Optimal. Leaf size=72 \[ \frac {1}{3} \sqrt {x^4-1} x+\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {x^4-1}} \]
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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 = {321, 222} \[ \frac {1}{3} \sqrt {x^4-1} x+\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {x^4-1}} \]
Antiderivative was successfully verified.
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Rule 222
Rule 321
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] time = 0.01, size = 44, normalized size = 0.61 \[ \frac {x \left (\sqrt {1-x^4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )+x^4-1\right )}{3 \sqrt {x^4-1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{\sqrt {x^{4} - 1}}, x\right ) \]
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 \[ \int \frac {x^{4}}{\sqrt {x^{4} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 45, normalized size = 0.62 \[ \frac {\sqrt {x^{4}-1}\, x}{3}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{3 \sqrt {x^{4}-1}} \]
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 \[ \int \frac {x^{4}}{\sqrt {x^{4} - 1}}\,{d x} \]
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 \[ \int \frac {x^4}{\sqrt {x^4-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.46, size = 27, normalized size = 0.38 \[ - \frac {i x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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