\(\int \frac {2-2 x+15 x^2+15 x^3+50 x^5+(-20 x^2-50 x^4) \log (2 x+5 x^3)+(-4 x^3-10 x^5+(4 x^2+10 x^4) \log (2 x+5 x^3)) \log (x-\log (2 x+5 x^3))}{10 x^2+25 x^4+(-10 x-25 x^3) \log (2 x+5 x^3)+(-2 x^2-5 x^4+(2 x+5 x^3) \log (2 x+5 x^3)) \log (x-\log (2 x+5 x^3))} \, dx\) [8368]

Optimal result

Integrand size = 172, antiderivative size = 24 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^2+\log \left (-5+\log \left (x-\log \left (x+x \left (1+5 x^2\right )\right )\right )\right ) \]

[Out]

ln(ln(x-ln(x+(5*x^2+1)*x))-5)+x^2

Rubi [F]

\[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=\int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx \]

[In]

Int[(2 - 2*x + 15*x^2 + 15*x^3 + 50*x^5 + (-20*x^2 - 50*x^4)*Log[2*x + 5*x^3] + (-4*x^3 - 10*x^5 + (4*x^2 + 10
*x^4)*Log[2*x + 5*x^3])*Log[x - Log[2*x + 5*x^3]])/(10*x^2 + 25*x^4 + (-10*x - 25*x^3)*Log[2*x + 5*x^3] + (-2*
x^2 - 5*x^4 + (2*x + 5*x^3)*Log[2*x + 5*x^3])*Log[x - Log[2*x + 5*x^3]]),x]

[Out]

x^2 + 10*Defer[Int][x/(-5 + Log[x - Log[x*(2 + 5*x^2)]]), x] + Defer[Int][1/((x - Log[x*(2 + 5*x^2)])*(-5 + Lo
g[x - Log[x*(2 + 5*x^2)]])), x] - Defer[Int][1/(x*(x - Log[x*(2 + 5*x^2)])*(-5 + Log[x - Log[x*(2 + 5*x^2)]]))
, x] - 10*Defer[Int][x^2/((x - Log[x*(2 + 5*x^2)])*(-5 + Log[x - Log[x*(2 + 5*x^2)]])), x] + Sqrt[5]*Defer[Int
][1/((I*Sqrt[2] - Sqrt[5]*x)*(x - Log[x*(2 + 5*x^2)])*(-5 + Log[x - Log[x*(2 + 5*x^2)]])), x] - Sqrt[5]*Defer[
Int][1/((I*Sqrt[2] + Sqrt[5]*x)*(x - Log[x*(2 + 5*x^2)])*(-5 + Log[x - Log[x*(2 + 5*x^2)]])), x] + 10*Defer[In
t][(x*Log[x*(2 + 5*x^2)])/((x - Log[x*(2 + 5*x^2)])*(-5 + Log[x - Log[x*(2 + 5*x^2)]])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{x \left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (5-\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx \\ & = \int \left (\frac {2}{\left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}-\frac {2}{x \left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}-\frac {15 x}{\left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}-\frac {15 x^2}{\left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}-\frac {50 x^4}{\left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}+\frac {10 x \log \left (x \left (2+5 x^2\right )\right )}{\left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}+\frac {2 x \log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )}{-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )}\right ) \, dx \\ & = 2 \int \frac {1}{\left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx-2 \int \frac {1}{x \left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx+2 \int \frac {x \log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )}{-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )} \, dx+10 \int \frac {x \log \left (x \left (2+5 x^2\right )\right )}{\left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx-15 \int \frac {x}{\left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx-15 \int \frac {x^2}{\left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx-50 \int \frac {x^4}{\left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx \\ & = 2 \int \left (x+\frac {5 x}{-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )}\right ) \, dx+2 \int \left (\frac {i}{2 \sqrt {2} \left (i \sqrt {2}-\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}+\frac {i}{2 \sqrt {2} \left (i \sqrt {2}+\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}\right ) \, dx-2 \int \left (\frac {1}{2 x \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}-\frac {5 x}{2 \left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}\right ) \, dx+10 \int \frac {x \log \left (x \left (2+5 x^2\right )\right )}{\left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx-15 \int \left (-\frac {1}{2 \sqrt {5} \left (i \sqrt {2}-\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}+\frac {1}{2 \sqrt {5} \left (i \sqrt {2}+\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}\right ) \, dx-15 \int \left (\frac {1}{5 \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}-\frac {2}{5 \left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}\right ) \, dx-50 \int \left (-\frac {2}{25 \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}+\frac {x^2}{5 \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}+\frac {4}{25 \left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}\right ) \, dx \\ & = x^2-3 \int \frac {1}{\left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx+4 \int \frac {1}{\left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx+5 \int \frac {x}{\left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx+6 \int \frac {1}{\left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx-8 \int \frac {1}{\left (2+5 x^2\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx+10 \int \frac {x}{-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )} \, dx-10 \int \frac {x^2}{\left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx+10 \int \frac {x \log \left (x \left (2+5 x^2\right )\right )}{\left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx+\frac {i \int \frac {1}{\left (i \sqrt {2}-\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx}{\sqrt {2}}+\frac {i \int \frac {1}{\left (i \sqrt {2}+\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx}{\sqrt {2}}+\frac {1}{2} \left (3 \sqrt {5}\right ) \int \frac {1}{\left (i \sqrt {2}-\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx-\frac {1}{2} \left (3 \sqrt {5}\right ) \int \frac {1}{\left (i \sqrt {2}+\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx-\int \frac {1}{x \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx \\ & = x^2-3 \int \frac {1}{\left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx+4 \int \frac {1}{\left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx+5 \int \left (-\frac {1}{2 \sqrt {5} \left (i \sqrt {2}-\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}+\frac {1}{2 \sqrt {5} \left (i \sqrt {2}+\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}\right ) \, dx+6 \int \left (\frac {i}{2 \sqrt {2} \left (i \sqrt {2}-\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}+\frac {i}{2 \sqrt {2} \left (i \sqrt {2}+\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}\right ) \, dx-8 \int \left (\frac {i}{2 \sqrt {2} \left (i \sqrt {2}-\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}+\frac {i}{2 \sqrt {2} \left (i \sqrt {2}+\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )}\right ) \, dx+10 \int \frac {x}{-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )} \, dx-10 \int \frac {x^2}{\left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx+10 \int \frac {x \log \left (x \left (2+5 x^2\right )\right )}{\left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx+\frac {i \int \frac {1}{\left (i \sqrt {2}-\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx}{\sqrt {2}}+\frac {i \int \frac {1}{\left (i \sqrt {2}+\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx}{\sqrt {2}}+\frac {1}{2} \left (3 \sqrt {5}\right ) \int \frac {1}{\left (i \sqrt {2}-\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx-\frac {1}{2} \left (3 \sqrt {5}\right ) \int \frac {1}{\left (i \sqrt {2}+\sqrt {5} x\right ) \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx-\int \frac {1}{x \left (x-\log \left (x \left (2+5 x^2\right )\right )\right ) \left (-5+\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^2+\log \left (5-\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right ) \]

[In]

Integrate[(2 - 2*x + 15*x^2 + 15*x^3 + 50*x^5 + (-20*x^2 - 50*x^4)*Log[2*x + 5*x^3] + (-4*x^3 - 10*x^5 + (4*x^
2 + 10*x^4)*Log[2*x + 5*x^3])*Log[x - Log[2*x + 5*x^3]])/(10*x^2 + 25*x^4 + (-10*x - 25*x^3)*Log[2*x + 5*x^3]
+ (-2*x^2 - 5*x^4 + (2*x + 5*x^3)*Log[2*x + 5*x^3])*Log[x - Log[2*x + 5*x^3]]),x]

[Out]

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

Maple [A] (verified)

Time = 2.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(-\frac {1}{5}+x^{2}+\ln \left (\ln \left (-\ln \left (5 x^{3}+2 x \right )+x \right )-5\right )\)	\(24\)

[In]

int((((10*x^4+4*x^2)*ln(5*x^3+2*x)-10*x^5-4*x^3)*ln(-ln(5*x^3+2*x)+x)+(-50*x^4-20*x^2)*ln(5*x^3+2*x)+50*x^5+15
*x^3+15*x^2-2*x+2)/(((5*x^3+2*x)*ln(5*x^3+2*x)-5*x^4-2*x^2)*ln(-ln(5*x^3+2*x)+x)+(-25*x^3-10*x)*ln(5*x^3+2*x)+
25*x^4+10*x^2),x,method=_RETURNVERBOSE)

[Out]

-1/5+x^2+ln(ln(-ln(5*x^3+2*x)+x)-5)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^{2} + \log \left (\log \left (x - \log \left (5 \, x^{3} + 2 \, x\right )\right ) - 5\right ) \]

[In]

integrate((((10*x^4+4*x^2)*log(5*x^3+2*x)-10*x^5-4*x^3)*log(-log(5*x^3+2*x)+x)+(-50*x^4-20*x^2)*log(5*x^3+2*x)
+50*x^5+15*x^3+15*x^2-2*x+2)/(((5*x^3+2*x)*log(5*x^3+2*x)-5*x^4-2*x^2)*log(-log(5*x^3+2*x)+x)+(-25*x^3-10*x)*l
og(5*x^3+2*x)+25*x^4+10*x^2),x, algorithm="fricas")

[Out]

x^2 + log(log(x - log(5*x^3 + 2*x)) - 5)

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^{2} + \log {\left (\log {\left (x - \log {\left (5 x^{3} + 2 x \right )} \right )} - 5 \right )} \]

[In]

integrate((((10*x**4+4*x**2)*ln(5*x**3+2*x)-10*x**5-4*x**3)*ln(-ln(5*x**3+2*x)+x)+(-50*x**4-20*x**2)*ln(5*x**3
+2*x)+50*x**5+15*x**3+15*x**2-2*x+2)/(((5*x**3+2*x)*ln(5*x**3+2*x)-5*x**4-2*x**2)*ln(-ln(5*x**3+2*x)+x)+(-25*x
**3-10*x)*ln(5*x**3+2*x)+25*x**4+10*x**2),x)

[Out]

x**2 + log(log(x - log(5*x**3 + 2*x)) - 5)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^{2} + \log \left (\log \left (x - \log \left (5 \, x^{2} + 2\right ) - \log \left (x\right )\right ) - 5\right ) \]

[In]

integrate((((10*x^4+4*x^2)*log(5*x^3+2*x)-10*x^5-4*x^3)*log(-log(5*x^3+2*x)+x)+(-50*x^4-20*x^2)*log(5*x^3+2*x)
+50*x^5+15*x^3+15*x^2-2*x+2)/(((5*x^3+2*x)*log(5*x^3+2*x)-5*x^4-2*x^2)*log(-log(5*x^3+2*x)+x)+(-25*x^3-10*x)*l
og(5*x^3+2*x)+25*x^4+10*x^2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^{2} + \log \left (\log \left (x - \log \left (5 \, x^{3} + 2 \, x\right )\right ) - 5\right ) \]

[In]

integrate((((10*x^4+4*x^2)*log(5*x^3+2*x)-10*x^5-4*x^3)*log(-log(5*x^3+2*x)+x)+(-50*x^4-20*x^2)*log(5*x^3+2*x)
+50*x^5+15*x^3+15*x^2-2*x+2)/(((5*x^3+2*x)*log(5*x^3+2*x)-5*x^4-2*x^2)*log(-log(5*x^3+2*x)+x)+(-25*x^3-10*x)*l
og(5*x^3+2*x)+25*x^4+10*x^2),x, algorithm="giac")

[Out]

x^2 + log(log(x - log(5*x^3 + 2*x)) - 5)

Mupad [B] (verification not implemented)

Time = 13.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=\ln \left (\ln \left (x-\ln \left (5\,x^3+2\,x\right )\right )-5\right )+x^2 \]

[In]

int(-(15*x^2 - log(2*x + 5*x^3)*(20*x^2 + 50*x^4) - log(x - log(2*x + 5*x^3))*(4*x^3 - log(2*x + 5*x^3)*(4*x^2
+ 10*x^4) + 10*x^5) - 2*x + 15*x^3 + 50*x^5 + 2)/(log(x - log(2*x + 5*x^3))*(2*x^2 - log(2*x + 5*x^3)*(2*x +
5*x^3) + 5*x^4) + log(2*x + 5*x^3)*(10*x + 25*x^3) - 10*x^2 - 25*x^4),x)

[Out]

log(log(x - log(2*x + 5*x^3)) - 5) + x^2