Optimal. Leaf size=72 \[ \frac {1}{3} x \sqrt {-1+x^4}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {-1+x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {327, 228}
\begin {gather*} \frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {x^4-1}}+\frac {1}{3} \sqrt {x^4-1} x \end {gather*}
Antiderivative was successfully verified.
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Rule 228
Rule 327
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {-1+x^4}} \, dx &=\frac {1}{3} x \sqrt {-1+x^4}+\frac {1}{3} \int \frac {1}{\sqrt {-1+x^4}} \, dx\\ &=\frac {1}{3} x \sqrt {-1+x^4}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {-1+x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 44, normalized size = 0.61 \begin {gather*} \frac {x \left (-1+x^4+\sqrt {1-x^4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )\right )}{3 \sqrt {-1+x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.17, size = 45, normalized size = 0.62
method | result | size |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, x^{5} \hypergeom \left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {9}{4}\right ], x^{4}\right )}{5 \sqrt {\mathrm {signum}\left (x^{4}-1\right )}}\) | \(33\) |
default | \(\frac {x \sqrt {x^{4}-1}}{3}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{3 \sqrt {x^{4}-1}}\) | \(45\) |
risch | \(\frac {x \sqrt {x^{4}-1}}{3}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{3 \sqrt {x^{4}-1}}\) | \(45\) |
elliptic | \(\frac {x \sqrt {x^{4}-1}}{3}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{3 \sqrt {x^{4}-1}}\) | \(45\) |
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 [A]
time = 0.08, size = 19, normalized size = 0.26 \begin {gather*} \frac {1}{3} \, \sqrt {x^{4} - 1} x - \frac {1}{3} \, F(\arcsin \left (\frac {1}{x}\right )\,|\,-1) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.33, size = 27, normalized size = 0.38 \begin {gather*} - \frac {i x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \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 = -0.01 \begin {gather*} \int \frac {x^4}{\sqrt {x^4-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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