Integrand size = 88, antiderivative size = 20 \[ \int \frac {2040 x+1008 x^2+144 x^3+16 x^4+\left (7225+2040 x+504 x^2+48 x^3+4 x^4\right ) \log \left (\frac {1}{4} \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx=x \log \left (5 x^2+\left (\frac {67}{2}+(3+x)^2\right )^2\right ) \] Output:
x*ln(5*x^2+(67/2+(3+x)^2)^2)
Time = 0.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {2040 x+1008 x^2+144 x^3+16 x^4+\left (7225+2040 x+504 x^2+48 x^3+4 x^4\right ) \log \left (\frac {1}{4} \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx=x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right ) \] Input:
Integrate[(2040*x + 1008*x^2 + 144*x^3 + 16*x^4 + (7225 + 2040*x + 504*x^2 + 48*x^3 + 4*x^4)*Log[(7225 + 2040*x + 504*x^2 + 48*x^3 + 4*x^4)/4])/(722 5 + 2040*x + 504*x^2 + 48*x^3 + 4*x^4),x]
Output:
x*Log[7225/4 + 510*x + 126*x^2 + 12*x^3 + x^4]
Result contains complex when optimal does not.
Time = 0.89 (sec) , antiderivative size = 202, normalized size of antiderivative = 10.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {7293, 2009}
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.
\(\displaystyle \int \frac {16 x^4+144 x^3+1008 x^2+\left (4 x^4+48 x^3+504 x^2+2040 x+7225\right ) \log \left (\frac {1}{4} \left (4 x^4+48 x^3+504 x^2+2040 x+7225\right )\right )+2040 x}{4 x^4+48 x^3+504 x^2+2040 x+7225} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {8 x \left (2 x^3+18 x^2+126 x+255\right )}{4 x^4+48 x^3+504 x^2+2040 x+7225}+\log \left (x^4+12 x^3+126 x^2+510 x+\frac {7225}{4}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\sqrt {139-12 i \sqrt {5}} \arctan \left (\frac {2 x+i \sqrt {5}+6}{\sqrt {139-12 i \sqrt {5}}}\right )-i \sqrt {139-12 i \sqrt {5}} \text {arctanh}\left (\frac {2 i x-\sqrt {5}+6 i}{\sqrt {139-12 i \sqrt {5}}}\right )+\frac {1}{2} \left (6+i \sqrt {5}\right ) \log \left (2 i x^2+2 \left (-\sqrt {5}+6 i\right ) x+85 i\right )-\frac {1}{2} \left (6+i \sqrt {5}\right ) \log \left (2 x^2+2 \left (6+i \sqrt {5}\right ) x+85\right )+x \log \left (x^4+12 x^3+126 x^2+510 x+\frac {7225}{4}\right )\) |
Input:
Int[(2040*x + 1008*x^2 + 144*x^3 + 16*x^4 + (7225 + 2040*x + 504*x^2 + 48* x^3 + 4*x^4)*Log[(7225 + 2040*x + 504*x^2 + 48*x^3 + 4*x^4)/4])/(7225 + 20 40*x + 504*x^2 + 48*x^3 + 4*x^4),x]
Output:
-(Sqrt[139 - (12*I)*Sqrt[5]]*ArcTan[(6 + I*Sqrt[5] + 2*x)/Sqrt[139 - (12*I )*Sqrt[5]]]) - I*Sqrt[139 - (12*I)*Sqrt[5]]*ArcTanh[(6*I - Sqrt[5] + (2*I) *x)/Sqrt[139 - (12*I)*Sqrt[5]]] + ((6 + I*Sqrt[5])*Log[85*I + 2*(6*I - Sqr t[5])*x + (2*I)*x^2])/2 - ((6 + I*Sqrt[5])*Log[85 + 2*(6 + I*Sqrt[5])*x + 2*x^2])/2 + x*Log[7225/4 + 510*x + 126*x^2 + 12*x^3 + x^4]
Time = 0.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10
method | result | size |
norman | \(\ln \left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right ) x\) | \(22\) |
risch | \(\ln \left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right ) x\) | \(22\) |
parallelrisch | \(\ln \left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right ) x\) | \(22\) |
default | \(-2 x \ln \left (2\right )+x \ln \left (4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225\right )\) | \(30\) |
parts | \(-2 x \ln \left (2\right )+x \ln \left (4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225\right )\) | \(30\) |
orering | \(\frac {x \left (\left (4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225\right ) \ln \left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right )+16 x^{4}+144 x^{3}+1008 x^{2}+2040 x \right )}{4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225}-\frac {\left (4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225\right ) \left (4 x^{5}-180 x^{3}-4026 x^{2}-18360 x -65025\right ) \left (\frac {\left (16 x^{3}+144 x^{2}+1008 x +2040\right ) \ln \left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right )+\frac {\left (4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225\right ) \left (4 x^{3}+36 x^{2}+252 x +510\right )}{x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}}+64 x^{3}+432 x^{2}+2016 x +2040}{4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225}-\frac {\left (\left (4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225\right ) \ln \left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right )+16 x^{4}+144 x^{3}+1008 x^{2}+2040 x \right ) \left (16 x^{3}+144 x^{2}+1008 x +2040\right )}{\left (4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225\right )^{2}}\right )}{4 \left (4 x^{7}+96 x^{6}+1692 x^{5}+16212 x^{4}+116837 x^{3}+517140 x^{2}+1625625 x +1842375\right )}\) | \(392\) |
Input:
int(((4*x^4+48*x^3+504*x^2+2040*x+7225)*ln(x^4+12*x^3+126*x^2+510*x+7225/4 )+16*x^4+144*x^3+1008*x^2+2040*x)/(4*x^4+48*x^3+504*x^2+2040*x+7225),x,met hod=_RETURNVERBOSE)
Output:
ln(x^4+12*x^3+126*x^2+510*x+7225/4)*x
Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {2040 x+1008 x^2+144 x^3+16 x^4+\left (7225+2040 x+504 x^2+48 x^3+4 x^4\right ) \log \left (\frac {1}{4} \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx=x \log \left (x^{4} + 12 \, x^{3} + 126 \, x^{2} + 510 \, x + \frac {7225}{4}\right ) \] Input:
integrate(((4*x^4+48*x^3+504*x^2+2040*x+7225)*log(x^4+12*x^3+126*x^2+510*x +7225/4)+16*x^4+144*x^3+1008*x^2+2040*x)/(4*x^4+48*x^3+504*x^2+2040*x+7225 ),x, algorithm="fricas")
Output:
x*log(x^4 + 12*x^3 + 126*x^2 + 510*x + 7225/4)
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {2040 x+1008 x^2+144 x^3+16 x^4+\left (7225+2040 x+504 x^2+48 x^3+4 x^4\right ) \log \left (\frac {1}{4} \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx=x \log {\left (x^{4} + 12 x^{3} + 126 x^{2} + 510 x + \frac {7225}{4} \right )} \] Input:
integrate(((4*x**4+48*x**3+504*x**2+2040*x+7225)*ln(x**4+12*x**3+126*x**2+ 510*x+7225/4)+16*x**4+144*x**3+1008*x**2+2040*x)/(4*x**4+48*x**3+504*x**2+ 2040*x+7225),x)
Output:
x*log(x**4 + 12*x**3 + 126*x**2 + 510*x + 7225/4)
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {2040 x+1008 x^2+144 x^3+16 x^4+\left (7225+2040 x+504 x^2+48 x^3+4 x^4\right ) \log \left (\frac {1}{4} \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx=-2 \, x \log \left (2\right ) + x \log \left (4 \, x^{4} + 48 \, x^{3} + 504 \, x^{2} + 2040 \, x + 7225\right ) \] Input:
integrate(((4*x^4+48*x^3+504*x^2+2040*x+7225)*log(x^4+12*x^3+126*x^2+510*x +7225/4)+16*x^4+144*x^3+1008*x^2+2040*x)/(4*x^4+48*x^3+504*x^2+2040*x+7225 ),x, algorithm="maxima")
Output:
-2*x*log(2) + x*log(4*x^4 + 48*x^3 + 504*x^2 + 2040*x + 7225)
Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {2040 x+1008 x^2+144 x^3+16 x^4+\left (7225+2040 x+504 x^2+48 x^3+4 x^4\right ) \log \left (\frac {1}{4} \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx=x \log \left (x^{4} + 12 \, x^{3} + 126 \, x^{2} + 510 \, x + \frac {7225}{4}\right ) \] Input:
integrate(((4*x^4+48*x^3+504*x^2+2040*x+7225)*log(x^4+12*x^3+126*x^2+510*x +7225/4)+16*x^4+144*x^3+1008*x^2+2040*x)/(4*x^4+48*x^3+504*x^2+2040*x+7225 ),x, algorithm="giac")
Output:
x*log(x^4 + 12*x^3 + 126*x^2 + 510*x + 7225/4)
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {2040 x+1008 x^2+144 x^3+16 x^4+\left (7225+2040 x+504 x^2+48 x^3+4 x^4\right ) \log \left (\frac {1}{4} \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx=x\,\ln \left (x^4+12\,x^3+126\,x^2+510\,x+\frac {7225}{4}\right ) \] Input:
int((2040*x + log(510*x + 126*x^2 + 12*x^3 + x^4 + 7225/4)*(2040*x + 504*x ^2 + 48*x^3 + 4*x^4 + 7225) + 1008*x^2 + 144*x^3 + 16*x^4)/(2040*x + 504*x ^2 + 48*x^3 + 4*x^4 + 7225),x)
Output:
x*log(510*x + 126*x^2 + 12*x^3 + x^4 + 7225/4)
Time = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.30 \[ \int \frac {2040 x+1008 x^2+144 x^3+16 x^4+\left (7225+2040 x+504 x^2+48 x^3+4 x^4\right ) \log \left (\frac {1}{4} \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx=-3 \,\mathrm {log}\left (4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225\right )+\mathrm {log}\left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right ) x +3 \,\mathrm {log}\left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right ) \] Input:
int(((4*x^4+48*x^3+504*x^2+2040*x+7225)*log(x^4+12*x^3+126*x^2+510*x+7225/ 4)+16*x^4+144*x^3+1008*x^2+2040*x)/(4*x^4+48*x^3+504*x^2+2040*x+7225),x)
Output:
- 3*log(4*x**4 + 48*x**3 + 504*x**2 + 2040*x + 7225) + log((4*x**4 + 48*x **3 + 504*x**2 + 2040*x + 7225)/4)*x + 3*log((4*x**4 + 48*x**3 + 504*x**2 + 2040*x + 7225)/4)