Optimal. Leaf size=53 \[ \frac {1}{2 (1-x)}-\frac {1}{3 x^3}-\frac {1}{x^2}-\frac {2}{x}-\frac {5}{2} \log (1-x)+2 \log (x)+\frac {1}{4} \log \left (1+x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2083, 266}
\begin {gather*} -\frac {1}{3 x^3}-\frac {1}{x^2}+\frac {1}{4} \log \left (x^2+1\right )+\frac {1}{2 (1-x)}-\frac {2}{x}-\frac {5}{2} \log (1-x)+2 \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 2083
Rubi steps
\begin {align*} \int \frac {1}{x^4-2 x^5+2 x^6-2 x^7+x^8} \, dx &=\int \left (\frac {1}{2 (-1+x)^2}-\frac {5}{2 (-1+x)}+\frac {1}{x^4}+\frac {2}{x^3}+\frac {2}{x^2}+\frac {2}{x}+\frac {x}{2 \left (1+x^2\right )}\right ) \, dx\\ &=\frac {1}{2 (1-x)}-\frac {1}{3 x^3}-\frac {1}{x^2}-\frac {2}{x}-\frac {5}{2} \log (1-x)+2 \log (x)+\frac {1}{2} \int \frac {x}{1+x^2} \, dx\\ &=\frac {1}{2 (1-x)}-\frac {1}{3 x^3}-\frac {1}{x^2}-\frac {2}{x}-\frac {5}{2} \log (1-x)+2 \log (x)+\frac {1}{4} \log \left (1+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 51, normalized size = 0.96 \begin {gather*} -\frac {1}{2 (-1+x)}-\frac {1}{3 x^3}-\frac {1}{x^2}-\frac {2}{x}-\frac {5}{2} \log (1-x)+2 \log (x)+\frac {1}{4} \log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 42, normalized size = 0.79
method | result | size |
default | \(-\frac {1}{3 x^{3}}-\frac {1}{x^{2}}-\frac {2}{x}+2 \ln \left (x \right )-\frac {1}{2 \left (-1+x \right )}-\frac {5 \ln \left (-1+x \right )}{2}+\frac {\ln \left (x^{2}+1\right )}{4}\) | \(42\) |
norman | \(\frac {\frac {1}{3}+x^{2}-\frac {5}{2} x^{3}+\frac {2}{3} x}{x^{3} \left (-1+x \right )}+2 \ln \left (x \right )-\frac {5 \ln \left (-1+x \right )}{2}+\frac {\ln \left (x^{2}+1\right )}{4}\) | \(42\) |
risch | \(\frac {\frac {1}{3}+x^{2}-\frac {5}{2} x^{3}+\frac {2}{3} x}{x^{3} \left (-1+x \right )}+2 \ln \left (x \right )-\frac {5 \ln \left (-1+x \right )}{2}+\frac {\ln \left (x^{2}+1\right )}{4}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 2.90, size = 47, normalized size = 0.89 \begin {gather*} -\frac {15 \, x^{3} - 6 \, x^{2} - 4 \, x - 2}{6 \, {\left (x^{4} - x^{3}\right )}} + \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {5}{2} \, \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.14, size = 73, normalized size = 1.38 \begin {gather*} -\frac {30 \, x^{3} - 12 \, x^{2} - 3 \, {\left (x^{4} - x^{3}\right )} \log \left (x^{2} + 1\right ) + 30 \, {\left (x^{4} - x^{3}\right )} \log \left (x - 1\right ) - 24 \, {\left (x^{4} - x^{3}\right )} \log \left (x\right ) - 8 \, x - 4}{12 \, {\left (x^{4} - x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.07, size = 46, normalized size = 0.87 \begin {gather*} 2 \log {\left (x \right )} - \frac {5 \log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x^{2} + 1 \right )}}{4} + \frac {- 15 x^{3} + 6 x^{2} + 4 x + 2}{6 x^{4} - 6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.46, size = 46, normalized size = 0.87 \begin {gather*} -\frac {15 \, x^{3} - 6 \, x^{2} - 4 \, x - 2}{6 \, {\left (x - 1\right )} x^{3}} + \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {5}{2} \, \log \left ({\left | x - 1 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 45, normalized size = 0.85 \begin {gather*} \frac {\ln \left (x^2+1\right )}{4}-\frac {5\,\ln \left (x-1\right )}{2}+2\,\ln \left (x\right )-\frac {-\frac {5\,x^3}{2}+x^2+\frac {2\,x}{3}+\frac {1}{3}}{x^3-x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________