Optimal. Leaf size=60 \[ \frac {169 x^5}{625}+\frac {1}{5} x^5 \log ^3(x)-\frac {3}{25} x^5 \log ^2(x)-\frac {44}{125} x^5 \log (x)-3 x+x \log ^3(x)-3 x \log ^2(x)+4 x \log (x) \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 73, normalized size of antiderivative = 1.22, number of steps used = 13, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6742, 2313, 12, 2330, 2296, 2295, 2305, 2304} \[ \frac {169 x^5}{625}+\frac {1}{5} x^5 \log ^3(x)-\frac {3}{25} x^5 \log ^2(x)+\frac {6}{125} x^5 \log (x)-\frac {2}{5} \left (x^5+5 x\right ) \log (x)-3 x+x \log ^3(x)-3 x \log ^2(x)+6 x \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2313
Rule 2330
Rule 6742
Rubi steps
\begin {align*} \int \left (1+x^4\right ) \left (1-2 \log (x)+\log ^3(x)\right ) \, dx &=\int \left (1+x^4-2 \left (1+x^4\right ) \log (x)+\left (1+x^4\right ) \log ^3(x)\right ) \, dx\\ &=x+\frac {x^5}{5}-2 \int \left (1+x^4\right ) \log (x) \, dx+\int \left (1+x^4\right ) \log ^3(x) \, dx\\ &=x+\frac {x^5}{5}-\frac {2}{5} \left (5 x+x^5\right ) \log (x)+2 \int \frac {1}{5} \left (5+x^4\right ) \, dx+\int \left (\log ^3(x)+x^4 \log ^3(x)\right ) \, dx\\ &=x+\frac {x^5}{5}-\frac {2}{5} \left (5 x+x^5\right ) \log (x)+\frac {2}{5} \int \left (5+x^4\right ) \, dx+\int \log ^3(x) \, dx+\int x^4 \log ^3(x) \, dx\\ &=3 x+\frac {7 x^5}{25}-\frac {2}{5} \left (5 x+x^5\right ) \log (x)+x \log ^3(x)+\frac {1}{5} x^5 \log ^3(x)-\frac {3}{5} \int x^4 \log ^2(x) \, dx-3 \int \log ^2(x) \, dx\\ &=3 x+\frac {7 x^5}{25}-\frac {2}{5} \left (5 x+x^5\right ) \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)+\frac {6}{25} \int x^4 \log (x) \, dx+6 \int \log (x) \, dx\\ &=-3 x+\frac {169 x^5}{625}+6 x \log (x)+\frac {6}{125} x^5 \log (x)-\frac {2}{5} \left (5 x+x^5\right ) \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)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 60, normalized size = 1.00 \[ \frac {169 x^5}{625}+\frac {1}{5} x^5 \log ^3(x)-\frac {3}{25} x^5 \log ^2(x)-\frac {44}{125} x^5 \log (x)-3 x+x \log ^3(x)-3 x \log ^2(x)+4 x \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1+x^4\right ) \left (1-2 \log (x)+\log ^3(x)\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.01, size = 48, normalized size = 0.80 \[ \frac {169}{625} \, x^{5} + \frac {1}{5} \, {\left (x^{5} + 5 \, x\right )} \log \relax (x)^{3} - \frac {3}{25} \, {\left (x^{5} + 25 \, x\right )} \log \relax (x)^{2} - \frac {4}{125} \, {\left (11 \, x^{5} - 125 \, x\right )} \log \relax (x) - 3 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.62, size = 52, normalized size = 0.87 \[ \frac {1}{5} \, x^{5} \log \relax (x)^{3} - \frac {3}{25} \, x^{5} \log \relax (x)^{2} - \frac {44}{125} \, x^{5} \log \relax (x) + \frac {169}{625} \, x^{5} + x \log \relax (x)^{3} - 3 \, x \log \relax (x)^{2} + 4 \, x \log \relax (x) - 3 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 48, normalized size = 0.80
method | result | size |
risch | \(\left (\frac {1}{5} x^{5}+x \right ) \ln \relax (x )^{3}+\left (-\frac {3}{25} x^{5}-3 x \right ) \ln \relax (x )^{2}+\left (-\frac {44}{125} x^{5}+4 x \right ) \ln \relax (x )+\frac {169 x^{5}}{625}-3 x\) | \(48\) |
default | \(-3 x +\frac {169 x^{5}}{625}+4 x \ln \relax (x )-\frac {44 x^{5} \ln \relax (x )}{125}-3 x \ln \relax (x )^{2}-\frac {3 x^{5} \ln \relax (x )^{2}}{25}+x \ln \relax (x )^{3}+\frac {x^{5} \ln \relax (x )^{3}}{5}\) | \(53\) |
norman | \(-3 x +\frac {169 x^{5}}{625}+4 x \ln \relax (x )-\frac {44 x^{5} \ln \relax (x )}{125}-3 x \ln \relax (x )^{2}-\frac {3 x^{5} \ln \relax (x )^{2}}{25}+x \ln \relax (x )^{3}+\frac {x^{5} \ln \relax (x )^{3}}{5}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 66, normalized size = 1.10 \[ \frac {1}{625} \, {\left (125 \, \log \relax (x)^{3} - 75 \, \log \relax (x)^{2} + 30 \, \log \relax (x) - 6\right )} x^{5} - \frac {2}{25} \, x^{5} {\left (5 \, \log \relax (x) - 1\right )} + \frac {1}{5} \, x^{5} + {\left (\log \relax (x)^{3} - 3 \, \log \relax (x)^{2} + 6 \, \log \relax (x) - 6\right )} x - 2 \, x {\left (\log \relax (x) - 1\right )} + x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.37, size = 51, normalized size = 0.85 \[ \frac {x\,\left (125\,x^4\,{\ln \relax (x)}^3-75\,x^4\,{\ln \relax (x)}^2-220\,x^4\,\ln \relax (x)+169\,x^4+625\,{\ln \relax (x)}^3-1875\,{\ln \relax (x)}^2+2500\,\ln \relax (x)-1875\right )}{625} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.16, size = 51, normalized size = 0.85 \[ \frac {169 x^{5}}{625} - 3 x + \left (- \frac {44 x^{5}}{125} + 4 x\right ) \log {\relax (x )} + \left (- \frac {3 x^{5}}{25} - 3 x\right ) \log {\relax (x )}^{2} + \left (\frac {x^{5}}{5} + x\right ) \log {\relax (x )}^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________