ei}(3 \log (x))+8 e^5 \left (1-2 e^5\right ) x \log \left (\frac {5 e^x \log (x)}{x}\right )+e^5 \left (2-7 e^5\right ) x^2 \log \left (\frac {5 e^x \log (x)}{x}\right )-\frac {2}{3} e^{10} x^3 \log \left (\frac {5 e^x \log (x)}{x}\right )+8 e^5 \text {li}(x)-8 e^5 \left (1-2 e^5\right ) \text {li}(x)+16 e^{10} x \text {li}(x)-16 e^{10} \log (x) \text {li}(x)-16 e^{10} \log \left (\frac {5 e^x \log (x)}{x}\right ) \text {li}(x)-\left (2 e^{10}\right ) \int \frac {x \log \left (\frac {5 e^x \log (x)}{x}\right )}{\log (x)} \, dx-\left (2 e^{10}\right ) \int x \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx-\left (8 e^{10}\right ) \int \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \, dx+\left (16 e^{10}\right ) \int \frac {\text {li}(x)}{x \log (x)} \, dx+\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x} \, dx-\left (32 e^{10}\right ) \int \frac {\log \left (\frac {5 e^x \log (x)}{x}\right )}{x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(180\) vs. \(2(28)=56\).
time = 0.15, size = 180, normalized size = 6.43 \begin {gather*} -x^2+16 e^{10} x^2-16 e^{10} \log ^2\left (\frac {\log (x)}{x}\right )-32 e^{10} \log (x) \left (x+\log \left (\frac {\log (x)}{x}\right )-\log \left (\frac {5 e^x \log (x)}{x}\right )\right )+32 e^{10} \log (\log (x)) \left (x+\log \left (\frac {\log (x)}{x}\right )-\log \left (\frac {5 e^x \log (x)}{x}\right )\right )+8 e^5 x \log \left (\frac {5 e^x \log (x)}{x}\right )-32 e^{10} x \log \left (\frac {5 e^x \log (x)}{x}\right )+2 e^5 x^2 \log \left (\frac {5 e^x \log (x)}{x}\right )-8 e^{10} x \log ^2\left (\frac {5 e^x \log (x)}{x}\right )-e^{10} x^2 \log ^2\left (\frac {5 e^x \log (x)}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^2*Log[x]*Log[(5*E^x*Log[x])/x] + E^10*Log[(5*E^x*Log[x])/x]^2*(-32 - 16*x - 2*x^2 + (32 - 16*x
- 14*x^2 - 2*x^3)*Log[x] + (-8*x - 2*x^2)*Log[x]*Log[(5*E^x*Log[x])/x]) + E^5*Log[(5*E^x*Log[x])/x]*(8*x + 2*
x^2 + (-8*x + 6*x^2 + 2*x^3)*Log[x] + (8*x + 4*x^2)*Log[x]*Log[(5*E^x*Log[x])/x]))/(x*Log[x]*Log[(5*E^x*Log[x]
)/x]),x]

[Out]

-x^2 + 16*E^10*x^2 - 16*E^10*Log[Log[x]/x]^2 - 32*E^10*Log[x]*(x + Log[Log[x]/x] - Log[(5*E^x*Log[x])/x]) + 32
*E^10*Log[Log[x]]*(x + Log[Log[x]/x] - Log[(5*E^x*Log[x])/x]) + 8*E^5*x*Log[(5*E^x*Log[x])/x] - 32*E^10*x*Log[
(5*E^x*Log[x])/x] + 2*E^5*x^2*Log[(5*E^x*Log[x])/x] - 8*E^10*x*Log[(5*E^x*Log[x])/x]^2 - E^10*x^2*Log[(5*E^x*L
og[x])/x]^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.55, size = 5540, normalized size = 197.86

method	result	size
risch	\(\text {Expression too large to display}\)	\(5540\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^2-8*x)*ln(x)*ln(5*exp(x)*ln(x)/x)+(-2*x^3-14*x^2-16*x+32)*ln(x)-2*x^2-16*x-32)*exp(ln(ln(5*exp(x)*
ln(x)/x))+5)^2+((4*x^2+8*x)*ln(x)*ln(5*exp(x)*ln(x)/x)+(2*x^3+6*x^2-8*x)*ln(x)+2*x^2+8*x)*exp(ln(ln(5*exp(x)*l
n(x)/x))+5)-2*x^2*ln(x)*ln(5*exp(x)*ln(x)/x))/x/ln(x)/ln(5*exp(x)*ln(x)/x),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2-8*x)*log(x)*log(5*exp(x)*log(x)/x)+(-2*x^3-14*x^2-16*x+32)*log(x)-2*x^2-16*x-32)*exp(log(l
og(5*exp(x)*log(x)/x))+5)^2+((4*x^2+8*x)*log(x)*log(5*exp(x)*log(x)/x)+(2*x^3+6*x^2-8*x)*log(x)+2*x^2+8*x)*exp
(log(log(5*exp(x)*log(x)/x))+5)-2*x^2*log(x)*log(5*exp(x)*log(x)/x))/x/log(x)/log(5*exp(x)*log(x)/x),x, algori
thm="maxima")

[Out]

-x^4*e^10 - 2/3*(3*(log(5) + 4)*e^10 - 2*e^5)*x^3 + 2/3*x^3*e^5 - ((log(5)^2 + 16*log(5) + 16)*e^10 - (2*log(5
) + 5)*e^5)*x^2 + 3*x^2*e^5 - (x^2*e^10 + 8*x*e^10 + 16*e^10)*log(x)^2 - (x^2*e^10 + 8*x*e^10 + 16*e^10)*log(l
og(x))^2 - 8*((log(5)^2 + 4*log(5))*e^10 - (log(5) + 1)*e^5)*x - x^2 - 8*x*e^5 + 2*Ei(2*log(x))*e^5 + 8*Ei(log
(x))*e^5 + 2*(x^3*e^10 + ((log(5) + 8)*e^10 - e^5)*x^2 + 4*(2*(log(5) + 2)*e^10 - e^5)*x + 16*e^10*log(5))*log
(x) - 2*(x^3*e^10 + ((log(5) + 8)*e^10 - e^5)*x^2 + 4*(2*(log(5) + 2)*e^10 - e^5)*x - (x^2*e^10 + 8*x*e^10 + 1
6*e^10)*log(x))*log(log(x)) - 2*integrate((x^2*e^5 + 4*x*e^5 + 16*e^10*log(5))/(x*log(x)), x)

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Fricas [A]
time = 0.37, size = 51, normalized size = 1.82 \begin {gather*} -{\left (x^{2} + 8 \, x + 16\right )} e^{10} \log \left (\frac {5 \, e^{x} \log \left (x\right )}{x}\right )^{2} + 2 \, {\left (x^{2} + 4 \, x\right )} e^{5} \log \left (\frac {5 \, e^{x} \log \left (x\right )}{x}\right ) - x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2-8*x)*log(x)*log(5*exp(x)*log(x)/x)+(-2*x^3-14*x^2-16*x+32)*log(x)-2*x^2-16*x-32)*exp(log(l
og(5*exp(x)*log(x)/x))+5)^2+((4*x^2+8*x)*log(x)*log(5*exp(x)*log(x)/x)+(2*x^3+6*x^2-8*x)*log(x)+2*x^2+8*x)*exp
(log(log(5*exp(x)*log(x)/x))+5)-2*x^2*log(x)*log(5*exp(x)*log(x)/x))/x/log(x)/log(5*exp(x)*log(x)/x),x, algori
thm="fricas")

[Out]

-(x^2 + 8*x + 16)*e^10*log(5*e^x*log(x)/x)^2 + 2*(x^2 + 4*x)*e^5*log(5*e^x*log(x)/x) - x^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (22) = 44\).
time = 0.37, size = 63, normalized size = 2.25 \begin {gather*} - x^{2} + \left (2 x^{2} e^{5} + 8 x e^{5}\right ) \log {\left (\frac {5 e^{x} \log {\left (x \right )}}{x} \right )} + \left (- x^{2} e^{10} - 8 x e^{10} - 16 e^{10}\right ) \log {\left (\frac {5 e^{x} \log {\left (x \right )}}{x} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**2-8*x)*ln(x)*ln(5*exp(x)*ln(x)/x)+(-2*x**3-14*x**2-16*x+32)*ln(x)-2*x**2-16*x-32)*exp(ln(ln
(5*exp(x)*ln(x)/x))+5)**2+((4*x**2+8*x)*ln(x)*ln(5*exp(x)*ln(x)/x)+(2*x**3+6*x**2-8*x)*ln(x)+2*x**2+8*x)*exp(l
n(ln(5*exp(x)*ln(x)/x))+5)-2*x**2*ln(x)*ln(5*exp(x)*ln(x)/x))/x/ln(x)/ln(5*exp(x)*ln(x)/x),x)

[Out]

-x**2 + (2*x**2*exp(5) + 8*x*exp(5))*log(5*exp(x)*log(x)/x) + (-x**2*exp(10) - 8*x*exp(10) - 16*exp(10))*log(5
*exp(x)*log(x)/x)**2

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (28) = 56\).
time = 0.52, size = 338, normalized size = 12.07 \begin {gather*} -x^{4} e^{10} - 2 \, x^{3} e^{10} \log \left (5\right ) - x^{2} e^{10} \log \left (5\right )^{2} + 2 \, x^{3} e^{10} \log \left (x\right ) + 2 \, x^{2} e^{10} \log \left (5\right ) \log \left (x\right ) - x^{2} e^{10} \log \left (x\right )^{2} - 2 \, x^{3} e^{10} \log \left (\log \left (x\right )\right ) - 2 \, x^{2} e^{10} \log \left (5\right ) \log \left (\log \left (x\right )\right ) + 2 \, x^{2} e^{10} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - x^{2} e^{10} \log \left (\log \left (x\right )\right )^{2} - 8 \, x^{3} e^{10} + 2 \, x^{3} e^{5} - 16 \, x^{2} e^{10} \log \left (5\right ) + 2 \, x^{2} e^{5} \log \left (5\right ) - 8 \, x e^{10} \log \left (5\right )^{2} + 16 \, x^{2} e^{10} \log \left (x\right ) - 2 \, x^{2} e^{5} \log \left (x\right ) + 16 \, x e^{10} \log \left (5\right ) \log \left (x\right ) - 8 \, x e^{10} \log \left (x\right )^{2} - 16 \, x^{2} e^{10} \log \left (\log \left (x\right )\right ) + 2 \, x^{2} e^{5} \log \left (\log \left (x\right )\right ) - 16 \, x e^{10} \log \left (5\right ) \log \left (\log \left (x\right )\right ) + 16 \, x e^{10} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 8 \, x e^{10} \log \left (\log \left (x\right )\right )^{2} - 16 \, x^{2} e^{10} + 8 \, x^{2} e^{5} - 32 \, x e^{10} \log \left (5\right ) + 8 \, x e^{5} \log \left (5\right ) + 32 \, x e^{10} \log \left (x\right ) - 8 \, x e^{5} \log \left (x\right ) + 32 \, e^{10} \log \left (5\right ) \log \left (x\right ) - 16 \, e^{10} \log \left (x\right )^{2} - 32 \, x e^{10} \log \left (\log \left (x\right )\right ) + 8 \, x e^{5} \log \left (\log \left (x\right )\right ) - 32 \, e^{10} \log \left (5\right ) \log \left (\log \left (x\right )\right ) + 32 \, e^{10} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 16 \, e^{10} \log \left (\log \left (x\right )\right )^{2} - x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2-8*x)*log(x)*log(5*exp(x)*log(x)/x)+(-2*x^3-14*x^2-16*x+32)*log(x)-2*x^2-16*x-32)*exp(log(l
og(5*exp(x)*log(x)/x))+5)^2+((4*x^2+8*x)*log(x)*log(5*exp(x)*log(x)/x)+(2*x^3+6*x^2-8*x)*log(x)+2*x^2+8*x)*exp
(log(log(5*exp(x)*log(x)/x))+5)-2*x^2*log(x)*log(5*exp(x)*log(x)/x))/x/log(x)/log(5*exp(x)*log(x)/x),x, algori
thm="giac")

[Out]

-x^4*e^10 - 2*x^3*e^10*log(5) - x^2*e^10*log(5)^2 + 2*x^3*e^10*log(x) + 2*x^2*e^10*log(5)*log(x) - x^2*e^10*lo
g(x)^2 - 2*x^3*e^10*log(log(x)) - 2*x^2*e^10*log(5)*log(log(x)) + 2*x^2*e^10*log(x)*log(log(x)) - x^2*e^10*log
(log(x))^2 - 8*x^3*e^10 + 2*x^3*e^5 - 16*x^2*e^10*log(5) + 2*x^2*e^5*log(5) - 8*x*e^10*log(5)^2 + 16*x^2*e^10*
log(x) - 2*x^2*e^5*log(x) + 16*x*e^10*log(5)*log(x) - 8*x*e^10*log(x)^2 - 16*x^2*e^10*log(log(x)) + 2*x^2*e^5*
log(log(x)) - 16*x*e^10*log(5)*log(log(x)) + 16*x*e^10*log(x)*log(log(x)) - 8*x*e^10*log(log(x))^2 - 16*x^2*e^
10 + 8*x^2*e^5 - 32*x*e^10*log(5) + 8*x*e^5*log(5) + 32*x*e^10*log(x) - 8*x*e^5*log(x) + 32*e^10*log(5)*log(x)
- 16*e^10*log(x)^2 - 32*x*e^10*log(log(x)) + 8*x*e^5*log(log(x)) - 32*e^10*log(5)*log(log(x)) + 32*e^10*log(x
)*log(log(x)) - 16*e^10*log(log(x))^2 - x^2

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Mupad [B]
time = 1.59, size = 36, normalized size = 1.29 \begin {gather*} -{\left (4\,\ln \left (\frac {5\,{\mathrm {e}}^x\,\ln \left (x\right )}{x}\right )\,{\mathrm {e}}^5-x+x\,\ln \left (\frac {5\,{\mathrm {e}}^x\,\ln \left (x\right )}{x}\right )\,{\mathrm {e}}^5\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*log(log((5*exp(x)*log(x))/x)) + 10)*(16*x + 2*x^2 + log(x)*(16*x + 14*x^2 + 2*x^3 - 32) + log((5*e
xp(x)*log(x))/x)*log(x)*(8*x + 2*x^2) + 32) - exp(log(log((5*exp(x)*log(x))/x)) + 5)*(8*x + 2*x^2 + log(x)*(6*
x^2 - 8*x + 2*x^3) + log((5*exp(x)*log(x))/x)*log(x)*(8*x + 4*x^2)) + 2*x^2*log((5*exp(x)*log(x))/x)*log(x))/(
x*log((5*exp(x)*log(x))/x)*log(x)),x)

[Out]

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

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