Optimal. Leaf size=13 \[ -\frac {x^3}{3}-x+\tanh ^{-1}(x) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1584, 302, 206} \begin {gather*} -\frac {x^3}{3}-x+\tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 302
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^5}{x-x^3} \, dx &=\int \frac {x^4}{1-x^2} \, dx\\ &=\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx\\ &=-x-\frac {x^3}{3}+\int \frac {1}{1-x^2} \, dx\\ &=-x-\frac {x^3}{3}+\tanh ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.00, size = 29, normalized size = 2.23 \begin {gather*} -\frac {x^3}{3}-x-\frac {1}{2} \log (1-x)+\frac {1}{2} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5}{x-x^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 21, normalized size = 1.62 \begin {gather*} -\frac {1}{3} \, x^{3} - x + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.17, size = 23, normalized size = 1.77 \begin {gather*} -\frac {1}{3} \, x^{3} - x + \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 22, normalized size = 1.69 \begin {gather*} -\frac {x^{3}}{3}-x -\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.38, size = 21, normalized size = 1.62 \begin {gather*} -\frac {1}{3} \, x^{3} - x + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.98, size = 11, normalized size = 0.85 \begin {gather*} \mathrm {atanh}\relax (x)-x-\frac {x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.11, size = 19, normalized size = 1.46 \begin {gather*} - \frac {x^{3}}{3} - x - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________