Optimal. Leaf size=23 \[ \frac {2 x^4-1}{3 x^3 \sqrt [4]{x^4+1}} \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.35, 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, 191} \begin {gather*} \frac {2 x}{3 \sqrt [4]{x^4+1}}-\frac {1}{3 x^3 \sqrt [4]{x^4+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 453
Rubi steps
\begin {align*} \int \frac {1+2 x^4}{x^4 \left (1+x^4\right )^{5/4}} \, dx &=-\frac {1}{3 x^3 \sqrt [4]{1+x^4}}+\frac {2}{3} \int \frac {1}{\left (1+x^4\right )^{5/4}} \, dx\\ &=-\frac {1}{3 x^3 \sqrt [4]{1+x^4}}+\frac {2 x}{3 \sqrt [4]{1+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 x^4-1}{3 x^3 \sqrt [4]{x^4+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 23, normalized size = 1.00 \begin {gather*} \frac {-1+2 x^4}{3 x^3 \sqrt [4]{1+x^4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 25, normalized size = 1.09 \begin {gather*} \frac {{\left (2 \, x^{4} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}}}{3 \, {\left (x^{7} + x^{3}\right )}} \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 {5}{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 = 20, normalized size = 0.87
method | result | size |
gosper | \(\frac {2 x^{4}-1}{3 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(20\) |
trager | \(\frac {2 x^{4}-1}{3 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(20\) |
risch | \(\frac {2 x^{4}-1}{3 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(20\) |
meijerg | \(-\frac {4 x^{4}+1}{3 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}+\frac {2 x}{\left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 22, normalized size = 0.96 \begin {gather*} \frac {x}{{\left (x^{4} + 1\right )}^{\frac {1}{4}}} - \frac {{\left (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 = 0.07, size = 19, normalized size = 0.83 \begin {gather*} \frac {2\,x^4-1}{3\,x^3\,{\left (x^4+1\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.02, size = 97, normalized size = 4.22 \begin {gather*} \frac {4 x^{4} \left (x^{4} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{16 x^{7} \Gamma \left (\frac {5}{4}\right ) + 16 x^{3} \Gamma \left (\frac {5}{4}\right )} + \frac {x \Gamma \left (\frac {1}{4}\right )}{2 \sqrt [4]{x^{4} + 1} \Gamma \left (\frac {5}{4}\right )} + \frac {\left (x^{4} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{16 x^{7} \Gamma \left (\frac {5}{4}\right ) + 16 x^{3} \Gamma \left (\frac {5}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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