Optimal. Leaf size=126 \[ \frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1+x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {312, 228, 1199}
\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 )}{\sqrt {2} \sqrt {x^4-1}}-\frac {\sqrt {2} \sqrt {x^2-1} \sqrt {x^2+1} E\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{\sqrt {x^4-1}}+\frac {x \left (x^2+1\right )}{\sqrt {x^4-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 228
Rule 312
Rule 1199
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {-1+x^4}} \, dx &=\int \frac {1}{\sqrt {-1+x^4}} \, dx-\int \frac {1-x^2}{\sqrt {-1+x^4}} \, dx\\ &=\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\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 )}{\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 = 10.03, size = 40, normalized size = 0.32 \begin {gather*} \frac {x^3 \sqrt {1-x^4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^4\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 = 0.16, size = 44, normalized size = 0.35
method | result | size |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, x^{3} \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], x^{4}\right )}{3 \sqrt {\mathrm {signum}\left (x^{4}-1\right )}}\) | \(33\) |
default | \(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\EllipticF \left (i x , i\right )-\EllipticE \left (i x , i\right )\right )}{\sqrt {x^{4}-1}}\) | \(44\) |
elliptic | \(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\EllipticF \left (i x , i\right )-\EllipticE \left (i x , i\right )\right )}{\sqrt {x^{4}-1}}\) | \(44\) |
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 = 0.07, size = 29, normalized size = 0.23 \begin {gather*} \frac {x E(\arcsin \left (\frac {1}{x}\right )\,|\,-1) - x F(\arcsin \left (\frac {1}{x}\right )\,|\,-1) + \sqrt {x^{4} - 1}}{x} \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 = 0.31, size = 27, normalized size = 0.21 \begin {gather*} - \frac {i x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 \Gamma \left (\frac {7}{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^2}{\sqrt {x^4-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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