Integrand size = 16, antiderivative size = 60 \[ \int \left (1+x^4\right ) \left (1-2 \log (x)+\log ^3(x)\right ) \, dx=-3 x+\frac {169 x^5}{625}+4 x \log (x)-\frac {44}{125} x^5 \log (x)-3 x \log ^2(x)-\frac {3}{25} x^5 \log ^2(x)+x \log ^3(x)+\frac {1}{5} x^5 \log ^3(x) \] Output:
-3*x+169/625*x^5+4*x*ln(x)-44/125*x^5*ln(x)-3*x*ln(x)^2-3/25*x^5*ln(x)^2+x *ln(x)^3+1/5*x^5*ln(x)^3
Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \left (1+x^4\right ) \left (1-2 \log (x)+\log ^3(x)\right ) \, dx=-3 x+\frac {169 x^5}{625}+4 x \log (x)-\frac {44}{125} x^5 \log (x)-3 x \log ^2(x)-\frac {3}{25} x^5 \log ^2(x)+x \log ^3(x)+\frac {1}{5} x^5 \log ^3(x) \] Input:
Integrate[(1 + x^4)*(1 - 2*Log[x] + Log[x]^3),x]
Output:
-3*x + (169*x^5)/625 + 4*x*Log[x] - (44*x^5*Log[x])/125 - 3*x*Log[x]^2 - ( 3*x^5*Log[x]^2)/25 + x*Log[x]^3 + (x^5*Log[x]^3)/5
Time = 0.35 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
\(\displaystyle \int \left (x^4+1\right ) \left (\log ^3(x)-2 \log (x)+1\right ) \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (x^4+\left (x^4+1\right ) \log ^3(x)-2 \left (x^4+1\right ) \log (x)+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {169 x^5}{625}+\frac {1}{5} x^5 \log ^3(x)-\frac {3}{25} x^5 \log ^2(x)-\frac {44}{125} x^5 \log (x)-3 x+x \log ^3(x)-3 x \log ^2(x)+4 x \log (x)\) |
Input:
Int[(1 + x^4)*(1 - 2*Log[x] + Log[x]^3),x]
Output:
-3*x + (169*x^5)/625 + 4*x*Log[x] - (44*x^5*Log[x])/125 - 3*x*Log[x]^2 - ( 3*x^5*Log[x]^2)/25 + x*Log[x]^3 + (x^5*Log[x]^3)/5
Time = 0.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\left (\frac {1}{5} x^{5}+x \right ) \ln \left (x \right )^{3}+\left (-\frac {3}{25} x^{5}-3 x \right ) \ln \left (x \right )^{2}+\left (-\frac {44}{125} x^{5}+4 x \right ) \ln \left (x \right )+\frac {169 x^{5}}{625}-3 x\) | \(48\) |
default | \(-3 x +\frac {169 x^{5}}{625}+4 x \ln \left (x \right )-\frac {44 x^{5} \ln \left (x \right )}{125}-3 x \ln \left (x \right )^{2}-\frac {3 x^{5} \ln \left (x \right )^{2}}{25}+\ln \left (x \right )^{3} x +\frac {x^{5} \ln \left (x \right )^{3}}{5}\) | \(53\) |
norman | \(-3 x +\frac {169 x^{5}}{625}+4 x \ln \left (x \right )-\frac {44 x^{5} \ln \left (x \right )}{125}-3 x \ln \left (x \right )^{2}-\frac {3 x^{5} \ln \left (x \right )^{2}}{25}+\ln \left (x \right )^{3} x +\frac {x^{5} \ln \left (x \right )^{3}}{5}\) | \(53\) |
parallelrisch | \(-3 x +\frac {169 x^{5}}{625}+4 x \ln \left (x \right )-\frac {44 x^{5} \ln \left (x \right )}{125}-3 x \ln \left (x \right )^{2}-\frac {3 x^{5} \ln \left (x \right )^{2}}{25}+\ln \left (x \right )^{3} x +\frac {x^{5} \ln \left (x \right )^{3}}{5}\) | \(53\) |
parts | \(-3 x +\frac {169 x^{5}}{625}+4 x \ln \left (x \right )-\frac {44 x^{5} \ln \left (x \right )}{125}-3 x \ln \left (x \right )^{2}-\frac {3 x^{5} \ln \left (x \right )^{2}}{25}+\ln \left (x \right )^{3} x +\frac {x^{5} \ln \left (x \right )^{3}}{5}\) | \(53\) |
orering | \(\frac {x \left (369 x^{16}+53444 x^{12}-152666 x^{8}+34500 x^{4}+625\right ) \left (1-2 \ln \left (x \right )+\ln \left (x \right )^{3}\right )}{625 \left (x^{4}+1\right )^{3}}-\frac {x^{2} \left (97 x^{12}+28147 x^{8}-31229 x^{4}+625\right ) \left (4 x^{3} \left (1-2 \ln \left (x \right )+\ln \left (x \right )^{3}\right )+\left (x^{4}+1\right ) \left (-\frac {2}{x}+\frac {3 \ln \left (x \right )^{2}}{x}\right )\right )}{625 \left (x^{4}+1\right )^{3}}+\frac {2 x^{3} \left (7 x^{8}+3126 x^{4}-625\right ) \left (12 x^{2} \left (1-2 \ln \left (x \right )+\ln \left (x \right )^{3}\right )+8 x^{3} \left (-\frac {2}{x}+\frac {3 \ln \left (x \right )^{2}}{x}\right )+\left (x^{4}+1\right ) \left (\frac {2}{x^{2}}+\frac {6 \ln \left (x \right )}{x^{2}}-\frac {3 \ln \left (x \right )^{2}}{x^{2}}\right )\right )}{625 \left (x^{4}+1\right )^{2}}-\frac {x^{4} \left (x^{4}+625\right ) \left (24 x \left (1-2 \ln \left (x \right )+\ln \left (x \right )^{3}\right )+36 x^{2} \left (-\frac {2}{x}+\frac {3 \ln \left (x \right )^{2}}{x}\right )+12 x^{3} \left (\frac {2}{x^{2}}+\frac {6 \ln \left (x \right )}{x^{2}}-\frac {3 \ln \left (x \right )^{2}}{x^{2}}\right )+\left (x^{4}+1\right ) \left (\frac {2}{x^{3}}-\frac {18 \ln \left (x \right )}{x^{3}}+\frac {6 \ln \left (x \right )^{2}}{x^{3}}\right )\right )}{625 \left (x^{4}+1\right )}\) | \(304\) |
Input:
int((x^4+1)*(1-2*ln(x)+ln(x)^3),x,method=_RETURNVERBOSE)
Output:
(1/5*x^5+x)*ln(x)^3+(-3/25*x^5-3*x)*ln(x)^2+(-44/125*x^5+4*x)*ln(x)+169/62 5*x^5-3*x
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.80 \[ \int \left (1+x^4\right ) \left (1-2 \log (x)+\log ^3(x)\right ) \, dx=\frac {169}{625} \, x^{5} + \frac {1}{5} \, {\left (x^{5} + 5 \, x\right )} \log \left (x\right )^{3} - \frac {3}{25} \, {\left (x^{5} + 25 \, x\right )} \log \left (x\right )^{2} - \frac {4}{125} \, {\left (11 \, x^{5} - 125 \, x\right )} \log \left (x\right ) - 3 \, x \] Input:
integrate((x^4+1)*(1-2*log(x)+log(x)^3),x, algorithm="fricas")
Output:
169/625*x^5 + 1/5*(x^5 + 5*x)*log(x)^3 - 3/25*(x^5 + 25*x)*log(x)^2 - 4/12 5*(11*x^5 - 125*x)*log(x) - 3*x
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int \left (1+x^4\right ) \left (1-2 \log (x)+\log ^3(x)\right ) \, dx=\frac {169 x^{5}}{625} - 3 x + \left (- \frac {44 x^{5}}{125} + 4 x\right ) \log {\left (x \right )} + \left (- \frac {3 x^{5}}{25} - 3 x\right ) \log {\left (x \right )}^{2} + \left (\frac {x^{5}}{5} + x\right ) \log {\left (x \right )}^{3} \] Input:
integrate((x**4+1)*(1-2*ln(x)+ln(x)**3),x)
Output:
169*x**5/625 - 3*x + (-44*x**5/125 + 4*x)*log(x) + (-3*x**5/25 - 3*x)*log( x)**2 + (x**5/5 + x)*log(x)**3
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.10 \[ \int \left (1+x^4\right ) \left (1-2 \log (x)+\log ^3(x)\right ) \, dx=\frac {1}{625} \, {\left (125 \, \log \left (x\right )^{3} - 75 \, \log \left (x\right )^{2} + 30 \, \log \left (x\right ) - 6\right )} x^{5} - \frac {2}{25} \, x^{5} {\left (5 \, \log \left (x\right ) - 1\right )} + \frac {1}{5} \, x^{5} + {\left (\log \left (x\right )^{3} - 3 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) - 6\right )} x - 2 \, x {\left (\log \left (x\right ) - 1\right )} + x \] Input:
integrate((x^4+1)*(1-2*log(x)+log(x)^3),x, algorithm="maxima")
Output:
1/625*(125*log(x)^3 - 75*log(x)^2 + 30*log(x) - 6)*x^5 - 2/25*x^5*(5*log(x ) - 1) + 1/5*x^5 + (log(x)^3 - 3*log(x)^2 + 6*log(x) - 6)*x - 2*x*(log(x) - 1) + x
Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87 \[ \int \left (1+x^4\right ) \left (1-2 \log (x)+\log ^3(x)\right ) \, dx=\frac {1}{5} \, x^{5} \log \left (x\right )^{3} - \frac {3}{25} \, x^{5} \log \left (x\right )^{2} - \frac {44}{125} \, x^{5} \log \left (x\right ) + \frac {169}{625} \, x^{5} + x \log \left (x\right )^{3} - 3 \, x \log \left (x\right )^{2} + 4 \, x \log \left (x\right ) - 3 \, x \] Input:
integrate((x^4+1)*(1-2*log(x)+log(x)^3),x, algorithm="giac")
Output:
1/5*x^5*log(x)^3 - 3/25*x^5*log(x)^2 - 44/125*x^5*log(x) + 169/625*x^5 + x *log(x)^3 - 3*x*log(x)^2 + 4*x*log(x) - 3*x
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int \left (1+x^4\right ) \left (1-2 \log (x)+\log ^3(x)\right ) \, dx=\frac {x\,\left (125\,x^4\,{\ln \left (x\right )}^3-75\,x^4\,{\ln \left (x\right )}^2-220\,x^4\,\ln \left (x\right )+169\,x^4+625\,{\ln \left (x\right )}^3-1875\,{\ln \left (x\right )}^2+2500\,\ln \left (x\right )-1875\right )}{625} \] Input:
int((x^4 + 1)*(log(x)^3 - 2*log(x) + 1),x)
Output:
(x*(2500*log(x) - 220*x^4*log(x) - 1875*log(x)^2 + 625*log(x)^3 - 75*x^4*l og(x)^2 + 125*x^4*log(x)^3 + 169*x^4 - 1875))/625
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int \left (1+x^4\right ) \left (1-2 \log (x)+\log ^3(x)\right ) \, dx=\frac {x \left (125 \mathrm {log}\left (x \right )^{3} x^{4}+625 \mathrm {log}\left (x \right )^{3}-75 \mathrm {log}\left (x \right )^{2} x^{4}-1875 \mathrm {log}\left (x \right )^{2}-220 \,\mathrm {log}\left (x \right ) x^{4}+2500 \,\mathrm {log}\left (x \right )+169 x^{4}-1875\right )}{625} \] Input:
int((x^4+1)*(1-2*log(x)+log(x)^3),x)
Output:
(x*(125*log(x)**3*x**4 + 625*log(x)**3 - 75*log(x)**2*x**4 - 1875*log(x)** 2 - 220*log(x)*x**4 + 2500*log(x) + 169*x**4 - 1875))/625