Optimal. Leaf size=23 \[ \frac {\left (1-9 x^4\right ) \sqrt [4]{x^4+1}}{5 x^5} \]
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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {453, 264} \begin {gather*} \frac {\sqrt [4]{x^4+1}}{5 x^5}-\frac {9 \sqrt [4]{x^4+1}}{5 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {-1+x^4}{x^6 \left (1+x^4\right )^{3/4}} \, dx &=\frac {\sqrt [4]{1+x^4}}{5 x^5}+\frac {9}{5} \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx\\ &=\frac {\sqrt [4]{1+x^4}}{5 x^5}-\frac {9 \sqrt [4]{1+x^4}}{5 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {\left (1-9 x^4\right ) \sqrt [4]{x^4+1}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 23, normalized size = 1.00 \begin {gather*} \frac {\left (1-9 x^4\right ) \sqrt [4]{1+x^4}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 19, normalized size = 0.83 \begin {gather*} -\frac {{\left (9 \, x^{4} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \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^{4} - 1}{{\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 20, normalized size = 0.87
method | result | size |
gosper | \(-\frac {\left (9 x^{4}-1\right ) \left (x^{4}+1\right )^{\frac {1}{4}}}{5 x^{5}}\) | \(20\) |
trager | \(-\frac {\left (9 x^{4}-1\right ) \left (x^{4}+1\right )^{\frac {1}{4}}}{5 x^{5}}\) | \(20\) |
risch | \(-\frac {9 x^{8}+8 x^{4}-1}{5 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}}\) | \(25\) |
meijerg | \(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}+\frac {\left (-4 x^{4}+1\right ) \left (x^{4}+1\right )^{\frac {1}{4}}}{5 x^{5}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 25, normalized size = 1.09 \begin {gather*} -\frac {2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + \frac {{\left (x^{4} + 1\right )}^{\frac {5}{4}}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 25, normalized size = 1.09 \begin {gather*} \frac {{\left (x^4+1\right )}^{1/4}-9\,x^4\,{\left (x^4+1\right )}^{1/4}}{5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.80, size = 71, normalized size = 3.09 \begin {gather*} \frac {\sqrt [4]{1 + \frac {1}{x^{4}}} \Gamma \left (- \frac {1}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} - \frac {\sqrt [4]{x^{4} + 1} \Gamma \left (- \frac {5}{4}\right )}{4 x \Gamma \left (\frac {3}{4}\right )} + \frac {\sqrt [4]{x^{4} + 1} \Gamma \left (- \frac {5}{4}\right )}{16 x^{5} \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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