Optimal. Leaf size=18 \[ -\frac {2 \left (x^2+x^4\right )^{3/4}}{3 x^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2039}
\begin {gather*} -\frac {2 \left (x^4+x^2\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt [4]{x^2+x^4}} \, dx &=-\frac {2 \left (x^2+x^4\right )^{3/4}}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2 \left (x^2+x^4\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 13, normalized size = 0.72
method | result | size |
meijerg | \(-\frac {2 \left (x^{2}+1\right )^{\frac {3}{4}}}{3 x^{\frac {3}{2}}}\) | \(13\) |
trager | \(-\frac {2 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(15\) |
gosper | \(-\frac {2 \left (x^{2}+1\right )}{3 x \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}\) | \(20\) |
risch | \(-\frac {2 \left (x^{2}+1\right )}{3 x \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 17, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (x^{3} + x\right )}}{3 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 14, normalized size = 0.78 \begin {gather*} -\frac {2 \, {\left (x^{4} + x^{2}\right )}^{\frac {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 {1}{x^{2} \sqrt [4]{x^{2} \left (x^{2} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 9, normalized size = 0.50 \begin {gather*} -\frac {2}{3} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 14, normalized size = 0.78 \begin {gather*} -\frac {2\,{\left (x^4+x^2\right )}^{3/4}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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