Optimal. Leaf size=23 \[ \frac {\left (x^4-1\right )^{3/4} \left (10 x^4-3\right )}{21 x^7} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {453, 264} \begin {gather*} \frac {10 \left (x^4-1\right )^{3/4}}{21 x^3}-\frac {\left (x^4-1\right )^{3/4}}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {-1+2 x^4}{x^8 \sqrt [4]{-1+x^4}} \, dx &=-\frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {10}{7} \int \frac {1}{x^4 \sqrt [4]{-1+x^4}} \, dx\\ &=-\frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {10 \left (-1+x^4\right )^{3/4}}{21 x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {\left (x^4-1\right )^{3/4} \left (10 x^4-3\right )}{21 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 23, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^4\right )^{3/4} \left (-3+10 x^4\right )}{21 x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 19, normalized size = 0.83 \begin {gather*} \frac {{\left (10 \, x^{4} - 3\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}}}{21 \, x^{7}} \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*} \int \frac {2 \, x^{4} - 1}{{\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 20, normalized size = 0.87
method | result | size |
trager | \(\frac {\left (x^{4}-1\right )^{\frac {3}{4}} \left (10 x^{4}-3\right )}{21 x^{7}}\) | \(20\) |
risch | \(\frac {10 x^{8}-13 x^{4}+3}{21 x^{7} \left (x^{4}-1\right )^{\frac {1}{4}}}\) | \(25\) |
gosper | \(\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (10 x^{4}-3\right )}{21 x^{7} \left (x^{4}-1\right )^{\frac {1}{4}}}\) | \(31\) |
meijerg | \(\frac {\left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \left (1+\frac {4 x^{4}}{3}\right ) \left (-x^{4}+1\right )^{\frac {3}{4}}}{7 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{7}}-\frac {2 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \left (-x^{4}+1\right )^{\frac {3}{4}}}{3 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{3}}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 25, normalized size = 1.09 \begin {gather*} \frac {{\left (x^{4} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} + \frac {{\left (x^{4} - 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 27, normalized size = 1.17 \begin {gather*} -\frac {3\,{\left (x^4-1\right )}^{3/4}-10\,x^4\,{\left (x^4-1\right )}^{3/4}}{21\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.00, size = 190, normalized size = 8.26 \begin {gather*} 2 \left (\begin {cases} - \frac {\left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {3}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {\left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases}\right ) - \begin {cases} - \frac {\left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} - \frac {3 \left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{16 x^{4} \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\\frac {\left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} + \frac {3 \left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{16 x^{4} \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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