Optimal. Leaf size=150 \[ \frac {3 x \left (1+x^2\right )}{5 \sqrt {-1+x^4}}+\frac {1}{5} x^3 \sqrt {-1+x^4}-\frac {3 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{5 \sqrt {-1+x^4}}+\frac {3 \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{5 \sqrt {2} \sqrt {-1+x^4}} \]
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Rubi [A]
time = 0.02, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {327, 312, 228,
1199} \begin {gather*} \frac {3 \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 )}{5 \sqrt {2} \sqrt {x^4-1}}-\frac {3 \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 )}{5 \sqrt {x^4-1}}+\frac {1}{5} \sqrt {x^4-1} x^3+\frac {3 \left (x^2+1\right ) x}{5 \sqrt {x^4-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 228
Rule 312
Rule 327
Rule 1199
Rubi steps
\begin {align*} \int \frac {x^6}{\sqrt {-1+x^4}} \, dx &=\frac {1}{5} x^3 \sqrt {-1+x^4}+\frac {3}{5} \int \frac {x^2}{\sqrt {-1+x^4}} \, dx\\ &=\frac {1}{5} x^3 \sqrt {-1+x^4}+\frac {3}{5} \int \frac {1}{\sqrt {-1+x^4}} \, dx-\frac {3}{5} \int \frac {1-x^2}{\sqrt {-1+x^4}} \, dx\\ &=\frac {3 x \left (1+x^2\right )}{5 \sqrt {-1+x^4}}+\frac {1}{5} x^3 \sqrt {-1+x^4}-\frac {3 \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 )}{5 \sqrt {-1+x^4}}+\frac {3 \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 )}{5 \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 = 10.02, size = 46, normalized size = 0.31 \begin {gather*} \frac {x^3 \left (-1+x^4+\sqrt {1-x^4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^4\right )\right )}{5 \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 = 57, normalized size = 0.38
method | result | size |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, x^{7} \hypergeom \left (\left [\frac {1}{2}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], x^{4}\right )}{7 \sqrt {\mathrm {signum}\left (x^{4}-1\right )}}\) | \(33\) |
default | \(\frac {x^{3} \sqrt {x^{4}-1}}{5}-\frac {3 i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\EllipticF \left (i x , i\right )-\EllipticE \left (i x , i\right )\right )}{5 \sqrt {x^{4}-1}}\) | \(57\) |
risch | \(\frac {x^{3} \sqrt {x^{4}-1}}{5}-\frac {3 i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\EllipticF \left (i x , i\right )-\EllipticE \left (i x , i\right )\right )}{5 \sqrt {x^{4}-1}}\) | \(57\) |
elliptic | \(\frac {x^{3} \sqrt {x^{4}-1}}{5}-\frac {3 i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\EllipticF \left (i x , i\right )-\EllipticE \left (i x , i\right )\right )}{5 \sqrt {x^{4}-1}}\) | \(57\) |
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 = 37, normalized size = 0.25 \begin {gather*} \frac {3 \, x E(\arcsin \left (\frac {1}{x}\right )\,|\,-1) - 3 \, x F(\arcsin \left (\frac {1}{x}\right )\,|\,-1) + {\left (x^{4} + 3\right )} \sqrt {x^{4} - 1}}{5 \, 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 = 0.37, size = 27, normalized size = 0.18 \begin {gather*} - \frac {i x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 \Gamma \left (\frac {11}{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^6}{\sqrt {x^4-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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