Optimal. Leaf size=25 \[ -\frac {2 \left (x^4-x^2\right )^{3/4}}{x \left (x^2-1\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.64, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1146, 264} \begin {gather*} -\frac {2 x}{\sqrt [4]{x^4-x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 1146
Rubi steps
\begin {align*} \int \frac {1}{\left (-1+x^2\right ) \sqrt [4]{-x^2+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-1+x^2\right )^{5/4}} \, dx}{\sqrt [4]{-x^2+x^4}}\\ &=-\frac {2 x}{\sqrt [4]{-x^2+x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 16, normalized size = 0.64 \begin {gather*} -\frac {2 x}{\sqrt [4]{x^2 \left (x^2-1\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.11, size = 25, normalized size = 1.00 \begin {gather*} -\frac {2 \left (-x^2+x^4\right )^{3/4}}{x \left (-1+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 22, normalized size = 0.88 \begin {gather*} -\frac {2 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.33, size = 11, normalized size = 0.44 \begin {gather*} \frac {2}{{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 15, normalized size = 0.60
method | result | size |
gosper | \(-\frac {2 x}{\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\) | \(15\) |
risch | \(-\frac {2 x}{\left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}\) | \(15\) |
trager | \(-\frac {2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}}{x \left (x^{2}-1\right )}\) | \(24\) |
meijerg | \(-\frac {2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{4}} \sqrt {x}}{\mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{4}} \left (-x^{2}+1\right )^{\frac {1}{4}}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.16, size = 23, normalized size = 0.92 \begin {gather*} -\frac {2\,{\left (x^4-x^2\right )}^{3/4}}{x\,\left (x^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________