Optimal. Leaf size=20 \[ \frac {3 \left (x^3+x\right )^{2/3}}{2 \left (x^2+1\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2056, 264} \begin {gather*} \frac {3 x}{2 \sqrt [3]{x^3+x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{x+x^3}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \left (1+x^2\right )^{4/3}} \, dx}{\sqrt [3]{x+x^3}}\\ &=\frac {3 x}{2 \sqrt [3]{x+x^3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 0.70 \begin {gather*} \frac {3 x}{2 \sqrt [3]{x^3+x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 20, normalized size = 1.00 \begin {gather*} \frac {3 \left (x+x^3\right )^{2/3}}{2 \left (1+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 16, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 9, normalized size = 0.45 \begin {gather*} \frac {3}{2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 11, normalized size = 0.55
method | result | size |
gosper | \(\frac {3 x}{2 \left (x^{3}+x \right )^{\frac {1}{3}}}\) | \(11\) |
meijerg | \(\frac {3 x^{\frac {2}{3}}}{2 \left (x^{2}+1\right )^{\frac {1}{3}}}\) | \(13\) |
risch | \(\frac {3 x}{2 \left (\left (x^{2}+1\right ) x \right )^{\frac {1}{3}}}\) | \(13\) |
trager | \(\frac {3 \left (x^{3}+x \right )^{\frac {2}{3}}}{2 \left (x^{2}+1\right )}\) | \(17\) |
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 {3 \, {\left (x^{3} + x\right )}}{4 \, {\left (x^{\frac {7}{3}} + x^{\frac {1}{3}}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} + \int \frac {3 \, {\left (x^{2} + 1\right )}^{\frac {2}{3}}}{2 \, {\left (x^{\frac {13}{3}} + 2 \, x^{\frac {7}{3}} + x^{\frac {1}{3}}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 18, normalized size = 0.90 \begin {gather*} \frac {3\,{\left (x^3+x\right )}^{2/3}}{2\,\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 {1}{\sqrt [3]{x \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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