3.41.88 \(\int \frac {(-2 e^x x+2 x^2) \log ^2(x)+((e^x (-10-2 x)+10 x+2 x^2) \log ^2(x)+(10 x+12 x^2+2 x^3+e^{2 x} (10 x+2 x^2)+e^x (-10 x-12 x^2-2 x^3)) \log ^3(x)) \log (5+x)+(2 x \log (x)+((10-8 x-2 x^2+e^x (10+2 x)) \log (x)+(10 x+2 x^2+e^x (-10 x-2 x^2)) \log ^2(x)) \log (5+x)) \log (x \log (5+x))+(-10-2 x) \log (5+x) \log ^2(x \log (5+x))}{(5 x+x^2) \log ^3(x) \log (5+x)} \, dx\)

Optimal. Leaf size=27 \[ 2+2 x+\left (e^x-x-\frac {\log (x \log (5+x))}{\log (x)}\right )^2 \]

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Rubi [F]  time = 22.37, 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 {\left (-2 e^x x+2 x^2\right ) \log ^2(x)+\left (\left (e^x (-10-2 x)+10 x+2 x^2\right ) \log ^2(x)+\left (10 x+12 x^2+2 x^3+e^{2 x} \left (10 x+2 x^2\right )+e^x \left (-10 x-12 x^2-2 x^3\right )\right ) \log ^3(x)\right ) \log (5+x)+\left (2 x \log (x)+\left (\left (10-8 x-2 x^2+e^x (10+2 x)\right ) \log (x)+\left (10 x+2 x^2+e^x \left (-10 x-2 x^2\right )\right ) \log ^2(x)\right ) \log (5+x)\right ) \log (x \log (5+x))+(-10-2 x) \log (5+x) \log ^2(x \log (5+x))}{\left (5 x+x^2\right ) \log ^3(x) \log (5+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-2*E^x*x + 2*x^2)*Log[x]^2 + ((E^x*(-10 - 2*x) + 10*x + 2*x^2)*Log[x]^2 + (10*x + 12*x^2 + 2*x^3 + E^(2*
x)*(10*x + 2*x^2) + E^x*(-10*x - 12*x^2 - 2*x^3))*Log[x]^3)*Log[5 + x] + (2*x*Log[x] + ((10 - 8*x - 2*x^2 + E^
x*(10 + 2*x))*Log[x] + (10*x + 2*x^2 + E^x*(-10*x - 2*x^2))*Log[x]^2)*Log[5 + x])*Log[x*Log[5 + x]] + (-10 - 2
*x)*Log[5 + x]*Log[x*Log[5 + x]]^2)/((5*x + x^2)*Log[x]^3*Log[5 + x]),x]

[Out]

E^(2*x) + 4*x - 2*E^x*x + x^2 + 2*LogIntegral[x] - 2*Log[x]*LogIntegral[x] + 2*Log[x*Log[5 + x]]*LogIntegral[x
] - 2*Defer[Int][E^x/(x*Log[x]), x] + 2*Defer[Int][1/(Log[x]*Log[5 + x]), x] - 10*Defer[Int][1/((5 + x)*Log[x]
*Log[5 + x]), x] - 2*Defer[Int][E^x/((5 + x)*Log[x]*Log[5 + x]), x] - 2*Defer[Int][Log[x*Log[5 + x]]/Log[x]^2,
x] + 2*Defer[Int][Log[x*Log[5 + x]]/(x*Log[x]^2), x] + 2*Defer[Int][(E^x*Log[x*Log[5 + x]])/(x*Log[x]^2), x]
- 2*Defer[Int][(E^x*Log[x*Log[5 + x]])/Log[x], x] + 2*Defer[Int][Log[x*Log[5 + x]]/((5 + x)*Log[x]^2*Log[5 + x
]), x] - 2*Defer[Int][Log[x*Log[5 + x]]^2/(x*Log[x]^3), x] - 2*Defer[Int][LogIntegral[x]/((5 + x)*Log[5 + x]),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-2 e^x x+2 x^2\right ) \log ^2(x)+\left (\left (e^x (-10-2 x)+10 x+2 x^2\right ) \log ^2(x)+\left (10 x+12 x^2+2 x^3+e^{2 x} \left (10 x+2 x^2\right )+e^x \left (-10 x-12 x^2-2 x^3\right )\right ) \log ^3(x)\right ) \log (5+x)+\left (2 x \log (x)+\left (\left (10-8 x-2 x^2+e^x (10+2 x)\right ) \log (x)+\left (10 x+2 x^2+e^x \left (-10 x-2 x^2\right )\right ) \log ^2(x)\right ) \log (5+x)\right ) \log (x \log (5+x))+(-10-2 x) \log (5+x) \log ^2(x \log (5+x))}{x (5+x) \log ^3(x) \log (5+x)} \, dx\\ &=\int \frac {-2 x (5+x) \left (-1-e^{2 x}-x+e^x (1+x)\right ) \log ^3(x) \log (5+x)-2 \log (x) \left (-x-\left (1+e^x-x\right ) (5+x) \log (5+x)\right ) \log (x \log (5+x))-2 (5+x) \log (5+x) \log ^2(x \log (5+x))+2 \log ^2(x) \left (x \left (-e^x+x\right )-(5+x) \log (5+x) \left (e^x-x+\left (-1+e^x\right ) x \log (x \log (5+x))\right )\right )}{x (5+x) \log ^3(x) \log (5+x)} \, dx\\ &=\int \left (2 e^{2 x}-\frac {2 e^x \left (x \log (x)+5 \log (x) \log (5+x)+x \log (x) \log (5+x)+5 x \log ^2(x) \log (5+x)+6 x^2 \log ^2(x) \log (5+x)+x^3 \log ^2(x) \log (5+x)-5 \log (5+x) \log (x \log (5+x))-x \log (5+x) \log (x \log (5+x))+5 x \log (x) \log (5+x) \log (x \log (5+x))+x^2 \log (x) \log (5+x) \log (x \log (5+x))\right )}{x (5+x) \log ^2(x) \log (5+x)}+\frac {2 \left (x^2 \log ^2(x)+5 x \log ^2(x) \log (5+x)+x^2 \log ^2(x) \log (5+x)+5 x \log ^3(x) \log (5+x)+6 x^2 \log ^3(x) \log (5+x)+x^3 \log ^3(x) \log (5+x)+x \log (x) \log (x \log (5+x))+5 \log (x) \log (5+x) \log (x \log (5+x))-4 x \log (x) \log (5+x) \log (x \log (5+x))-x^2 \log (x) \log (5+x) \log (x \log (5+x))+5 x \log ^2(x) \log (5+x) \log (x \log (5+x))+x^2 \log ^2(x) \log (5+x) \log (x \log (5+x))-5 \log (5+x) \log ^2(x \log (5+x))-x \log (5+x) \log ^2(x \log (5+x))\right )}{x (5+x) \log ^3(x) \log (5+x)}\right ) \, dx\\ &=2 \int e^{2 x} \, dx-2 \int \frac {e^x \left (x \log (x)+5 \log (x) \log (5+x)+x \log (x) \log (5+x)+5 x \log ^2(x) \log (5+x)+6 x^2 \log ^2(x) \log (5+x)+x^3 \log ^2(x) \log (5+x)-5 \log (5+x) \log (x \log (5+x))-x \log (5+x) \log (x \log (5+x))+5 x \log (x) \log (5+x) \log (x \log (5+x))+x^2 \log (x) \log (5+x) \log (x \log (5+x))\right )}{x (5+x) \log ^2(x) \log (5+x)} \, dx+2 \int \frac {x^2 \log ^2(x)+5 x \log ^2(x) \log (5+x)+x^2 \log ^2(x) \log (5+x)+5 x \log ^3(x) \log (5+x)+6 x^2 \log ^3(x) \log (5+x)+x^3 \log ^3(x) \log (5+x)+x \log (x) \log (x \log (5+x))+5 \log (x) \log (5+x) \log (x \log (5+x))-4 x \log (x) \log (5+x) \log (x \log (5+x))-x^2 \log (x) \log (5+x) \log (x \log (5+x))+5 x \log ^2(x) \log (5+x) \log (x \log (5+x))+x^2 \log ^2(x) \log (5+x) \log (x \log (5+x))-5 \log (5+x) \log ^2(x \log (5+x))-x \log (5+x) \log ^2(x \log (5+x))}{x (5+x) \log ^3(x) \log (5+x)} \, dx\\ &=e^{2 x}+2 \int \frac {x \left (5+6 x+x^2\right ) \log ^3(x) \log (5+x)+\log (x) \left (x-\left (-5+4 x+x^2\right ) \log (5+x)\right ) \log (x \log (5+x))-(5+x) \log (5+x) \log ^2(x \log (5+x))+x \log ^2(x) (x+(5+x) \log (5+x) (1+\log (x \log (5+x))))}{x (5+x) \log ^3(x) \log (5+x)} \, dx-2 \int \frac {e^x \left (x \left (5+6 x+x^2\right ) \log ^2(x) \log (5+x)-(5+x) \log (5+x) \log (x \log (5+x))+\log (x) (x+(5+x) \log (5+x) (1+x \log (x \log (5+x))))\right )}{x (5+x) \log ^2(x) \log (5+x)} \, dx\\ &=e^{2 x}-2 \int \left (\frac {e^x \left (x+5 \log (5+x)+x \log (5+x)+5 x \log (x) \log (5+x)+6 x^2 \log (x) \log (5+x)+x^3 \log (x) \log (5+x)\right )}{x (5+x) \log (x) \log (5+x)}+\frac {e^x (-1+x \log (x)) \log (x \log (5+x))}{x \log ^2(x)}\right ) \, dx+2 \int \left (\frac {x+5 \log (5+x)+x \log (5+x)+5 \log (x) \log (5+x)+6 x \log (x) \log (5+x)+x^2 \log (x) \log (5+x)}{(5+x) \log (x) \log (5+x)}+\frac {\left (x+5 \log (5+x)-4 x \log (5+x)-x^2 \log (5+x)+5 x \log (x) \log (5+x)+x^2 \log (x) \log (5+x)\right ) \log (x \log (5+x))}{x (5+x) \log ^2(x) \log (5+x)}-\frac {\log ^2(x \log (5+x))}{x \log ^3(x)}\right ) \, dx\\ &=e^{2 x}+2 \int \frac {x+5 \log (5+x)+x \log (5+x)+5 \log (x) \log (5+x)+6 x \log (x) \log (5+x)+x^2 \log (x) \log (5+x)}{(5+x) \log (x) \log (5+x)} \, dx-2 \int \frac {e^x \left (x+5 \log (5+x)+x \log (5+x)+5 x \log (x) \log (5+x)+6 x^2 \log (x) \log (5+x)+x^3 \log (x) \log (5+x)\right )}{x (5+x) \log (x) \log (5+x)} \, dx-2 \int \frac {e^x (-1+x \log (x)) \log (x \log (5+x))}{x \log ^2(x)} \, dx+2 \int \frac {\left (x+5 \log (5+x)-4 x \log (5+x)-x^2 \log (5+x)+5 x \log (x) \log (5+x)+x^2 \log (x) \log (5+x)\right ) \log (x \log (5+x))}{x (5+x) \log ^2(x) \log (5+x)} \, dx-2 \int \frac {\log ^2(x \log (5+x))}{x \log ^3(x)} \, dx\\ &=e^{2 x}+2 \int \left (1+x+\frac {1+\frac {x}{(5+x) \log (5+x)}}{\log (x)}\right ) \, dx-2 \int \frac {e^x (x+(5+x) (1+x (1+x) \log (x)) \log (5+x))}{x (5+x) \log (x) \log (5+x)} \, dx+2 \int \frac {(x+(5+x) (1-x+x \log (x)) \log (5+x)) \log (x \log (5+x))}{x (5+x) \log ^2(x) \log (5+x)} \, dx-2 \int \frac {\log ^2(x \log (5+x))}{x \log ^3(x)} \, dx-2 \int \left (-\frac {e^x \log (x \log (5+x))}{x \log ^2(x)}+\frac {e^x \log (x \log (5+x))}{\log (x)}\right ) \, dx\\ &=e^{2 x}+2 x+x^2+2 \int \frac {1+\frac {x}{(5+x) \log (5+x)}}{\log (x)} \, dx-2 \int \left (\frac {e^x \left (1+x \log (x)+x^2 \log (x)\right )}{x \log (x)}+\frac {e^x}{(5+x) \log (x) \log (5+x)}\right ) \, dx+2 \int \frac {e^x \log (x \log (5+x))}{x \log ^2(x)} \, dx-2 \int \frac {e^x \log (x \log (5+x))}{\log (x)} \, dx-2 \int \frac {\log ^2(x \log (5+x))}{x \log ^3(x)} \, dx+2 \int \left (\frac {\left (x+5 \log (5+x)-4 x \log (5+x)-x^2 \log (5+x)+5 x \log (x) \log (5+x)+x^2 \log (x) \log (5+x)\right ) \log (x \log (5+x))}{5 x \log ^2(x) \log (5+x)}-\frac {\left (x+5 \log (5+x)-4 x \log (5+x)-x^2 \log (5+x)+5 x \log (x) \log (5+x)+x^2 \log (x) \log (5+x)\right ) \log (x \log (5+x))}{5 (5+x) \log ^2(x) \log (5+x)}\right ) \, dx\\ &=e^{2 x}+2 x+x^2+\frac {2}{5} \int \frac {\left (x+5 \log (5+x)-4 x \log (5+x)-x^2 \log (5+x)+5 x \log (x) \log (5+x)+x^2 \log (x) \log (5+x)\right ) \log (x \log (5+x))}{x \log ^2(x) \log (5+x)} \, dx-\frac {2}{5} \int \frac {\left (x+5 \log (5+x)-4 x \log (5+x)-x^2 \log (5+x)+5 x \log (x) \log (5+x)+x^2 \log (x) \log (5+x)\right ) \log (x \log (5+x))}{(5+x) \log ^2(x) \log (5+x)} \, dx-2 \int \frac {e^x \left (1+x \log (x)+x^2 \log (x)\right )}{x \log (x)} \, dx+2 \int \left (\frac {1}{\log (x)}+\frac {x}{(5+x) \log (x) \log (5+x)}\right ) \, dx-2 \int \frac {e^x}{(5+x) \log (x) \log (5+x)} \, dx+2 \int \frac {e^x \log (x \log (5+x))}{x \log ^2(x)} \, dx-2 \int \frac {e^x \log (x \log (5+x))}{\log (x)} \, dx-2 \int \frac {\log ^2(x \log (5+x))}{x \log ^3(x)} \, dx\\ &=e^{2 x}+2 x+x^2+\frac {2}{5} \int \frac {(x+(5+x) (1-x+x \log (x)) \log (5+x)) \log (x \log (5+x))}{x \log ^2(x) \log (5+x)} \, dx-\frac {2}{5} \int \frac {(x+(5+x) (1-x+x \log (x)) \log (5+x)) \log (x \log (5+x))}{(5+x) \log ^2(x) \log (5+x)} \, dx-2 \int e^x \left (1+x+\frac {1}{x \log (x)}\right ) \, dx+2 \int \frac {1}{\log (x)} \, dx-2 \int \frac {e^x}{(5+x) \log (x) \log (5+x)} \, dx+2 \int \frac {x}{(5+x) \log (x) \log (5+x)} \, dx+2 \int \frac {e^x \log (x \log (5+x))}{x \log ^2(x)} \, dx-2 \int \frac {e^x \log (x \log (5+x))}{\log (x)} \, dx-2 \int \frac {\log ^2(x \log (5+x))}{x \log ^3(x)} \, dx\\ &=e^{2 x}+2 x+x^2+2 \text {li}(x)+\frac {2}{5} \int \left (-\frac {4 \log (x \log (5+x))}{\log ^2(x)}+\frac {5 \log (x \log (5+x))}{x \log ^2(x)}-\frac {x \log (x \log (5+x))}{\log ^2(x)}+\frac {5 \log (x \log (5+x))}{\log (x)}+\frac {x \log (x \log (5+x))}{\log (x)}+\frac {\log (x \log (5+x))}{\log ^2(x) \log (5+x)}\right ) \, dx-\frac {2}{5} \int \left (\frac {5 \log (x \log (5+x))}{(5+x) \log ^2(x)}-\frac {4 x \log (x \log (5+x))}{(5+x) \log ^2(x)}-\frac {x^2 \log (x \log (5+x))}{(5+x) \log ^2(x)}+\frac {5 x \log (x \log (5+x))}{(5+x) \log (x)}+\frac {x^2 \log (x \log (5+x))}{(5+x) \log (x)}+\frac {x \log (x \log (5+x))}{(5+x) \log ^2(x) \log (5+x)}\right ) \, dx-2 \int \left (e^x+e^x x+\frac {e^x}{x \log (x)}\right ) \, dx+2 \int \left (\frac {1}{\log (x) \log (5+x)}-\frac {5}{(5+x) \log (x) \log (5+x)}\right ) \, dx-2 \int \frac {e^x}{(5+x) \log (x) \log (5+x)} \, dx+2 \int \frac {e^x \log (x \log (5+x))}{x \log ^2(x)} \, dx-2 \int \frac {e^x \log (x \log (5+x))}{\log (x)} \, dx-2 \int \frac {\log ^2(x \log (5+x))}{x \log ^3(x)} \, dx\\ &=e^{2 x}+2 x+x^2+2 \text {li}(x)-\frac {2}{5} \int \frac {x \log (x \log (5+x))}{\log ^2(x)} \, dx+\frac {2}{5} \int \frac {x^2 \log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx+\frac {2}{5} \int \frac {x \log (x \log (5+x))}{\log (x)} \, dx-\frac {2}{5} \int \frac {x^2 \log (x \log (5+x))}{(5+x) \log (x)} \, dx+\frac {2}{5} \int \frac {\log (x \log (5+x))}{\log ^2(x) \log (5+x)} \, dx-\frac {2}{5} \int \frac {x \log (x \log (5+x))}{(5+x) \log ^2(x) \log (5+x)} \, dx-\frac {8}{5} \int \frac {\log (x \log (5+x))}{\log ^2(x)} \, dx+\frac {8}{5} \int \frac {x \log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx-2 \int e^x \, dx-2 \int e^x x \, dx-2 \int \frac {e^x}{x \log (x)} \, dx+2 \int \frac {1}{\log (x) \log (5+x)} \, dx-2 \int \frac {e^x}{(5+x) \log (x) \log (5+x)} \, dx+2 \int \frac {\log (x \log (5+x))}{x \log ^2(x)} \, dx+2 \int \frac {e^x \log (x \log (5+x))}{x \log ^2(x)} \, dx-2 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx+2 \int \frac {\log (x \log (5+x))}{\log (x)} \, dx-2 \int \frac {e^x \log (x \log (5+x))}{\log (x)} \, dx-2 \int \frac {x \log (x \log (5+x))}{(5+x) \log (x)} \, dx-2 \int \frac {\log ^2(x \log (5+x))}{x \log ^3(x)} \, dx-10 \int \frac {1}{(5+x) \log (x) \log (5+x)} \, dx\\ &=-2 e^x+e^{2 x}+2 x-2 e^x x+x^2+2 \text {li}(x)+2 \log (x \log (5+x)) \text {li}(x)-\frac {2}{5} \int \frac {x \log (x \log (5+x))}{\log ^2(x)} \, dx+\frac {2}{5} \int \frac {x \log (x \log (5+x))}{\log (x)} \, dx+\frac {2}{5} \int \frac {\log (x \log (5+x))}{\log ^2(x) \log (5+x)} \, dx+\frac {2}{5} \int \left (-\frac {5 \log (x \log (5+x))}{\log ^2(x)}+\frac {x \log (x \log (5+x))}{\log ^2(x)}+\frac {25 \log (x \log (5+x))}{(5+x) \log ^2(x)}\right ) \, dx-\frac {2}{5} \int \left (-\frac {5 \log (x \log (5+x))}{\log (x)}+\frac {x \log (x \log (5+x))}{\log (x)}+\frac {25 \log (x \log (5+x))}{(5+x) \log (x)}\right ) \, dx-\frac {2}{5} \int \left (\frac {\log (x \log (5+x))}{\log ^2(x) \log (5+x)}-\frac {5 \log (x \log (5+x))}{(5+x) \log ^2(x) \log (5+x)}\right ) \, dx-\frac {8}{5} \int \frac {\log (x \log (5+x))}{\log ^2(x)} \, dx+\frac {8}{5} \int \left (\frac {\log (x \log (5+x))}{\log ^2(x)}-\frac {5 \log (x \log (5+x))}{(5+x) \log ^2(x)}\right ) \, dx+2 \int e^x \, dx-2 \int \frac {e^x}{x \log (x)} \, dx+2 \int \frac {1}{\log (x) \log (5+x)} \, dx-2 \int \frac {e^x}{(5+x) \log (x) \log (5+x)} \, dx+2 \int \frac {\log (x \log (5+x))}{x \log ^2(x)} \, dx+2 \int \frac {e^x \log (x \log (5+x))}{x \log ^2(x)} \, dx-2 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx-2 \int \frac {e^x \log (x \log (5+x))}{\log (x)} \, dx-2 \int \frac {\log ^2(x \log (5+x))}{x \log ^3(x)} \, dx-2 \int \left (\frac {\log (x \log (5+x))}{\log (x)}-\frac {5 \log (x \log (5+x))}{(5+x) \log (x)}\right ) \, dx-2 \int \left (\frac {1}{x}+\frac {1}{(5+x) \log (5+x)}\right ) \text {li}(x) \, dx-10 \int \frac {1}{(5+x) \log (x) \log (5+x)} \, dx\\ &=e^{2 x}+2 x-2 e^x x+x^2+2 \text {li}(x)+2 \log (x \log (5+x)) \text {li}(x)-2 \int \frac {e^x}{x \log (x)} \, dx+2 \int \frac {1}{\log (x) \log (5+x)} \, dx-2 \int \frac {e^x}{(5+x) \log (x) \log (5+x)} \, dx-2 \int \frac {\log (x \log (5+x))}{\log ^2(x)} \, dx+2 \int \frac {\log (x \log (5+x))}{x \log ^2(x)} \, dx+2 \int \frac {e^x \log (x \log (5+x))}{x \log ^2(x)} \, dx-2 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx-2 \int \frac {e^x \log (x \log (5+x))}{\log (x)} \, dx+2 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x) \log (5+x)} \, dx-2 \int \frac {\log ^2(x \log (5+x))}{x \log ^3(x)} \, dx-2 \int \left (\frac {\text {li}(x)}{x}+\frac {\text {li}(x)}{(5+x) \log (5+x)}\right ) \, dx-8 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx-10 \int \frac {1}{(5+x) \log (x) \log (5+x)} \, dx+10 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx\\ &=e^{2 x}+2 x-2 e^x x+x^2+2 \text {li}(x)+2 \log (x \log (5+x)) \text {li}(x)-2 \int \frac {e^x}{x \log (x)} \, dx+2 \int \frac {1}{\log (x) \log (5+x)} \, dx-2 \int \frac {e^x}{(5+x) \log (x) \log (5+x)} \, dx-2 \int \frac {\log (x \log (5+x))}{\log ^2(x)} \, dx+2 \int \frac {\log (x \log (5+x))}{x \log ^2(x)} \, dx+2 \int \frac {e^x \log (x \log (5+x))}{x \log ^2(x)} \, dx-2 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx-2 \int \frac {e^x \log (x \log (5+x))}{\log (x)} \, dx+2 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x) \log (5+x)} \, dx-2 \int \frac {\log ^2(x \log (5+x))}{x \log ^3(x)} \, dx-2 \int \frac {\text {li}(x)}{x} \, dx-2 \int \frac {\text {li}(x)}{(5+x) \log (5+x)} \, dx-8 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx-10 \int \frac {1}{(5+x) \log (x) \log (5+x)} \, dx+10 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx\\ &=e^{2 x}+4 x-2 e^x x+x^2+2 \text {li}(x)-2 \log (x) \text {li}(x)+2 \log (x \log (5+x)) \text {li}(x)-2 \int \frac {e^x}{x \log (x)} \, dx+2 \int \frac {1}{\log (x) \log (5+x)} \, dx-2 \int \frac {e^x}{(5+x) \log (x) \log (5+x)} \, dx-2 \int \frac {\log (x \log (5+x))}{\log ^2(x)} \, dx+2 \int \frac {\log (x \log (5+x))}{x \log ^2(x)} \, dx+2 \int \frac {e^x \log (x \log (5+x))}{x \log ^2(x)} \, dx-2 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx-2 \int \frac {e^x \log (x \log (5+x))}{\log (x)} \, dx+2 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x) \log (5+x)} \, dx-2 \int \frac {\log ^2(x \log (5+x))}{x \log ^3(x)} \, dx-2 \int \frac {\text {li}(x)}{(5+x) \log (5+x)} \, dx-8 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx-10 \int \frac {1}{(5+x) \log (x) \log (5+x)} \, dx+10 \int \frac {\log (x \log (5+x))}{(5+x) \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 52, normalized size = 1.93 \begin {gather*} e^{2 x}+2 x-2 e^x x+x^2-\frac {2 \left (e^x-x\right ) \log (x \log (5+x))}{\log (x)}+\frac {\log ^2(x \log (5+x))}{\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-2*E^x*x + 2*x^2)*Log[x]^2 + ((E^x*(-10 - 2*x) + 10*x + 2*x^2)*Log[x]^2 + (10*x + 12*x^2 + 2*x^3 +
E^(2*x)*(10*x + 2*x^2) + E^x*(-10*x - 12*x^2 - 2*x^3))*Log[x]^3)*Log[5 + x] + (2*x*Log[x] + ((10 - 8*x - 2*x^
2 + E^x*(10 + 2*x))*Log[x] + (10*x + 2*x^2 + E^x*(-10*x - 2*x^2))*Log[x]^2)*Log[5 + x])*Log[x*Log[5 + x]] + (-
10 - 2*x)*Log[5 + x]*Log[x*Log[5 + x]]^2)/((5*x + x^2)*Log[x]^3*Log[5 + x]),x]

[Out]

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

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fricas [B]  time = 0.56, size = 53, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (x - e^{x}\right )} \log \left (x \log \left (x + 5\right )\right ) \log \relax (x) + {\left (x^{2} - 2 \, x e^{x} + 2 \, x + e^{\left (2 \, x\right )}\right )} \log \relax (x)^{2} + \log \left (x \log \left (x + 5\right )\right )^{2}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-10)*log(5+x)*log(x*log(5+x))^2+((((-2*x^2-10*x)*exp(x)+2*x^2+10*x)*log(x)^2+((2*x+10)*exp(x)-
2*x^2-8*x+10)*log(x))*log(5+x)+2*x*log(x))*log(x*log(5+x))+(((2*x^2+10*x)*exp(x)^2+(-2*x^3-12*x^2-10*x)*exp(x)
+2*x^3+12*x^2+10*x)*log(x)^3+((-2*x-10)*exp(x)+2*x^2+10*x)*log(x)^2)*log(5+x)+(-2*exp(x)*x+2*x^2)*log(x)^2)/(x
^2+5*x)/log(x)^3/log(5+x),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.25, size = 84, normalized size = 3.11 \begin {gather*} \frac {x^{2} \log \relax (x)^{2} - 2 \, x e^{x} \log \relax (x)^{2} + 4 \, x \log \relax (x)^{2} + e^{\left (2 \, x\right )} \log \relax (x)^{2} - 2 \, e^{x} \log \relax (x)^{2} + 2 \, x \log \relax (x) \log \left (\log \left (x + 5\right )\right ) - 2 \, e^{x} \log \relax (x) \log \left (\log \left (x + 5\right )\right ) + 2 \, \log \relax (x) \log \left (\log \left (x + 5\right )\right ) + \log \left (\log \left (x + 5\right )\right )^{2}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-10)*log(5+x)*log(x*log(5+x))^2+((((-2*x^2-10*x)*exp(x)+2*x^2+10*x)*log(x)^2+((2*x+10)*exp(x)-
2*x^2-8*x+10)*log(x))*log(5+x)+2*x*log(x))*log(x*log(5+x))+(((2*x^2+10*x)*exp(x)^2+(-2*x^3-12*x^2-10*x)*exp(x)
+2*x^3+12*x^2+10*x)*log(x)^3+((-2*x-10)*exp(x)+2*x^2+10*x)*log(x)^2)*log(5+x)+(-2*exp(x)*x+2*x^2)*log(x)^2)/(x
^2+5*x)/log(x)^3/log(5+x),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.21, size = 696, normalized size = 25.78

method	result	size
risch	\(\frac {\ln \left (\ln \left (5+x \right )\right )^{2}}{\ln \relax (x )^{2}}+\frac {\left (-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (5+x \right )\right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i \ln \left (5+x \right )\right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{2}-i \pi \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{3}+2 x \ln \relax (x )-2 \,{\mathrm e}^{x} \ln \relax (x )+2 \ln \relax (x )\right ) \ln \left (\ln \left (5+x \right )\right )}{\ln \relax (x )^{2}}+\frac {-8 x \,{\mathrm e}^{x} \ln \relax (x )^{2}+4 x^{2} \ln \relax (x )^{2}-8 \,{\mathrm e}^{x} \ln \relax (x )^{2}+16 x \ln \relax (x )^{2}-\pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \left (5+x \right )\right )^{2} \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{2}+2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \left (5+x \right )\right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{3}+2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (5+x \right )\right )^{2} \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{3}-4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (5+x \right )\right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{4}-4 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{3}-\pi ^{2} \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{6}+4 \,{\mathrm e}^{2 x} \ln \relax (x )^{2}+4 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{2} \ln \relax (x )+4 i \pi x \,\mathrm {csgn}\left (i \ln \left (5+x \right )\right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{2} \ln \relax (x )-4 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (5+x \right )\right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )-4 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{2} {\mathrm e}^{x} \ln \relax (x )-4 i \pi \,\mathrm {csgn}\left (i \ln \left (5+x \right )\right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{2} {\mathrm e}^{x} \ln \relax (x )-\pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{4}+2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{5}-\pi ^{2} \mathrm {csgn}\left (i \ln \left (5+x \right )\right )^{2} \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{4}+2 \pi ^{2} \mathrm {csgn}\left (i \ln \left (5+x \right )\right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{5}-4 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (5+x \right )\right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right ) \ln \relax (x )+4 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (5+x \right )\right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right ) {\mathrm e}^{x} \ln \relax (x )+4 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i \ln \left (5+x \right )\right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{2}-4 i \pi x \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{3} \ln \relax (x )+4 i \pi \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{3} {\mathrm e}^{x} \ln \relax (x )+4 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (5+x \right )\right )^{2}}{4 \ln \relax (x )^{2}}\)	\(696\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x-10)*ln(5+x)*ln(x*ln(5+x))^2+((((-2*x^2-10*x)*exp(x)+2*x^2+10*x)*ln(x)^2+((2*x+10)*exp(x)-2*x^2-8*x+
10)*ln(x))*ln(5+x)+2*x*ln(x))*ln(x*ln(5+x))+(((2*x^2+10*x)*exp(x)^2+(-2*x^3-12*x^2-10*x)*exp(x)+2*x^3+12*x^2+1
0*x)*ln(x)^3+((-2*x-10)*exp(x)+2*x^2+10*x)*ln(x)^2)*ln(5+x)+(-2*exp(x)*x+2*x^2)*ln(x)^2)/(x^2+5*x)/ln(x)^3/ln(
5+x),x,method=_RETURNVERBOSE)

[Out]

1/ln(x)^2*ln(ln(5+x))^2+(-I*Pi*csgn(I*x)*csgn(I*ln(5+x))*csgn(I*x*ln(5+x))+I*Pi*csgn(I*x)*csgn(I*x*ln(5+x))^2+
I*Pi*csgn(I*ln(5+x))*csgn(I*x*ln(5+x))^2-I*Pi*csgn(I*x*ln(5+x))^3+2*x*ln(x)-2*exp(x)*ln(x)+2*ln(x))/ln(x)^2*ln
(ln(5+x))+1/4*(-8*x*exp(x)*ln(x)^2+4*x^2*ln(x)^2-8*exp(x)*ln(x)^2+16*x*ln(x)^2-Pi^2*csgn(I*x)^2*csgn(I*ln(5+x)
)^2*csgn(I*x*ln(5+x))^2+2*Pi^2*csgn(I*x)^2*csgn(I*ln(5+x))*csgn(I*x*ln(5+x))^3+2*Pi^2*csgn(I*x)*csgn(I*ln(5+x)
)^2*csgn(I*x*ln(5+x))^3-4*Pi^2*csgn(I*x)*csgn(I*ln(5+x))*csgn(I*x*ln(5+x))^4-4*I*ln(x)*Pi*csgn(I*x*ln(5+x))^3+
4*exp(2*x)*ln(x)^2-Pi^2*csgn(I*x*ln(5+x))^6+4*I*Pi*x*csgn(I*x)*csgn(I*x*ln(5+x))^2*ln(x)+4*I*Pi*x*csgn(I*ln(5+
x))*csgn(I*x*ln(5+x))^2*ln(x)-4*I*ln(x)*Pi*csgn(I*x)*csgn(I*ln(5+x))*csgn(I*x*ln(5+x))-4*I*Pi*csgn(I*x)*csgn(I
*x*ln(5+x))^2*exp(x)*ln(x)-4*I*Pi*csgn(I*ln(5+x))*csgn(I*x*ln(5+x))^2*exp(x)*ln(x)-Pi^2*csgn(I*x)^2*csgn(I*x*l
n(5+x))^4+2*Pi^2*csgn(I*x)*csgn(I*x*ln(5+x))^5-Pi^2*csgn(I*ln(5+x))^2*csgn(I*x*ln(5+x))^4+2*Pi^2*csgn(I*ln(5+x
))*csgn(I*x*ln(5+x))^5-4*I*Pi*x*csgn(I*x)*csgn(I*ln(5+x))*csgn(I*x*ln(5+x))*ln(x)+4*I*Pi*csgn(I*x)*csgn(I*ln(5
+x))*csgn(I*x*ln(5+x))*exp(x)*ln(x)+4*I*ln(x)*Pi*csgn(I*ln(5+x))*csgn(I*x*ln(5+x))^2-4*I*Pi*x*csgn(I*x*ln(5+x)
)^3*ln(x)+4*I*Pi*csgn(I*x*ln(5+x))^3*exp(x)*ln(x)+4*I*ln(x)*Pi*csgn(I*x)*csgn(I*x*ln(5+x))^2)/ln(x)^2

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maxima [B]  time = 0.44, size = 70, normalized size = 2.59 \begin {gather*} -\frac {2 \, {\left (x + 1\right )} e^{x} \log \relax (x)^{2} - {\left (x^{2} + 4 \, x\right )} \log \relax (x)^{2} - e^{\left (2 \, x\right )} \log \relax (x)^{2} - 2 \, {\left ({\left (x + 1\right )} \log \relax (x) - e^{x} \log \relax (x)\right )} \log \left (\log \left (x + 5\right )\right ) - \log \left (\log \left (x + 5\right )\right )^{2}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-10)*log(5+x)*log(x*log(5+x))^2+((((-2*x^2-10*x)*exp(x)+2*x^2+10*x)*log(x)^2+((2*x+10)*exp(x)-
2*x^2-8*x+10)*log(x))*log(5+x)+2*x*log(x))*log(x*log(5+x))+(((2*x^2+10*x)*exp(x)^2+(-2*x^3-12*x^2-10*x)*exp(x)
+2*x^3+12*x^2+10*x)*log(x)^3+((-2*x-10)*exp(x)+2*x^2+10*x)*log(x)^2)*log(5+x)+(-2*exp(x)*x+2*x^2)*log(x)^2)/(x
^2+5*x)/log(x)^3/log(5+x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.67, size = 49, normalized size = 1.81 \begin {gather*} 2\,x+{\mathrm {e}}^{2\,x}+\frac {{\ln \left (x\,\ln \left (x+5\right )\right )}^2}{{\ln \relax (x)}^2}-2\,x\,{\mathrm {e}}^x+x^2+\frac {2\,\ln \left (x\,\ln \left (x+5\right )\right )\,\left (x-{\mathrm {e}}^x\right )}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + 5)*(log(x)^3*(10*x + exp(2*x)*(10*x + 2*x^2) + 12*x^2 + 2*x^3 - exp(x)*(10*x + 12*x^2 + 2*x^3)) +
log(x)^2*(10*x - exp(x)*(2*x + 10) + 2*x^2)) + log(x*log(x + 5))*(2*x*log(x) - log(x + 5)*(log(x)*(8*x - exp(
x)*(2*x + 10) + 2*x^2 - 10) - log(x)^2*(10*x - exp(x)*(10*x + 2*x^2) + 2*x^2))) - log(x)^2*(2*x*exp(x) - 2*x^2
) - log(x + 5)*log(x*log(x + 5))^2*(2*x + 10))/(log(x + 5)*log(x)^3*(5*x + x^2)),x)

[Out]

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

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sympy [B]  time = 1.45, size = 70, normalized size = 2.59 \begin {gather*} x^{2} + 2 x + \frac {2 x \log {\left (x \log {\left (x + 5 \right )} \right )}}{\log {\relax (x )}} + \frac {\left (- 2 x \log {\relax (x )} - 2 \log {\left (x \log {\left (x + 5 \right )} \right )}\right ) e^{x} + e^{2 x} \log {\relax (x )}}{\log {\relax (x )}} + \frac {\log {\left (x \log {\left (x + 5 \right )} \right )}^{2}}{\log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-10)*ln(5+x)*ln(x*ln(5+x))**2+((((-2*x**2-10*x)*exp(x)+2*x**2+10*x)*ln(x)**2+((2*x+10)*exp(x)-
2*x**2-8*x+10)*ln(x))*ln(5+x)+2*x*ln(x))*ln(x*ln(5+x))+(((2*x**2+10*x)*exp(x)**2+(-2*x**3-12*x**2-10*x)*exp(x)
+2*x**3+12*x**2+10*x)*ln(x)**3+((-2*x-10)*exp(x)+2*x**2+10*x)*ln(x)**2)*ln(5+x)+(-2*exp(x)*x+2*x**2)*ln(x)**2)
/(x**2+5*x)/ln(x)**3/ln(5+x),x)

[Out]

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

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