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

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

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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 = {} $$\begin{array}{c}\int \frac{(-2e^{x}x+2x^2){\log }^2(x)+((e^x(-10-2x)+10x+2x^2){\log }^2(x)+(10x+12x^2+2x^3+e^{2x}(10x+2x^2)+e^x(-10x-12x^2-2x^3)){\log }^3(x))\log (5+x)+(2x\log (x)+((10-8x-2x^2+e^x(10+2x))\log (x)+(10x+2x^2+e^x(-10x-2x^2)){\log }^2(x))\log (5+x))\log (x\log (5+x))+(-10-2x)\log (5+x){\log }^2(x\log (5+x))}{(5x+x^2){\log }^3(x)\log (5+x)}dx\end{array}$$

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{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{(-2e^{x}x+2x^2){\log }^2(x)+((e^x(-10-2x)+10x+2x^2){\log }^2(x)+(10x+12x^2+2x^3+e^{2x}(10x+2x^2)+e^x(-10x-12x^2-2x^3)){\log }^3(x))\log (5+x)+(2x\log (x)+((10-8x-2x^2+e^x(10+2x))\log (x)+(10x+2x^2+e^x(-10x-2x^2)){\log }^2(x))\log (5+x))\log (x\log (5+x))+(-10-2x)\log (5+x){\log }^2(x\log (5+x))}{x(5+x){\log }^3(x)\log (5+x)}dx\\ & =\int \frac{-2x(5+x)(-1-e^{2x}-x+e^x(1+x)){\log }^3(x)\log (5+x)-2\log (x)(-x-(1+e^x-x)(5+x)\log (5+x))\log (x\log (5+x))-2(5+x)\log (5+x){\log }^2(x\log (5+x))+2{\log }^2(x)(x(-e^x+x)-(5+x)\log (5+x)(e^x-x+(-1+e^x)x\log (x\log (5+x))))}{x(5+x){\log }^3(x)\log (5+x)}dx\\ & =\int (2e^{2x}-\frac{2e^x(x\log (x)+5\log (x)\log (5+x)+x\log (x)\log (5+x)+5x{\log }^2(x)\log (5+x)+6x^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))+5x\log (x)\log (5+x)\log (x\log (5+x))+x^2\log (x)\log (5+x)\log (x\log (5+x)))}{x(5+x){\log }^2(x)\log (5+x)}+\frac{2(x^2{\log }^2(x)+5x{\log }^2(x)\log (5+x)+x^2{\log }^2(x)\log (5+x)+5x{\log }^3(x)\log (5+x)+6x^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))-4x\log (x)\log (5+x)\log (x\log (5+x))-x^2\log (x)\log (5+x)\log (x\log (5+x))+5x{\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\\ & =2\int e^{2x}dx-2\int \frac{e^x(x\log (x)+5\log (x)\log (5+x)+x\log (x)\log (5+x)+5x{\log }^2(x)\log (5+x)+6x^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))+5x\log (x)\log (5+x)\log (x\log (5+x))+x^2\log (x)\log (5+x)\log (x\log (5+x)))}{x(5+x){\log }^2(x)\log (5+x)}dx+2\int \frac{x^2{\log }^2(x)+5x{\log }^2(x)\log (5+x)+x^2{\log }^2(x)\log (5+x)+5x{\log }^3(x)\log (5+x)+6x^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))-4x\log (x)\log (5+x)\log (x\log (5+x))-x^2\log (x)\log (5+x)\log (x\log (5+x))+5x{\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^{2x}+2\int \frac{x(5+6x+x^2){\log }^3(x)\log (5+x)+\log (x)(x-(-5+4x+x^2)\log (5+x))\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(x(5+6x+x^2){\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)))))}{x(5+x){\log }^2(x)\log (5+x)}dx\\ & =e^{2x}-2\int (\frac{e^x(x+5\log (5+x)+x\log (5+x)+5x\log (x)\log (5+x)+6x^2\log (x)\log (5+x)+x^3\log (x)\log (5+x))}{x(5+x)\log (x)\log (5+x)}+\frac{e^x(-1+x\log (x))\log (x\log (5+x))}{x{\log }^2(x)})dx+2\int (\frac{x+5\log (5+x)+x\log (5+x)+5\log (x)\log (5+x)+6x\log (x)\log (5+x)+x^2\log (x)\log (5+x)}{(5+x)\log (x)\log (5+x)}+\frac{(x+5\log (5+x)-4x\log (5+x)-x^2\log (5+x)+5x\log (x)\log (5+x)+x^2\log (x)\log (5+x))\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)})dx\\ & =e^{2x}+2\int \frac{x+5\log (5+x)+x\log (5+x)+5\log (x)\log (5+x)+6x\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(x+5\log (5+x)+x\log (5+x)+5x\log (x)\log (5+x)+6x^2\log (x)\log (5+x)+x^3\log (x)\log (5+x))}{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{(x+5\log (5+x)-4x\log (5+x)-x^2\log (5+x)+5x\log (x)\log (5+x)+x^2\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\\ & =e^{2x}+2\int (1+x+\frac{1+\frac{x}{(5+x)\log (5+x)}}{\log (x)})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 (-\frac{e^x\log (x\log (5+x))}{x{\log }^2(x)}+\frac{e^x\log (x\log (5+x))}{\log (x)})dx\\ & =e^{2x}+2x+x^2+2\int \frac{1+\frac{x}{(5+x)\log (5+x)}}{\log (x)}dx-2\int (\frac{e^x(1+x\log (x)+x^2\log (x))}{x\log (x)}+\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+2\int (\frac{(x+5\log (5+x)-4x\log (5+x)-x^2\log (5+x)+5x\log (x)\log (5+x)+x^2\log (x)\log (5+x))\log (x\log (5+x))}{5x{\log }^2(x)\log (5+x)}-\frac{(x+5\log (5+x)-4x\log (5+x)-x^2\log (5+x)+5x\log (x)\log (5+x)+x^2\log (x)\log (5+x))\log (x\log (5+x))}{5(5+x){\log }^2(x)\log (5+x)})dx\\ & =e^{2x}+2x+x^2+\frac{2}{5}\int \frac{(x+5\log (5+x)-4x\log (5+x)-x^2\log (5+x)+5x\log (x)\log (5+x)+x^2\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\log (5+x)-4x\log (5+x)-x^2\log (5+x)+5x\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)}dx-2\int \frac{e^x(1+x\log (x)+x^2\log (x))}{x\log (x)}dx+2\int (\frac{1}{\log (x)}+\frac{x}{(5+x)\log (x)\log (5+x)})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^{2x}+2x+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(1+x+\frac{1}{x\log (x)})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^{2x}+2x+x^2+2\text{li}(x)+\frac{2}{5}\int (-\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)})dx-\frac{2}{5}\int (\frac{5\log (x\log (5+x))}{(5+x){\log }^2(x)}-\frac{4x\log (x\log (5+x))}{(5+x){\log }^2(x)}-\frac{x^2\log (x\log (5+x))}{(5+x){\log }^2(x)}+\frac{5x\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)})dx-2\int (e^x+e^{x}x+\frac{e^x}{x\log (x)})dx+2\int (\frac{1}{\log (x)\log (5+x)}-\frac{5}{(5+x)\log (x)\log (5+x)})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^{2x}+2x+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}xdx-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\\ & =-2e^x+e^{2x}+2x-2e^{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 (-\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)})dx-\frac{2}{5}\int (-\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)})dx-\frac{2}{5}\int (\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)})dx-\frac{8}{5}\int \frac{\log (x\log (5+x))}{{\log }^2(x)}dx+\frac{8}{5}\int (\frac{\log (x\log (5+x))}{{\log }^2(x)}-\frac{5\log (x\log (5+x))}{(5+x){\log }^2(x)})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 (\frac{\log (x\log (5+x))}{\log (x)}-\frac{5\log (x\log (5+x))}{(5+x)\log (x)})dx-2\int (\frac{1}{x}+\frac{1}{(5+x)\log (5+x)})\text{li}(x)dx-10\int \frac{1}{(5+x)\log (x)\log (5+x)}dx\\ & =e^{2x}+2x-2e^{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}+\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^{2x}+2x-2e^{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^{2x}+4x-2e^{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{array}\end{array}$$

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

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 $$\begin{array}{c}\frac{2(x-e^x)\log (x\log (x+5))\log (x)+(x^2-2x{e}^x+2x+e^{\left(2x\right)})\log (x{)}^2+\log {(x\log (x+5))}^2}{\log (x{)}^2}\end{array}$$

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 $$\begin{array}{c}\frac{x^2\log (x{)}^2-2x{e}^x\log (x{)}^2+4x\log (x{)}^2+e^{\left(2x\right)}\log (x{)}^2-2e^x\log (x{)}^2+2x\log (x)\log (\log (x+5))-2e^x\log (x)\log (\log (x+5))+2\log (x)\log (\log (x+5))+\log {(\log (x+5))}^2}{\log (x{)}^2}\end{array}$$

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 {(\ln (5+x))}^2}{\ln (x{)}^2}+\frac{(-i\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}(i\ln (5+x))\mathrm{csgn}(ix\ln (5+x))+i\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{(ix\ln (5+x))}^2+i\pi \mathrm{csgn}(i\ln (5+x))\mathrm{csgn}{(ix\ln (5+x))}^2-i\pi \mathrm{csgn}{(ix\ln (5+x))}^3+2x\ln (x)-2{\mathrm{e}}^x\ln (x)+2\ln (x))\ln (\ln (5+x))}{\ln (x{)}^2}+\frac{-8x{\mathrm{e}}^x\ln (x{)}^2+4x^2\ln (x{)}^2-8{\mathrm{e}}^x\ln (x{)}^2+16x\ln (x{)}^2-{\pi }^2\mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}{(i\ln (5+x))}^2\mathrm{csgn}{(ix\ln (5+x))}^2+2{\pi }^2\mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}(i\ln (5+x))\mathrm{csgn}{(ix\ln (5+x))}^3+2{\pi }^2\mathrm{csgn}\left(ix\right)\mathrm{csgn}{(i\ln (5+x))}^2\mathrm{csgn}{(ix\ln (5+x))}^3-4{\pi }^2\mathrm{csgn}\left(ix\right)\mathrm{csgn}(i\ln (5+x))\mathrm{csgn}{(ix\ln (5+x))}^4-4i\ln (x)\pi \mathrm{csgn}{(ix\ln (5+x))}^3-{\pi }^2\mathrm{csgn}{(ix\ln (5+x))}^6+4{\mathrm{e}}^{2x}\ln (x{)}^2+4i\pi x\mathrm{csgn}\left(ix\right)\mathrm{csgn}{(ix\ln (5+x))}^2\ln (x)+4i\pi x\mathrm{csgn}(i\ln (5+x))\mathrm{csgn}{(ix\ln (5+x))}^2\ln (x)-4i\ln (x)\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}(i\ln (5+x))\mathrm{csgn}(ix\ln (5+x))-4i\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{(ix\ln (5+x))}^2{\mathrm{e}}^x\ln (x)-4i\pi \mathrm{csgn}(i\ln (5+x))\mathrm{csgn}{(ix\ln (5+x))}^2{\mathrm{e}}^x\ln (x)-{\pi }^2\mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}{(ix\ln (5+x))}^4+2{\pi }^2\mathrm{csgn}\left(ix\right)\mathrm{csgn}{(ix\ln (5+x))}^5-{\pi }^2\mathrm{csgn}{(i\ln (5+x))}^2\mathrm{csgn}{(ix\ln (5+x))}^4+2{\pi }^2\mathrm{csgn}(i\ln (5+x))\mathrm{csgn}{(ix\ln (5+x))}^5-4i\pi x\mathrm{csgn}\left(ix\right)\mathrm{csgn}(i\ln (5+x))\mathrm{csgn}(ix\ln (5+x))\ln (x)+4i\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}(i\ln (5+x))\mathrm{csgn}(ix\ln (5+x)){\mathrm{e}}^x\ln (x)+4i\ln (x)\pi \mathrm{csgn}(i\ln (5+x))\mathrm{csgn}{(ix\ln (5+x))}^2-4i\pi x\mathrm{csgn}{(ix\ln (5+x))}^3\ln (x)+4i\pi \mathrm{csgn}{(ix\ln (5+x))}^3{\mathrm{e}}^x\ln (x)+4i\ln (x)\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{(ix\ln (5+x))}^2}{4\ln (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 $$\begin{array}{c}-\frac{2(x+1)e^x\log (x{)}^2-(x^2+4x)\log (x{)}^2-e^{\left(2x\right)}\log (x{)}^2-2((x+1)\log (x)-e^x\log (x))\log (\log (x+5))-\log {(\log (x+5))}^2}{\log (x{)}^2}\end{array}$$

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 $$\begin{array}{c}2x+{\mathrm{e}}^{2x}+\frac{{\ln (x\ln (x+5))}^2}{{\ln (x)}^2}-2x{\mathrm{e}}^x+x^2+\frac{2\ln (x\ln (x+5))(x-{\mathrm{e}}^x)}{\ln (x)}\end{array}$$

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 $$\begin{array}{c}{x}^2+2x+\frac{2x\log (x\log (x+5))}{\log (x)}+\frac{(-2x\log (x)-2\log (x\log (x+5)))e^x+e^{2x}\log (x)}{\log (x)}+\frac{\log {(x\log (x+5))}^2}{\log {(x)}^2}\end{array}$$

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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