Optimal. Leaf size=13 \[ -\frac {3}{2} \tan ^{-1}(x)-\frac {7}{2} \tanh ^{-1}(x) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1167, 207, 203} \begin {gather*} -\frac {3}{2} \tan ^{-1}(x)-\frac {7}{2} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 207
Rule 1167
Rubi steps
\begin {align*} \int \frac {5+2 x^2}{-1+x^4} \, dx &=-\left (\frac {3}{2} \int \frac {1}{1+x^2} \, dx\right )+\frac {7}{2} \int \frac {1}{-1+x^2} \, dx\\ &=-\frac {3}{2} \tan ^{-1}(x)-\frac {7}{2} \tanh ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 25, normalized size = 1.92 \begin {gather*} \frac {7}{4} \log (1-x)-\frac {7}{4} \log (x+1)-\frac {3}{2} \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5+2 x^2}{-1+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 17, normalized size = 1.31 \begin {gather*} -\frac {3}{2} \, \arctan \relax (x) - \frac {7}{4} \, \log \left (x + 1\right ) + \frac {7}{4} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 19, normalized size = 1.46 \begin {gather*} -\frac {3}{2} \, \arctan \relax (x) - \frac {7}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {7}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 18, normalized size = 1.38 \begin {gather*} -\frac {3 \arctan \relax (x )}{2}-\frac {7 \ln \left (x +1\right )}{4}+\frac {7 \ln \left (x -1\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.35, size = 17, normalized size = 1.31 \begin {gather*} -\frac {3}{2} \, \arctan \relax (x) - \frac {7}{4} \, \log \left (x + 1\right ) + \frac {7}{4} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 9, normalized size = 0.69 \begin {gather*} -\frac {3\,\mathrm {atan}\relax (x)}{2}-\frac {7\,\mathrm {atanh}\relax (x)}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.20, size = 22, normalized size = 1.69 \begin {gather*} \frac {7 \log {\left (x - 1 \right )}}{4} - \frac {7 \log {\left (x + 1 \right )}}{4} - \frac {3 \operatorname {atan}{\relax (x )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________