Optimal. Leaf size=15 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2137, 212}
\begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2137
Rubi steps
\begin {align*} \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt {-1+x^2+x^4}} \, dx &=\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2+x^4}}\right )\\ &=\tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 15, normalized size = 1.00 \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.59, size = 265, normalized size = 17.67
method | result | size |
elliptic | \(\arctanh \left (\frac {\sqrt {x^{4}+x^{2}-1}}{x}\right )\) | \(16\) |
trager | \(\frac {\ln \left (-\frac {x^{4}+2 \sqrt {x^{4}+x^{2}-1}\, x +2 x^{2}-1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\) | \(46\) |
default | \(-\frac {2 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {2-2 \sqrt {5}}}{2}, \frac {i}{2}+\frac {i \sqrt {5}}{2}\right )}{\sqrt {2-2 \sqrt {5}}\, \sqrt {x^{4}+x^{2}-1}}+\frac {\sqrt {1-\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {x^{2}}{2}-\frac {\sqrt {5}\, x^{2}}{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , -\frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}}}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}+x^{2}-1}}+\frac {\sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}}}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}+x^{2}-1}}\) | \(265\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs.
\(2 (13) = 26\).
time = 0.37, size = 34, normalized size = 2.27 \begin {gather*} \frac {1}{2} \, \log \left (\frac {x^{4} + 2 \, x^{2} + 2 \, \sqrt {x^{4} + x^{2} - 1} x - 1}{x^{4} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{4}}{x^{4} \sqrt {x^{4} + x^{2} - 1} - \sqrt {x^{4} + x^{2} - 1}}\, dx - \int \frac {1}{x^{4} \sqrt {x^{4} + x^{2} - 1} - \sqrt {x^{4} + x^{2} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int -\frac {x^4+1}{\left (x^4-1\right )\,\sqrt {x^4+x^2-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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