Optimal. Leaf size=16 \[ \frac {\left (x^4-1\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1479, 449} \begin {gather*} \frac {\left (x^4-1\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 1479
Rubi steps
\begin {align*} \int \frac {-1+x^8}{x^4 \sqrt {-1+x^4}} \, dx &=\int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^4} \, dx\\ &=\frac {\left (-1+x^4\right )^{3/2}}{3 x^3}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 76, normalized size = 4.75 \begin {gather*} \frac {x^4 \left (\sqrt {1-x^4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )+x^4-1\right )+\sqrt {1-x^4} \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};x^4\right )}{3 x^3 \sqrt {x^4-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^4\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{4} - 1\right )}^{\frac {3}{2}}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.65, size = 25, normalized size = 1.56 \begin {gather*} \frac {1}{3} \, \sqrt {x^{4} - 1} x - \frac {\sqrt {-\frac {1}{x^{4}} + 1}}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 13, normalized size = 0.81
method | result | size |
trager | \(\frac {\left (x^{4}-1\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(13\) |
elliptic | \(\frac {\left (x^{4}-1\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(13\) |
risch | \(\frac {x^{8}-2 x^{4}+1}{3 x^{3} \sqrt {x^{4}-1}}\) | \(23\) |
gosper | \(\frac {\sqrt {x^{4}-1}\, \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}{3 x^{3}}\) | \(24\) |
default | \(\frac {x \sqrt {x^{4}-1}}{3}-\frac {\sqrt {x^{4}-1}}{3 x^{3}}\) | \(24\) |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, \hypergeom \left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {9}{4}\right ], x^{4}\right ) x^{5}}{5 \sqrt {\mathrm {signum}\left (x^{4}-1\right )}}+\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, \hypergeom \left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [\frac {1}{4}\right ], x^{4}\right )}{3 \sqrt {\mathrm {signum}\left (x^{4}-1\right )}\, x^{3}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.96, size = 27, normalized size = 1.69 \begin {gather*} \frac {{\left (x^{4} - 1\right )} \sqrt {x^{2} + 1} \sqrt {x + 1} \sqrt {x - 1}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^4-1\right )}^{3/2}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.85, size = 56, normalized size = 3.50 \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 )} + \frac {i \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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