Optimal. Leaf size=124 \[ \frac {1}{5} \sqrt {x^4+1} x^3-\frac {3 \sqrt {x^4+1} x}{5 \left (x^2+1\right )}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{10 \sqrt {x^4+1}}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {x^4+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 124, 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 = {321, 305, 220, 1196} \[ \frac {1}{5} \sqrt {x^4+1} x^3-\frac {3 \sqrt {x^4+1} x}{5 \left (x^2+1\right )}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{10 \sqrt {x^4+1}}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 321
Rule 1196
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 {1}{5} x^3 \sqrt {1+x^4}-\frac {3 x \sqrt {1+x^4}}{5 \left (1+x^2\right )}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {1+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{10 \sqrt {1+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.27 \[ \frac {1}{5} x^3 \left (\sqrt {x^4+1}-\, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-x^4\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{\sqrt {x^{4} + 1}}, x\right ) \]
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 \[ \int \frac {x^{6}}{\sqrt {x^{4} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 95, normalized size = 0.77 \[ \frac {\sqrt {x^{4}+1}\, x^{3}}{5}-\frac {3 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (-\EllipticE \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )+\EllipticF \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )\right )}{5 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}} \]
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 \[ \int \frac {x^{6}}{\sqrt {x^{4} + 1}}\,{d x} \]
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 \[ \int \frac {x^6}{\sqrt {x^4+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.91, size = 29, normalized size = 0.23 \[ \frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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