Optimal. Leaf size=8 \[ \frac {1}{x}-\tanh ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1593, 325, 207} \[ \frac {1}{x}-\tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 207
Rule 325
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{-x^2+x^4} \, dx &=\int \frac {1}{x^2 \left (-1+x^2\right )} \, dx\\ &=\frac {1}{x}+\int \frac {1}{-1+x^2} \, dx\\ &=\frac {1}{x}-\tanh ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.00, size = 22, normalized size = 2.75 \[ \frac {1}{x}+\frac {1}{2} \log (1-x)-\frac {1}{2} \log (x+1) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 20, normalized size = 2.50 \[ -\frac {x \log \left (x + 1\right ) - x \log \left (x - 1\right ) - 2}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.89, size = 18, normalized size = 2.25 \[ \frac {1}{x} - \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 17, normalized size = 2.12 \[ \frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}+\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 16, normalized size = 2.00 \[ \frac {1}{x} - \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.17, size = 8, normalized size = 1.00 \[ \frac {1}{x}-\mathrm {atanh}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.11, size = 15, normalized size = 1.88 \[ \frac {\log {\left (x - 1 \right )}}{2} - \frac {\log {\left (x + 1 \right )}}{2} + \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________