Optimal. Leaf size=32 \[ \frac {2 \left (7 x^2+1\right ) \left (x^4+x^2\right )^{3/4}}{3 x^3 \left (x^2+1\right )} \]
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Rubi [A] time = 0.16, antiderivative size = 35, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2056, 453, 264} \begin {gather*} \frac {14 x}{3 \sqrt [4]{x^4+x^2}}+\frac {2}{3 \sqrt [4]{x^4+x^2} x} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rule 2056
Rubi steps
\begin {align*} \int \frac {-1+x^2}{x^2 \left (1+x^2\right ) \sqrt [4]{x^2+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {-1+x^2}{x^{5/2} \left (1+x^2\right )^{5/4}} \, dx}{\sqrt [4]{x^2+x^4}}\\ &=\frac {2}{3 x \sqrt [4]{x^2+x^4}}+\frac {\left (7 \sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1+x^2\right )^{5/4}} \, dx}{3 \sqrt [4]{x^2+x^4}}\\ &=\frac {2}{3 x \sqrt [4]{x^2+x^4}}+\frac {14 x}{3 \sqrt [4]{x^2+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.78 \begin {gather*} \frac {14 x^2+2}{3 x \sqrt [4]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 32, normalized size = 1.00 \begin {gather*} \frac {2 \left (1+7 x^2\right ) \left (x^2+x^4\right )^{3/4}}{3 x^3 \left (1+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 27, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (7 \, x^{2} + 1\right )}}{3 \, {\left (x^{5} + x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 19, normalized size = 0.59 \begin {gather*} \frac {2}{3} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}} + \frac {4}{{\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 22, normalized size = 0.69
method | result | size |
gosper | \(\frac {\frac {14 x^{2}}{3}+\frac {2}{3}}{x \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}\) | \(22\) |
risch | \(\frac {\frac {14 x^{2}}{3}+\frac {2}{3}}{x \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\) | \(24\) |
trager | \(\frac {2 \left (7 x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {3}{4}}}{3 x^{3} \left (x^{2}+1\right )}\) | \(29\) |
meijerg | \(\frac {\frac {8 x^{2}}{3}+\frac {2}{3}}{\left (x^{2}+1\right )^{\frac {1}{4}} x^{\frac {3}{2}}}+\frac {2 \sqrt {x}}{\left (x^{2}+1\right )^{\frac {1}{4}}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {2 \, {\left (8 \, x^{5} + 7 \, {\left (x^{3} + x\right )} x^{2} + 9 \, x^{3} + x\right )}}{21 \, {\left (x^{\frac {9}{2}} + x^{\frac {5}{2}}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} + \int \frac {8 \, {\left (4 \, x^{4} + x^{2} - 3\right )}}{21 \, {\left (x^{\frac {13}{2}} + 2 \, x^{\frac {9}{2}} + x^{\frac {5}{2}}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 28, normalized size = 0.88 \begin {gather*} \frac {2\,{\left (x^4+x^2\right )}^{3/4}\,\left (7\,x^2+1\right )}{3\,x^3\,\left (x^2+1\right )} \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 {\left (x - 1\right ) \left (x + 1\right )}{x^{2} \sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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