Optimal. Leaf size=19 \[ \frac {\left (x^8+x^4-1\right )^{3/4}}{3 x^3} \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1590} \begin {gather*} \frac {\left (x^8+x^4-1\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin {align*} \int \frac {1+x^8}{x^4 \sqrt [4]{-1+x^4+x^8}} \, dx &=\frac {\left (-1+x^4+x^8\right )^{3/4}}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 19, normalized size = 1.00 \begin {gather*} \frac {\left (x^8+x^4-1\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 19, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^4+x^8\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 15, normalized size = 0.79 \begin {gather*} \frac {{\left (x^{8} + x^{4} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \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 {x^{8} + 1}{{\left (x^{8} + x^{4} - 1\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 16, normalized size = 0.84
method | result | size |
gosper | \(\frac {\left (x^{8}+x^{4}-1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
trager | \(\frac {\left (x^{8}+x^{4}-1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
risch | \(\frac {\left (x^{8}+x^{4}-1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 15, normalized size = 0.79 \begin {gather*} \frac {{\left (x^{8} + x^{4} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 15, normalized size = 0.79 \begin {gather*} \frac {{\left (x^8+x^4-1\right )}^{3/4}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} + 1}{x^{4} \sqrt [4]{x^{8} + x^{4} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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