Integrand size = 172, antiderivative size = 24 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^2+\log \left (-5+\log \left (x-\log \left (x+x \left (1+5 x^2\right )\right )\right )\right ) \]
Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^2+\log \left (5-\log \left (x-\log \left (x \left (2+5 x^2\right )\right )\right )\right ) \]
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]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {50 x^5+15 x^3+15 x^2+\left (-50 x^4-20 x^2\right ) \log \left (5 x^3+2 x\right )+\left (-10 x^5-4 x^3+\left (10 x^4+4 x^2\right ) \log \left (5 x^3+2 x\right )\right ) \log \left (x-\log \left (5 x^3+2 x\right )\right )-2 x+2}{25 x^4+\left (-25 x^3-10 x\right ) \log \left (5 x^3+2 x\right )+10 x^2+\left (-5 x^4+\left (5 x^3+2 x\right ) \log \left (5 x^3+2 x\right )-2 x^2\right ) \log \left (x-\log \left (5 x^3+2 x\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {50 x^5+15 x^3+15 x^2+\left (-50 x^4-20 x^2\right ) \log \left (5 x^3+2 x\right )+\left (-10 x^5-4 x^3+\left (10 x^4+4 x^2\right ) \log \left (5 x^3+2 x\right )\right ) \log \left (x-\log \left (5 x^3+2 x\right )\right )-2 x+2}{x \left (5 x^2+2\right ) \left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (5-\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {15 x^2}{\left (5 x^2+2\right ) \left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}+\frac {2 x \log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )}{\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5}+\frac {10 x \log \left (x \left (5 x^2+2\right )\right )}{\left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}-\frac {15 x}{\left (5 x^2+2\right ) \left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}+\frac {2}{\left (5 x^2+2\right ) \left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}-\frac {2}{\left (5 x^2+2\right ) x \left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}-\frac {50 x^4}{\left (5 x^2+2\right ) \left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 10 \int \frac {x}{\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5}dx+\int \frac {1}{\left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}dx-\int \frac {1}{x \left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}dx-10 \int \frac {x^2}{\left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}dx+\sqrt {5} \int \frac {1}{\left (i \sqrt {2}-\sqrt {5} x\right ) \left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}dx-\sqrt {5} \int \frac {1}{\left (\sqrt {5} x+i \sqrt {2}\right ) \left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}dx+10 \int \frac {x \log \left (x \left (5 x^2+2\right )\right )}{\left (x-\log \left (x \left (5 x^2+2\right )\right )\right ) \left (\log \left (x-\log \left (x \left (5 x^2+2\right )\right )\right )-5\right )}dx+x^2\) |
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]
3.24.68.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 2.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
| method | result | size |
| parallelrisch | \(-\frac {1}{5}+x^{2}+\ln \left (\ln \left (-\ln \left (5 x^{3}+2 x \right )+x \right )-5\right )\) | \(24\) |
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)
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^{2} + \log \left (\log \left (x - \log \left (5 \, x^{3} + 2 \, x\right )\right ) - 5\right ) \]
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)*log (5*x^3+2*x)+25*x^4+10*x^2),x, algorithm=\
Time = 0.32 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^{2} + \log {\left (\log {\left (x - \log {\left (5 x^{3} + 2 x \right )} \right )} - 5 \right )} \]
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)
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^{2} + \log \left (\log \left (x - \log \left (5 \, x^{2} + 2\right ) - \log \left (x\right )\right ) - 5\right ) \]
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)*log (5*x^3+2*x)+25*x^4+10*x^2),x, algorithm=\
Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=x^{2} + \log \left (\log \left (x - \log \left (5 \, x^{3} + 2 \, x\right )\right ) - 5\right ) \]
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)*log (5*x^3+2*x)+25*x^4+10*x^2),x, algorithm=\
Time = 13.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {2-2 x+15 x^2+15 x^3+50 x^5+\left (-20 x^2-50 x^4\right ) \log \left (2 x+5 x^3\right )+\left (-4 x^3-10 x^5+\left (4 x^2+10 x^4\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )}{10 x^2+25 x^4+\left (-10 x-25 x^3\right ) \log \left (2 x+5 x^3\right )+\left (-2 x^2-5 x^4+\left (2 x+5 x^3\right ) \log \left (2 x+5 x^3\right )\right ) \log \left (x-\log \left (2 x+5 x^3\right )\right )} \, dx=\ln \left (\ln \left (x-\ln \left (5\,x^3+2\,x\right )\right )-5\right )+x^2 \]
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)